Patent Publication Number: US-8531368-B2

Title: Transmissive liquid crystal display device having color saturation conversion section

Description:
This Nonprovisional application claims priority under U.S.C. §119(a) on Patent Application No. 252699/2007 filed in Japan on Sep. 27, 2007, the entire contents of which are hereby incorporated by reference. 
     FIELD OF THE INVENTION 
     The present invention relates to a transmissive liquid crystal display device constituted by a backlight and a liquid crystal panel. 
     BACKGROUND OF THE INVENTION 
     There are various types of color displays that have been in practical use. Thin displays are classified broadly into self-luminous displays, such as PDPs (plasma display panels), and nonluminous displays typified by LCDs (liquid crystal displays). Known as an LCD, which is a nonluminous display, is a transmissive LCD having a backlight disposed on the rear of a liquid crystal panel. 
       FIG. 41  is a cross-sectional view of a typical structure of a transmissive LCD. The transmissive LCD has a backlight  210  disposed on the rear of a liquid crystal panel  200 . The liquid crystal panel  200  is arranged such that a liquid crystal layer  203  is disposed between a pair of transparent substrates  201  and  202 , and that polarizers  204  and  205  are provided on the outer sides of the transparent substrates  201  and  202 , respectively. Further, a color filter  206  is provided in the liquid crystal panel  200 , so that color displays become available. 
     Although not shown, an electrode layer and an alignment layer are provided inside of the transparent substrates  201  and  202 . The amount of light that is transmitted through the liquid crystal panel  200  is controlled for each pixel by controlling the application of voltage to the liquid crystal layer  203 . That is, the transmissive LCD controls emitted light from the backlight  210  in such a manner that the amount of light that is transmitted through is controlled at the liquid crystal panel  200 , thereby controlling displays. 
     The backlight  210  emits light that contains wavelengths of three colors RGB necessary for color displays. In combination with the color filter  206 , the respective transmittances of red, green, and blue light are adjusted, so that the luminance and hue of each pixel can be arbitrarily set. Used commonly as such a backlight  210  is a white light source such as electroluminescence (EL), a cold-cathode fluorescent lamp (CCFL), or a light-emitting diode (LED). 
     As shown in  FIG. 42 , the liquid crystal panel  200  has a plurality of pixels, arranged in a matrix manner, each of which is usually constituted by three subpixels. The subpixels are arranged so as to correspond to red (R), green (G), and blue (B) filter layers of the color filter  206 , respectively. Hereinafter, the subpixels will be referred to as R, G, and B subpixels, respectively. 
     Each of the R, G, and B subpixels selectively transmits, from among the white light emitted from the backlight  210 , light falling within the corresponding wavelength band (i.e., red, green, or blue), and absorbs light falling within the other wavelength bands. 
     In the transmissive LCD thus arranged, the light emitted from the backlight  210  is controlled in such a manner that the amount of light that is transmitted through is controlled at each pixel of the liquid crystal panel  200 . This naturally causes some of the light to be absorbed by the liquid crystal panel  200 . Further, also in the color filter  206 , each of the R, G, and B subpixels absorbs, from among the white light emitted from the backlight  210 , light falling outside the corresponding wavelength band. Thus, in an ordinary transmissive LCD, a liquid crystal panel and a color filter absorb so large an amount of light as to reduce the efficiency of use of light emitted from a backlight. Accordingly, the ordinary transmissive LCD suffers from an increase in amount of electricity that is used by the backlight. 
     Known as a technique for reducing the amount of electricity that is used by a transmissive LCD is a method that involves the use of an active backlight capable of adjusting its light emission luminance in accordance with a displayed image (e.g., Japanese Unexamined Patent Application Publication No. 65531/11999 (Tokukaihei 11-65531 (published on Mar. 9, 1999); hereinafter referred to as “Patent Document 1”). 
     That is, Patent Document 1 discloses a technique for, by using a luminance-adjustable active backlight to perform display control (luminance control) of an LCD by controlling the transmittance of a liquid crystal panel and the luminance of the active backlight, reducing the amount of electricity that is used by the backlight. 
     In Patent Document 1, the luminance of the backlight is controlled so as to be identical to the maximum luminance value of an input image (input signal). Further, the transmittance of the liquid crystal panel is adjusted in accordance with the current luminance of the backlight. 
     At this time, the transmittance of a subpixel, i.e., the maximum value of the input signal becomes 100%. Further, the transmittances of other subpixels are calculated from the backlight value to be not more than 100% each. This makes it possible to darken the backlight when the image is entirely dark, thereby enabling a reduction in the amount of electricity that is used by the backlight. 
     Thus, in Patent Document 1, the brightness of the backlight is minimized on the basis of the input signal RGB of the input image, and the transmittance of the liquid crystals is increased to the extent that the backlight is darkened. This makes it possible to reduce the amount of light that is absorbed by the liquid crystal panel, thereby enabling a reduction in the amount of electricity that is used by the backlight. 
     With the foregoing conventional arrangement, the amount of electricity that is used by the backlight can be reduced by reducing the amount of light that is absorbed by the liquid crystal panel. However, the amount of light that is absorbed by the color filter cannot be reduced. If the amount of light that is absorbed by the color filter can be reduced, the amount of electricity that is used can be further reduced. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a transmissive liquid crystal display device capable of achieving a further reduction in power consumption by reducing the amount of light that is absorbed by a color filter as well as a liquid crystal panel. 
     In order to solve the foregoing problems, a transmissive liquid crystal device according to the present invention includes: a liquid crystal panel whose pixels are each divided into four subpixels, namely red (R), green (G), blue (B), and white (W) subpixels; a white active backlight whose light emission luminance is controllable; a color-saturation conversion section that performs a color-saturation reduction process on pixel data high in luminance and color saturation, from among pixel data contained in a first input RGB signal serving as an input image, and thereby converts the first input RGB signal into a second input RGB signal; a gamma-correction section that converts the second input RGB signal into a third input RGB signal by performing a gamma-correction process on the second input RGB signal; an output signal generation section that generates, from the third input RGB signal, respective transmittance signals of the R, G, B, and W subpixels of each pixel of the liquid crystal panel and calculates a backlight value of the active backlight; a liquid crystal panel control section that drives and controls the liquid crystal panel in accordance with the transmittance signals generated by the output signal generation section; and a backlight control section that controls the light emission luminance of the backlight in accordance with the backlight value calculated by the output signal generation section. 
     Alternatively, in order to solve the foregoing problems, another transmissive liquid crystal device according to the present invention includes: a liquid crystal panel whose pixels are each divided into four subpixels, namely red (R), green (G), blue (B), and white (W) subpixels; a white active backlight whose light emission luminance is controllable; a color-saturation conversion and gamma-correction section that performs a color-saturation reduction process and a gamma-correction process in this order on pixel data high in luminance and color saturation, from among pixel data contained in a first input RGB signal serving as an input image, and performs a gamma-correction process on pixel data low in at least either luminance or color saturation, and thereby converts the first input RGB signal into a third input RGB signal; an output signal generation section that generates, from the third input RGB signal, respective transmittance signals of the R, G, B, and W subpixels of each pixel of the liquid crystal panel and calculates a backlight value of the active backlight; a liquid crystal panel control section that drives and controls the liquid crystal panel in accordance with the transmittance signals generated by the output signal generation section; and a backlight control section that controls the light emission luminance of the backlight in accordance with the backlight value calculated by the output signal generation section. 
     According to the foregoing arrangement, the use of a liquid crystal panel whose pixels are each divided into four subpixels, namely R, G, B, and W subpixels, makes it possible to transfer a part of each of the R, G, and B components to the subpixel W, which suffers no loss (or little loss) of light due to absorption by a filter. This makes it possible to reduce the amount of light that is absorbed by the color filter and to reduce the backlight value accordingly, thereby enabling a reduction in the amount of electricity that is used by the transmissive liquid crystal display device. 
     Furthermore, the backlight value can be more surely reduced by calculating the backlight value and the RGBW transmittances after performing the color-saturation reduction process on the first input RGB signal, which is an original input. 
     Further, as for the color-saturation reduction process and the gamma-correction process, the gamma-correction process is performed after the color-saturation reduction process. In cases where the color-saturation reduction process and the gamma-correction process are performed in an inverse order, there is a difference in luminance between before the processes and after the processes. However, since the gamma-correction process is performed after the color-saturation reduction process, the luminance after the processes stays the same as the luminance before the processes. 
     Additional objects, features, and strengths of the present invention will be made clear by the description below. Further, the advantages of the present invention will be evident from the following explanation in reference to the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram showing an arrangement of a main part of a liquid crystal display device according to an embodiment of the present invention. 
         FIGS. 2(   a ) and  2 ( b ) show example arrangements of subpixels in the transmissive liquid crystal display device. 
         FIG. 3(   a ) shows how a backlight luminance value is calculated in the liquid crystal display device.  FIG. 3(   b ) shows, for comparison, how a backlight luminance value is calculated in Patent Document 1. 
         FIG. 4(   a ) shows how a backlight luminance value is calculated in the liquid crystal display device.  FIG. 4(   b ) shows, for comparisons how a backlight luminance value is calculated in Patent Document 1. 
         FIGS. 5(   a ) through  5 ( e ) show how a backlight luminance value and the transmittances of subpixels are determined in the liquid crystal display device. 
         FIG. 6  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 1 of the liquid crystal display device. 
         FIG. 7  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 6 . 
         FIG. 8  is a block diagram showing an example arrangement of an output signal generation section of the liquid crystal display device. 
         FIG. 9  is a flow chart showing a sequence of operations that are performed by the output signal generation section. 
         FIG. 10  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 2 of the liquid crystal display device. 
         FIG. 11  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 10 . 
         FIG. 12  is a block diagram showing another example arrangement of the color-saturation conversion section in Embodiment 2 of the liquid crystal display device. 
         FIG. 13  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 12 . 
         FIG. 14  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 4 of the liquid crystal display device. 
         FIG. 15  is a flow chart showing a sequence of operations that are performed by a gamma-curve normalization and piecewise-linear approximation section of the color-saturation conversion section of  FIG. 14 . 
         FIG. 16  is a flow chart showing a sequence of operations that are performed by a third color-saturation conversion rate calculation section of the color-saturation conversion section of  FIG. 14 . 
         FIG. 17  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 5 of the liquid crystal display device. 
         FIG. 18  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 17 . 
         FIG. 19  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 6 of the liquid crystal display device. 
         FIG. 20  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 19 . 
         FIG. 21  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 7 of the liquid crystal display device. 
         FIG. 22  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 21 . 
         FIG. 23  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 8 of the liquid crystal display device. 
         FIG. 24  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 23 . 
         FIG. 25  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 9 of the liquid crystal display device. 
         FIG. 26  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 25 . 
         FIG. 27  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 10 of the liquid crystal display device. 
         FIG. 28  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 27 . 
         FIG. 29  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 11 of the liquid crystal display device. 
         FIG. 30  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 29 . 
         FIG. 31  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 12 of the liquid crystal display device. 
         FIG. 32  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 31 . 
         FIG. 33  is a block diagram showing an example arrangement of a color-saturation conversion section in Embodiment 13 of the liquid crystal display device. 
         FIG. 34  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion section of  FIG. 33 . 
         FIG. 35  is a block diagram showing an arrangement of a main part of a transmissive liquid crystal display device according to another embodiment of the present invention. 
         FIG. 36  shows a system configuration in which a display control process of the present invention is realized by software. 
         FIG. 37  shows a modified example of the system configuration in which the display control process of the present invention is realized by software. 
         FIG. 38  is a block diagram showing an arrangement of a main part of a liquid crystal display device according to another embodiment of the present invention. 
         FIG. 39  is a block diagram showing an example arrangement of a color-saturation conversion and gamma-correction section in Embodiment 3 of the liquid crystal display device. 
         FIG. 40  is a flow chart showing a sequence of operations that are performed by the color-saturation conversion and gamma-correction section of  FIG. 39 . 
         FIG. 41  is a cross-sectional view of a typical arrangement of a transmissive liquid crystal display device. 
         FIG. 42  shows a typical example arrangement of subpixels in the transmissive liquid crystal display device. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     An embodiment of the present invention will be described below with reference to the drawings. First, a schematic arrangement of a liquid crystal display device according to the present embodiment (hereinafter referred to as a “present liquid crystal display device”) will be described with reference to  FIG. 1 . 
     The present liquid crystal display device includes a color-saturation conversion section  11 , a gamma-correction section  12 , an output signal generation section  13 , a liquid crystal panel control section  14 , an RGBW liquid crystal panel (hereinafter referred to simply as “liquid crystal panel”)  15 , a backlight control section  16 , and a white backlight (hereinafter referred to simply as “backlight”)  17 . 
     The liquid crystal panel  15  is constituted by Np pixels arranged in a matrix manner. As shown in  FIGS. 2(   a ) and  2 ( b ), each of the pixels is constituted by four subpixels, namely R (red), G (green), B (blue), and W (white) subpixels. It should be noted that the shapes and arrangement of the R, G, B, and W subpixels in each pixel are not particularly limited. Further, the backlight  17  is an active backlight, capable of adjusting the brightness of light that it emits, which employs a white light source such as a cold-cathode fluorescent lamp (CCFL) or a white light-emitting diode (white LED). 
     The R, G, and B subpixels in each pixel of the liquid crystal panel  15  are arranged so as to correspond to R, G, and B filter layers of a color filter (not shown), respectively. Therefore, each of the R, G, and B subpixels selectively transmits, from among the white light emitted from the backlight  17 , light falling within the corresponding wavelength band, and absorbs light falling within the other wavelength bands. Further, the W subpixel basically has no corresponding absorption filter layer in the color filter. That is, the light transmitted through the W subpixel is not absorbed by the color filter at all, and is emitted from the liquid crystal panel  15  as white light. It should be noted, however, that the W subpixel may have a filter layer that absorbs the light from the backlight less than the R, G, and B color filters do. 
     The present liquid crystal display device receives an RGB signal, which carries information on an image to be displayed, from an external device such as a personal computer or a television tuner, and processes the RGB signal as an input RGB signal (R[i], G[i], B[i]) (i=1, 2, Np). 
     The color-saturation conversion section  11  performs a color-saturation conversion process on the input RGB signal (first input RGB signal), and then outputs, to the subsequent gamma-correction section  12 , a saturation-converted RGB signal (second input RGB signal) finished with the color-saturation conversion process. The gamma-correction section  12  performs a gamma-correction process on the saturation-converted RGB signal, and then outputs, to the subsequent output signal generation section  13 , a gamma-corrected RGB signal (third input RGB signal) finished with the gamma-correction conversion process. 
     The output signal generation section  13  calculates a backlight value of the backlight  17  and the respective transmittances of the R, G, B, and W subpixels in each pixel of the liquid crystal panel  15  from the gamma-corrected RGB signal, and then outputs the backlight value and the transmittances. That is, the output signal generation section  13  calculates the backlight value from the gamma-corrected RGB signal, and converts the gamma-corrected RGB signal into a transmittance signal compatible with the backlight. 
     The backlight value thus calculated is outputted to the backlight control section  16 , and the backlight control section  16  adjusts the luminance of the backlight  17  in accordance with the backlight value. The backlight  17  employs a white light source such as a CCFL and a white LED, and can be controlled by the backlight control section  16  so as to have brightness proportional to the backlight value. How the brightness of the backlight  17  is controlled varies depending on the type of light source that is used. For example, the brightness of the backlight  17  can be controlled by applying voltage proportional to the backlight value or by passing an electrical current proportional to the backlight value. In cases where the backlight is an LED, the brightness can also be controlled by changing the duty ratio with pulse width modulation (PWM). In cases where the brightness of the backlight&#39;s light source has nonlinear characteristics, it is also possible to use a look-up table to calculate, from the backlight value, voltage, an electrical current, or the like that is to be applied to the light source, and then control the brightness of the backlight so that the brightness reaches a desired level. 
     The transmittance signal obtained at the output signal generation section  13  is outputted to the liquid crystal panel control section  14 . Based on the transmittance signal, the liquid crystal panel control section  14  controls the transmittance of each subpixel of the liquid crystal panel  15  so that the transmittance reaches a desired level. The liquid crystal panel control section  14  includes a scanning line driving circuit, a signal line driving circuit, and the like. The liquid crystal panel control section  14  generates a scan signal and a data signal, and drives the liquid crystal panel  15  with use of panel control signals such as the scan signal and the data signal. The transmittance signal is used for the signal line driving circuit to generate the data signal. The transmittance of the liquid crystal panel  15  can be controlled, for example, by (i) a method for controlling the transmittance of a liquid crystal panel by applying voltage proportional to the transmittance of a subpixel, or (ii) a method for, in order to linearize nonlinear characteristics, looking up voltage to be applied to a liquid crystal panel in a look-up table on the basis of the transmittance of a subpixel, and for controlling the liquid crystal panel so that the liquid crystal panel has a desired transmittance. 
     In the liquid crystal display device of the present invention, the input signal is not limited to such an RGB signal as described above. The input signal may be a color signal such as a YUV signal. In cases where a color signal other than an RGB signal is inputted, the color signal may be converted into an RGB signal and then supplied to the output signal generation section  13 . Alternatively, the output signal generation section  13  may be arranged so as be able to convert a color input signal other than an RGB signal into an RGBW signal. 
     In the present liquid crystal display device, the display luminance of each subpixel of the liquid crystal panel  15  is represented by the product of the brightness of (luminance of light emitted by) the backlight and the transmittance of the subpixel. The following fully describes the display principles of the present liquid crystal device and the effect of reduction in the amount of electricity that is used by the present liquid crystal display device. In the present liquid crystal display device, the output signal generation section  13  calculates a backlight value and the transmittance of a subpixel. Therefore, the following process for calculating a backlight value and the transmittance of a subpixel is a process that is to be performed on the gamma-corrected RGB signal inputted from the gamma-correction section  12  to the output signal generation section  13 . In the following description, the gamma-corrected RGB signal outputted from the gamma-correction section  12  is expressed as (Rsi, Gsi, Bsi), the transmittance signal obtained at the output signal generation section  13  is expressed as (rsi, gsi, bsi), and the backlight value calculated by the output signal generation section  13  is expressed as Wbs. 
     In the present liquid crystal display device, a backlight value and the transmittance of a subpixel are determined in the following manner. First, a minimum necessary backlight value is calculated for every pixel falling within a display area corresponding to the backlight. How the backlight value is calculated varies between the following two ways, depending on the content of data to be displayed by a pixel. Specifically, how the backlight value is calculated varies depending on the relationship between the maximum luminance (i.e., max (Rsi, Gsi, Bsi)) and minimum luminance (i.e., min (Rsi, Gsi, Bsi)) of subpixels in a target pixel. 
     First, the following describes, with reference to  FIGS. 3(   a ) and  3 ( b ), how a backlight value is calculated for a pixel where min (Rsi, Gsi, Bsi)≧max (Rsi, Gsi, Bsi)/2.  FIG. 3(   a ) shows how a backlight value is calculated in the present liquid crystal display device.  FIG. 3(   b ) shows, for comparison, how a backlight value is calculated in Patent Document 1. 
     Let it be assumed that the target panel-output luminance of a target pixel is (R, G, B)=(50, 60, 40) in  FIGS. 3(   a ) and  3 ( b ). In this case, the luminance value 60 of G is max (Rsi, Gsi, Bsi), and the luminance value 40 of B is min (Rsi, Gsi, Bsi). Therefore, the following relationship is satisfied:
 
min( Rsi,Gsi,Bsi )≧max( Rsi,Gsi,Bsi )/2.
 
     According to the display method of Patent Document 1, as shown in  FIG. 3(   b ), the backlight luminance value is set to max (Rsi, Gsi, Bsi)=60, and the transmittance of each subpixel is determined according to the backlight value. That is, the transmittances of the R, G, and B subpixels are set to 83% (=50/60), 100% (=60/60), and 67% (−40/60), respectively. 
     Meanwhile, in the present liquid crystal display device, each of the R, G, and B components of the input signal (Rsi, Gsi, Bsi) is transferred to the luminance value of the W component in an amount corresponding to max (Rsi, Gsi, Bsi)/2. As a result, the input signal (R, G, B) (50, 60, 40), which is expressed as an RGB signal, is converted into an output signal (R, G, B, W)=(20, 30, 10, 30), which is expressed as an RGBW signal. Further, the backlight luminance value is set to max (Rsi, Gsi, Bsi)/2=30 for the target pixel. Further, the respective transmittances of the R, G, B, and W subpixels are determined in accordance with the backlight value Specifically, the transmittances of the R, G, B, and W subpixels are set to 67% (=20/30), 100% (=30/30), 33% (=10/30), and 100% (=30/30), respectively. It should be noted, however, that the transmittances shown in  FIG. 3(   a ) are examples of transmittances that are obtained in cases where the backlight value calculated for the target pixel is the greatest of those backlight values calculated for all the pixels and is adopted as the luminance value of the backlight. 
     Further, in order to make a comparison between the aforementioned backlight value calculated in the present liquid crystal display device and the backlight value calculated according to the method of Patent Document 1, it is necessary to also consider an area ratio of subpixels. That is, whereas a single pixel is divided into three subpixels in Patent Document 1, a single pixel is divided into four subpixels in the present liquid crystal display device. For this reason, the area of each subpixel in the present liquid crystal display device is only ¾ of the area of each subpixel in Patent Document 1. In order to make up for such a reduction in the area of a subpixel, the present liquid crystal display device multiplies the backlight luminance value by 4/3, thereby making it possible to make a commensurate comparison between the backlight value calculated in the present liquid crystal display device and the backlight value calculated in Patent Document 1. 
     As a result, making a correction to the backlight value of  FIG. 3(   a ) in order to make it commensurate with the backlight value of  FIG. 3(   b ) yields (4/3)×60/2=40. In the example of  FIG. 3(   b ) where a similar display is performed, the backlight value is 60. From this, it is obvious that the present invention brings about an effect of reduction in the amount of electricity that is used by the target pixel. 
     Next, the following describes, with reference to  FIGS. 4(   a ) and  4 ( b ), how a backlight value is calculated for a pixel where min (Rsi, Gsi, Bsi)&lt;max (Rsi, Gsi, Bsi)/2.  FIG. 4(   a ) shows how a backlight value is calculated in the present liquid crystal display device.  FIG. 4(   b ) shows, for comparison, how a backlight value is calculated in Patent Document 1. 
     Let it be assumed that the target panel-output luminance of a target pixel is (R, G, B)=(50, 60, 20) in  FIGS. 4(   a ) and  4 ( b ). In this case, the luminance value 60 of G is max (Rsi, Gsi, Bsi), and the luminance value 20 of B is min (Rsi, Gsi, Bsi). Therefore, the following relationship is satisfied:
 
min( Rsi,Gsi,Bsi )&lt;max( Rsi,Gsi,Bsi )/2.
 
     According to the display method of Patent Document 1, as shown in  FIG. 4(   b ), the backlight luminance value is set to max (Rsi, Gsi, Bsi)=60, and the transmittance of each subpixel is determined in accordance with the backlight value. That, the transmittances of the R, G, and B subpixels are set to 83% (=50/60), 100% (=60/60), and 33% (=20/60), respectively. 
     Meanwhile, in the present liquid crystal display device, each of the R, G, and B components of the input signal (Rsi, Gsi, Bsi) is transferred to the luminance value of the W component in an amount corresponding to min (Rsi, Gsi, Bsi). As a result, the input signal (R, G, B)=(50, 60, 20), which is expressed as an RGB signal, is converted into an output signal (R, G, B, W)=(30, 40, 0, 20), which is expressed as an RGBW signal. Further, the backlight luminance value is set to (max (Rsi, Gsi, Bsi)−min (Rsi, Gsi, Bsi))=40 for the target pixel. Further, the respective transmittances of the R, G, B, and W subpixels are determined in accordance with the backlight value. Specifically, the transmittances of the R, G, B, and W subpixels are set to 75% (=30/40), 100% (=40/40), 0% (=0/40), and 50% (=20/40), respectively. 
     It should be noted, however, that the transmittances shown in  FIG. 4(   a ) are examples of transmittances that are obtained in cases where the backlight value calculated for the target pixel is the greatest of those backlight values calculated for all the pixels and is adopted as the luminance value of the backlight. Further, also in the example of  FIG. 4(   a ), the backlight value is multiplied by 4/3, so that it becomes possible to make a commensurate comparison between the backlight value calculated in the present liquid crystal display device and the backlight value calculated in Patent Document 1. 
     As a result, in the example of  FIG. 4(   a ), the backlight value is (4/3)×(60−20)=53.3. In the example of  FIG. 4(   b ) where a similar display is performed, the backlight value is 60. From this, it is obvious that the present invention brings about an effect of reduction in the amount of electricity that is used by the target pixel. 
       FIGS. 3(   a ),  3 ( b ),  4 ( a ), and  4 ( b ) show how a minimum necessary backlight value is calculated for each pixel. Minimum necessary backlight values are calculated according to the above-described method for all the pixels falling within a display area corresponding to the backlight. The value greatest of all those backlight values thus calculated is set as the luminance value of the backlight. 
     The following describes, with reference to  FIGS. 5(   a ) through  5 ( e ), how a backlight value and the transmittances of subpixels are determined in the present liquid crystal display device according to the method described above. 
       FIG. 5(   a ) shows input signals (Rsi, Gsi, Bsi) inputted to a display area corresponding to a single backlight. For convenience of explanation, let it be assumed that the display area is constituted by four pixels A to D. 
       FIG. 5(   b ) shows results obtained for the pixels A to D by converting the input signals (Rsi, Gsi, Bsi) into output signals (Rtsi, Gtsi, Btsi, Wtsi), which are expressed as RGBW signals. Further,  FIG. 5(   c ) shows backlight values respectively calculated for the pixels. With this, the value greatest of those backlight values respectively calculated for the pixels, i.e., 100 is set as the backlight value. 
     The respective transmittances (rsi, gsi, bsi, wsi) of the pixels with respect to the backlight value of 100 thus calculated are calculated from the values of the output signals (Rtsi, Gtsi, Btsi, Wtsi) shown in  FIG. 5(   b ). The results are shown in  FIG. 5(   d ). Moreover, the respective final display luminances of the pixels are shown in  FIG. 5(   e ). From this, it can be confirmed that the respective final display luminances of the pixels are identical to the luminance values of the input signals (Rsi, Gsi, Bsi) shown in  FIG. 5(   a ). 
     Thus, according to the aforementioned process for calculating a backlight value and the transmittances of subpixels in the output signal generation section  13 , the amount of electricity that is used by the backlight  17  can be reduced by suppressing the absorption of light by the color filer by causing the W subpixel to bear part of the amount of light of the white component. For this reason, the transferability of the amount of light of the white component to the W subpixel in display image data is a prerequisite for bringing about the effect of reduction in the amount of electricity that is used by the backlight. 
     That is, according to the aforementioned process for calculating a backlight value and the transmittances of subpixels in the output signal generation section  13 , the larger the amount of white-component light that is transferred to the W subpixel of every pixel falling within a display area corresponding to the backlight (i.e., the lower the color saturation), the greater the effect of reduction in the amount of electricity that is used by the backlight. Meanwhile, if the display area corresponding to the backlight contains a pixel where the amount of white-component light that is transferred to its W subpixel is small (i.e., the color saturation is high), the effect of reduction in the amount of electricity that is used by the backlight becomes small. If the luminance is higher, there may be even an increase in power consumption in comparison with the display method of Patent Document 1. 
     The following describes an example of how backlight values are set for two pixels identical in luminance to each other but different in color saturation from each other. 
     First, the backlight value for a pixel A (Luminance=208; Color Saturation=0.533) where (R, G, B)=(176, 240, 112) is calculated in the following manner. 
     In the pixel A, the amount of light that is transferred to the W subpixel is (112). Then, the amount of light that is transferred to the W subpixel is subtracted from each of the amounts of light of the R, G, and B subpixels, so that the respective amounts of light of the R, G, and B subpixels become (64, 128, 0). As a result, the backlight value for the pixel A is (128). 
     Meanwhile, the backlight value for a pixel B (Luminance=208; Color Saturation=0.75) where (R, G, B)=(160, 256, 64) is calculated in the following manner. 
     In the pixel B, the amount of light that is transferred to the W subpixel is (64). Then, the amount of light that is transferred to the W subpixel is subtracted from each of the amounts of light of the R, G, and B subpixels, so that the respective amounts of light of the R, G, and B subpixels become (96, 192, 0). As a result, the backlight value for the pixel B is (192). 
     Thus, a comparison between the pixels A and B shows that although the pixels A and B are identical in luminance to each other, the pixel B, which is higher in color saturation than the pixel A, has a greater backlight value set therefor than the pixel A. This shows that the effect of reduction in the amount of electricity that is used by the backlight is small. 
     The output signal generation section  13  can also calculate a backlight value and the transmittances of subpixels by performing the above-described process on the original image data (i.e. the first input RGB signal) originally inputted to the present liquid crystal display device. However, in this case, the effect of reduction in power consumption is not necessarily achieved with every image for the aforementioned reason (It should be noted that, in reality, the effect of reduction in power consumption is often achieved with normal gradation-display screens considered to be most likely to be displayed). 
     For this reason, in the present liquid crystal display device, the color-saturation conversion section  11  is disposed in front of the output signal generation section  13  so as to perform a color-saturation conversion process of converting the first input RGB signal into a saturation-converted RGB signal. This makes it possible to more surely obtain, in the process performed by the output signal generation section  13 , a greater effect of reducing the amount of electricity that is used by the backlight. 
     Further, in the present liquid crystal display device, the gamma-correction section  12  is disposed between the color-saturation conversion section  11  and the output signal generation section  13  so as to perform a gamma-correction process of converting the saturation-converted RGB signal into a gamma-corrected RGB signal. In an ordinary liquid crystal display device, an input RGB signal on which gamma correction has been performed is outputted to a liquid crystal panel. However, in the present invention, the color-saturation conversion process and the gamma-correction process exert an influence on each other. Therefore, the present embodiment is characterized so that the color-saturation conversion process and the gamma-correction process are both suitably performed. 
     The following fully describes, with reference to each of Embodiments 1 to 13, a color-saturation reduction process and a gamma-correction process that are performed in a liquid crystal display device of the present invention. 
     Embodiment 1 
       FIG. 6  shows an arrangement of a color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 1. The color-saturation conversion section  11  includes a backlight upper-limit calculation section  21 , a calculation section  22  for calculating the maximum and minimum values of a gamma-corrected RGB signal, a first color-saturation conversion rate calculation section  23 , and a saturation-converted RGB signal calculation section  25 . Further,  FIG. 7  is a flow chart that explains the operation of the color-saturation conversion section  11 . 
     First, the backlight upper-limit calculation section  21  calculates a backlight upper limit according to Formula (1) below (S 11 ). The color-saturation conversion section  11  performs a color-saturation reduction process only on a pixel low in amount of light that is directly transferred to its W subpixel (i.e., high in color saturation) and high in luminance, but does not perform a color-saturation reduction process on a pixel low in at least either color saturation or luminance. The reason for this is as follows: The backlight value for a pixel low in color saturation can be greatly reduced by transferring a large amount of light to its W subpixel even if the pixel is high in luminance, and a pixel low in luminance does not require a high backlight value for a display in the first place. The backlight upper limit is used in determining a pixel on which to perform a color-saturation reduction process. The following fully describes a procedure for calculating a backlight upper limit. 
     First, let it be assumed that no color-saturation reduction process is performed on image data (i.e., an input RGB signal) and a maximum backlight value is obtained. This is a case where there exists a pixel where the color saturation is 1 (the amount of light cannot be transferred to the W subpixel) and at least one of the RGB values is MAX (indicative of the upper limit of the input RGB signal). Further, at this time, the backlight value also becomes MAX. 
     Next, let it be assumed that a color-saturation reduction process is performed on image data (i.e., an input RGB signal) and a maximum backlight value is obtained. It should be noted here that the color-saturation reduction process is a process by which the color saturation of a processing pixel is minimized without causing a difference in luminance of the pixel between before and after the process. In this case, a maximum backlight value is obtained when there exists a pixel where the color saturation is 0 (the backlight value is irreducible because the color saturation cannot be reduced any further) and all the RGB values are MAX. In the pixel, half the amount of light of each of the KGB values is transferred to the W subpixel (provided the area of each of the R, G, and B subpixels is equal to the area of the W subpixel). Therefore, at this time, the backlight value becomes MAX/2. 
     Therefore, the backlight upper limit MAXw falls within a range of MAX/2 to MAX. The backlight upper limit MAXw can be represented by Formula (1) as follows:
 
MAX w =MAX× B 1 Ratio  (1)
 
     where B1 Ratio falls within a range of 0.5 to 1.0. 
     MAX here denotes the upper limit of the input RGB signal. MAX may not take on a single value, but may take on a plurality of values. That is, the lower limit of MAX is the maximum value (MAXi) of all the RGB values of the input RGB signal. The reason for this is that it is impossible to guarantee a desired backlight value when MAX is smaller than MAXi. On the other hand, the upper limit of MAX is the maximum value (MAXs) on which the input RGB signal can take, The reason for this is that there is no need for a backlight value greater than MAXs. 
     MAXs is expressed as follows:
 
MAX s =2 Bw −1
 
     where Bw is the bit width of the input RGB signal. For example, if Bw is 8, MAXs is calculated as 2 8 −1=255. Therefore, the effective range of MAX is expressed as follows:
 
MAX i ≦MAX≦MAX s.  
 
     Basically, MAX may take on any value as long as the value satisfies MAXi≦MAX≦MAXs. If MAX=MAXi, the backlight value can be minimized. However, this makes it necessary to calculate MAX for each image. On the other hand, if MAX=MAXs, the backlight upper limit (MAXw) becomes higher than MAXi; however, MAX becomes a value that remains constant regardless of images. This makes it unnecessary to calculate MAX for each image. 
     Further, in Formula (1), B1 Ratio is a constant that denotes the degree of a color-saturation reduction process. That is, a B1 Ratio of 1 corresponds to a case where no color-saturation reduction process is performed, and a B1 Ratio of 0.5 corresponds to a case where such a process is performed as to minimize the color saturation. In the color-saturation reduction process, the more the color saturation is reduced, the greater the effect of reduction in the amount of electricity that is used by the backlight. However, this naturally causes an increase in deterioration in image quality due to the reduction in color saturation. For this reason, in consideration of a balance between the effect of reduction in power consumption and the deterioration in image quality, it is only necessary to arbitrarily set B1 Ratio within a range of 0.5 to 1 in accordance with a required level of color-saturation reduction. 
     Once the backlight upper limit MAXw is thus determined, it is judged for each pixel, in Steps S 12  to S 15 , according to the following procedure whether or not a color-saturation reduction process is performed. 
     First, the calculation section  22  uses Formulae (2) and (3) below to calculate, from the input RGB signal and a gamma coefficient, the maximum and minimum values maxRGBg and minRGBg of an RGB signal that is obtained after gamma correction in a target pixel (to be processed) and then outputs the maximum and minimum values maxRGBg and minRGBg (S 12 ):
 
max RGBg=fg (max RGB ,γ)  (2)
 
min RGBg=fg (min RGB ,γ)  (3)
 
     where 
     maxRGB=max (R[i], G[i], B[i]), 
     minRGB=min (R[i], G[i], B[i]), 
     max (A, B, . . . )=the maximum value of A, B, . . . , 
     min (A, B, . . . ) the minimum value of A, B, . . . , 
     γ=the gamma coefficient (&gt;0), and 
     fg(x,g)=a gamma-correction function (e.g., fg(x,g)=(x/MAX) g ×MAX). 
     Next, the first color-saturation conversion rate calculation section  23  judges whether or not the luminance and color saturation of the target pixel are both high, and calculates a color-saturation conversion rate α in accordance with a result of the judgment (S 13 ). In this judgment process, it is confirmed whether or not the maximum and minimum values maxRGBg and minRGBg calculated for the gamma-corrected RGB signal according to Formulae (2) and (3) satisfy Formula (4):
 
MAX w &lt;max RGBg− min RGBg.   (4)
 
See below for how to derive Formula (4).
 
     In cases where Formula (4) is satisfied, it is deemed that the luminance and the color saturation are both high. In this case, the color-saturation conversion rate α is calculated to satisfy Formula (5): 
     
       
         
           
             
               
                 
                   
                     
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     where Y[i]=the luminance of the input RGB signal (R[i], G[i], B[i]) (e.g., Y[i]=(2×R[i]+5×G[i]+B[i])/8). See below for how to derive Formula (5). 
     In cases where Formula (4) is not satisfied, it is deemed that at least either the luminance or the color saturation is low. In this case, α=1. 
     Next, the saturation-converted RGB signal calculation section  25  calculates a saturation-converted RGB signal (Rs[i], Gs[i], Bs[i]) according to Formulae (6) to (8) (S 15 ):
 
 Rs[i]=α×R[i ]+(1−α)× Y[i]   (6)
 
 Gs[i]=α×G[i ]+(1−α)× Y[i]   (7)
 
 Bs[i]=α×B[i ]+(1−α)× Y[i].   (8)
 
See below for how to derive Formulae (6) to (8).
 
     The following explains how to derive a calculating formula in each step. 
     First, the following explains how to derive Formula (4), which serves as a condition for judging whether or not to perform a color-saturation conversion process. 
     As shown below, an algorithm up to the calculation of a backlight value that is obtained in cases where no color-saturation conversion is performed is a sequence of a step of performing a gamma-correction process, a step of calculating a transmission amount of a W subpixel, a step of calculating transmission amounts of RGB subpixels, and a step of calculating a backlight value, the steps being arranged in the order named. 
     According to the gamma-correction process, an RGB signal (Rg[i], Gg[i], Bg[i]) that is obtained after gamma correction in cases where no color-saturation conversion process is performed on the input RGB signal (R[i], G[i], B[i]) is given by Formulae (101) to (103):
 
 Rg[i]=fg ( R[i],γ )  (101)
 
 Gg[i]=fg ( G[i],γ )  (102)
 
 Bg[i]=fg ( B[i],γ ).  (103)
 
     For the gamma-corrected RGB signal (Rg[i], Gg[i], Bg[i]), the transmission amount Wtg[i] of the W subpixel is given by Formula (104):
 
 Wtg[i] =min(max RGBg/ 2,min RGBg )  (104)
 
     where 
     maxRGBg=max(Rg[i], Gg[i], Bg[i]), and 
     minRGBg=min(Rg[i], Gg[i], Bg[i]). 
     For the gamma-corrected RGB signal (Rg[i], Gg[i], Bg[i]) and the transmission amount Wtg[i] of the W subpixel, the transmission amounts (Rtg[i], Gtg[i], Btg[i]) of the RGB subpixels are given by Formulae (105) to (107):
 
 Rtg[i]=Rg[i]−Wtg[i]   (105)
 
 Gtg[i]=Gg[i]−Wtg[i]   (106)
 
 Btg[i]=Bg[i]−Wtg[i].   (107)
 
     Then, the backlight value Wbg is given by Formula (108):
 
 Wbg =max( Rtg[ 1 ],Gtg[ 1], Btg[ 1 ],Wtg[ 1], . . .  Rtg[Np],Gtg[Np],Btg[Np],Wtg[Np ]).  (108)
 
     In Formulae (104) to (107), each of the RGBW transmission amounts does not take on a value that falls short of 0. Next, from Formula (104), a condition in which the W transmission amount does not exceed MAXw is given as Formula (109):
 
min(max RGBg/ 2,min RGBg )≦MAX w.   (109)
 
     Furthermore, form Formulae (105) to (107), a condition in which each of the RGB transmission amounts does not exceed MAXw is given as Formulae (110) to (112):
 
 Rtg[i ]≦MAX w   (110)
 
 Gtg[i ]≦MAX w   (111)
 
 Btg[i ]≦MAX w.   (112)
 
     From Formulae (104) to (107) and Formulae (110) to (112), the condition in which each of the RGB transmission amounts does not exceed MAXw is given as Formula (113):
 
max( Rtg[i],Gtg[i],Btg[i] )≦MAX w  
 
max RGBg−Wtg[i]≦ MAX w  
 
∵max RGBg −min(max RGBg/ 2,min RGBg )≦MAX w   (113)
 
     where, if (A) maxRGBg/2≦minRGBg, the condition in which the W transmission amount does not exceed MAXw is calculated from Formula (109) as follows:
 
max RGBg/ 2≦MAX w.   (114)
 
Further, since MAXw falls within a range of MAX/2≦MAXw≦MAX, maxRGBg/2≦MAX/2≦MAXw, so that Formula (114) always holds.
 
     Next, the condition in which the RGB transmission amounts do not exceed MAXw is calculated from Formula (113) as follows:
 
max RGBg −max RGBg/ 2≦MAX w  
 
∵max RGBg/ 2≦MAX w.  
 
This formula is identical to Formula (114), and therefore always holds.
 
     Meanwhile, if (B) minRGBg&lt;maxRGBg/2, the condition in which the W transmission amount does not exceed MAXw is calculated from Formula (109) as follows:
 
min RGBg ≦MAX w.   (115)
 
In this case, since MAX/2≦MAXw≦MAX and minRGBg&lt;maxRGBg/2, minRGBg&lt;maxRGBg/2≦MAX/2≦MAXw, so that Formula (115) always holds.
 
     Next, the condition in which the RGB transmission amounts do not exceed MAXw is calculated from Formula (113) as follows:
 
max RGBg −min RGBg ≦MAX w.   (116)
 
     Since Formula (116) does not always hold, a condition in which none of the RGBW transmission amounts exceeds MAXw is given as Formula (116) if (B) minRGBg&lt;maxRGBg/2. 
     On the other hand, a condition in which at least one of the RGBW transmission amounts exceeds MAXw is given as Formula (4) if (B) minRGBg&lt;maxRGBg/2. 
     In cases where Formula (4) holds, it is found from MAX/2≦MAXw≦MAX that:
 
max RGBg/ 2≦MAX/2≦MAX w &lt;max RGBg −min RGBg  
 
max RGBg/ 2&lt;max RGBg −min RGBg  
 
∵min RGBg &lt;max RGBg/ 2.
 
That is, (B) minRGBg&lt;maxRGBg/2 always holds.
 
     Therefore, the condition in which at least one of the RGBW transmission amounts exceeds MAXw is unconditionally given as Formula (4). 
     That is, in cases where R[i], G[i], and B[i] satisfy Formula (4), a color-saturation conversion process is performed to prevent the backlight value from exceeding MAXw. 
     The following explains how to derive Formulae (5) to (8) for calculation in color-saturation conversion. 
     First, formulae for converting an RGB signal so that the luminance and the hue are constant and only the color saturation is reduced are written as Formulae (6) to (8) above where Formula (117) is satisfied as follows:
 
0≦α&lt;1.  (117)
 
     It should be noted, however, that α=1 only when no color-saturation conversion is performed. 
     It is proved as below that Formulae (6) to (8) do not cause a difference in luminance and hue of the RGB signal between before and after the color-saturation conversion process. 
     First, let it be assumed that the luminance is calculated according to the formula (2×R+5×G+B)/8 when the RGB values are (R, G, B). In this case, the luminance Ys[i] after the color-saturation conversion is given by Formula (118):
 
 Ys[i ]=(2 ×Rs[i]+ 5 ×Gs[i]+Bs[i ])/8.  (118)
 
     Substituting Formulae (6) to (8) in Formula (118) yields Formula (119): 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                             
                           
                             
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     From Formula (119), it is found that there is no difference in luminance value between before and after the color-saturation conversion. 
     Meanwhile, for the hue, let it be assumed that the R value is at maximum. First, when the R value is at maximum, the hue H[i] before the color-saturation conversion is given by Formula (120):
 
 H[i ]=( Cb−Cg )×60  (120)
 
     where 
     Cb=(maxRGB B[i])/(maxRGB−minRGB), and 
     Cg=(maxRGB−G[i])/(maxRGB−minRGB). 
     Next, the hue Hs[i] after the color-saturation conversion is given by Formula (121):
 
 Hs[i ]=( Cbs−Cgs )×60  (121)
 
where
 
     Cbs=(maxRGBs−Bs[i])/(maxRGBs−minRGBs), 
     Cgs=(maxRGBs−Gs[i])/(maxRGBs−minRGBs), 
     minRGBs=minRGB after the color-saturation conversion, and 
     maxRGBs=maxRGB after the color-saturation conversion. 
     Transforming Formula (121) and substituting Formulae (6) to (8) in Formula (121) thus transformed yield Formula (122): 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                             
                           
                             
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                                           max 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           RGB 
                                         
                                         - 
                                         
                                           G 
                                           ⁡ 
                                           
                                             [ 
                                             i 
                                             ] 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   } 
                                 
                                 
                                   ( 
                                   
                                     
                                       max 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       RGB 
                                     
                                     - 
                                     
                                       min 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       RGB 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             × 
                             60 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 Cb 
                                 - 
                                 Cg 
                               
                               ) 
                             
                             × 
                             60 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               H 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     122 
                     ) 
                   
                 
               
             
           
         
       
     
     From Formula (122), it is found that there is no difference in hue between before and after the color-saturation conversion. The same applies to a case where the G or B value is at maximum. 
     Next, when Formulae (6) to (8) are calculating formulae for color-saturation conversion, such a relational expression of α is derived that the backlight value is MAXw. 
     If all those pixels which satisfy Formula (4) are subjected to color-saturation conversion so as to satisfy Formula (123)
 
MAX w =max RGBsg −min RGBsg,   (123)
 
the backlight value is necessarily not more than MAXw.
 
     From Formulae (6) to (8) and Formula (123), Formula (5) is derived. Solving Formula (5) derives such a value of α that the backlight value becomes MAXw. However, the formula is a nonlinear equation, and therefore cannot by easily solved. Accordingly, it is conceivable that Formula (5) may be solved with use of an approximation method. The solving method will be fully described below in each of Embodiments 4 to 13. 
     Finally, in the case of a pixel that does not satisfy Formula (4), setting α to 1 in Formulae (6) to (8) causes the RGB values (R[i], G[i], B[i]) before the color-saturation conversion and the RGB values (Rs[i], Gs[i], Bs[i]) after the color-saturation conversion to be equal, with the same result as in the case where no color-saturation conversion is performed. 
     The color-saturation conversion process causes the input image to deteriorate in image quality in terms of the vividness of colors. However, ordinary images do not have so many portions high in luminance and color saturation. Thus, in many cases, only limited portions of the images decrease in color saturation. Further, humans have such a visual property as not to be as sensitive to a change in color as a change in brightness. Thus, in many cases, such deterioration in image quality is hard for humans to recognize. On the other hand, humans have such a visual property as to be more sensitive to a change in brightness than to a change in color. Therefore, a reduction in luminance is recognized as great deterioration in image quality. Therefore, it is important in the color-saturation conversion process to maintain the luminance so that there is no difference in luminance between before and after the conversion. 
     The saturation-converted RGB signal calculated by the color-saturation conversion section  11  is then subjected to gamma correction by the gamma-correction section  12 , and the gamma-correction section  12  outputs a gamma-corrected RGB signal. The gamma-correction process in the gamma-correction section  12  is identical to gamma correction that is performed in an ordinary display device. Generally, an output signal (the gamma-corrected RGB signal here) corresponding to an input signal (saturation-converted RGB signal here) is read out from an LUT (look-up table), or is calculated according to a conversion formula. Further, the LUT or the conversion formula for use in the gamma-correction process of the present embodiment can be identical to that for use in gamma correction that is performed in an ordinary display device. 
     However, in the present embodiment, it is important to perform a gamma-correction process after performing a color-saturation conversion process. That is, the present embodiment is characterized in one aspect by disposing the gamma-correction section  12  behind the color-saturation conversion section  11 . The reason for this will be described below. 
       FIG. 8  shows an arrangement of an output signal generation section  13  of the liquid crystal display device according to Embodiment 1. The output signal generation section  13  includes a W transmission amount calculation section  41 , an RGB transmission amount calculation section  42 , a backlight value calculation section  43 , and a transmittance calculation section  44 . Further,  FIG. 9  is a flow chart that explains the operation of the output signal generating section  13 . 
     First, the W transmission amount calculation section  41  calculates the W transmission amount Wtsg[i] of a target pixel from a gamma-corrected RGB signal according to Formula (55) (S 41 ):
 
 Wtsg[i] =min(max RGBsg/ 2,min RGBsg ).  (55)
 
     Next, the RGB transmission amount calculation section  42  calculates the RGB transmission amounts (Rtsg[i], Gtsg[i], Btsg[i]) of the target pixel from the gamma-corrected RGB signal and the W transmission amount according to Formulae (56) to (58) (S 42 ):
 
 Rtsg[i]=Rsg[i]−Wtsg[i]   (56)
 
 Gtsg[i]=Gsg[i]−Wtsg[i]   (57)
 
 Btsg[i]=Bsg[i]−Wtsg[i].   (58)
 
     Steps S 41  and S 42  are repeated as many times as the number of pixels of the input RGB signal. 
     The backlight value calculation section  43  calculates a backlight value Wbsg from the RGBW transmission amounts of each pixel according to Formula (59) (S 43 ):
 
 Wbsg =max( Rtsg[ 1], Gtsg[ 1 ],Btsg[ 1 ],Wtsg[ 1], . . .  Rtsg[Np],Gtsg[Np],Btsg[Np],Wtsg[Np] )  (59)
 
     Next, the transmittance calculation section  44  calculates the RGBW transmittances of the target pixel from the RGBW transmission amounts and the backlight value according to Formulae (60) to (63) (S 44 ):
 
 rsg[i]=Rtsg[i]/Wbsg   (60)
 
 gsg[i]=Gtsg[i]/Wbsg   (61)
 
 bsg[i]=Btsg[i]/Wbsg   (62)
 
 Wsg[i]=Wtsg[i]/Wbsg   (63)
 
     where rsg[i]=gsg[i]=bsg[i]=wsg[i]=0 if Wbsg=0. Step S 44  is repeated as many times as the number of pixels of the input RGB signal. 
     The following shows an example of calculation of a backlight value according to Embodiment 1. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). It should be noted that the following explanation assumes that MAX=255, γ=2, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the following manner. Performing gamma correction on the pixel values of the input image yields gamma-corrected pixel values (Rg[1], Gg[1], Bg[1]) as follows: 
     
       
         
           
             
               
                 
                   
                     Rg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             R 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           MAX 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           159 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   99 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             G 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           MAX 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           255 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   255 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             B 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           MAX 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           63 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   16. 
                 
               
             
           
         
       
     
     Based on the gamma-corrected pixel values, the W transmission amount Wtg[1] is calculated as: 
     
       
         
           
               
             
               
                 
                   
                     
                       Wtg 
                       ⁡ 
                       
                         [ 
                         1 
                         ] 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       min 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               max 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               RGBg 
                             
                             2 
                           
                           , 
                           
                             min 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             RGBg 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       min 
                       ⁡ 
                       
                         ( 
                         
                           
                             255 
                             2 
                           
                           , 
                           16 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     16. 
                   
                 
               
             
           
         
       
     
     Further, at this time, the RGB transmission amounts (Rtg[1], Gtg[1], Btg[1]) are given by:
 
 Rtg[ 1]= Rg[ 1]− Wtg[ 1]=99−16=83
 
 Gtg[ 1]= Gg[ 1] −Wtg[ 1]=255−16=239
 
 Btg[ 1]= Bg[ 1 ]−Wtg[ 1]=16−16=0.
 
     From the RGBW transmission amounts, a backlight value Wbg is calculated as: 
                         Wbg   =       ⁢     max   ⁡     (       Rtg   ⁡     [   1   ]       ,     Gtg   ⁡     [   1   ]       ,     Btg   ⁡     [   1   ]       ,     Wtg   ⁡     [   1   ]         )                   =       ⁢     max   ⁡     (     83   ,   239   ,   0   ,   16     )                   =       ⁢   239.                 
That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239.
 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. According to the calculating procedure described above in the present embodiment, the backlight upper limit MAXw is calculated from Formula (1) as follows:
 
MAX w =MAX× B 1 Ratio=255×0.5=127.5.
 
     From Formulae (2) and (3), the maximum and minimum values of the gamma-corrected RGB signal are given by: 
     
       
         
           
             
               
                 
                   
                     max 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     RGBg 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             max 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             RGB 
                           
                           MAX 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           255 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   255 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     min 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     RGBg 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             min 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             RGB 
                           
                           MAX 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           63 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   16. 
                 
               
             
           
         
       
     
     Since the target pixel satisfies Formula (4), color-saturation conversion is performed. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated as: 
     
       
         
           
               
             
               
                 
                   
                     
                       Y 
                       ⁡ 
                       
                         [ 
                         1 
                         ] 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             2 
                             × 
                             
                               R 
                               ⁡ 
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                           
                           + 
                           
                             5 
                             × 
                             
                               G 
                               ⁡ 
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                           
                           + 
                           
                             B 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         ) 
                       
                       8 
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             2 
                             × 
                             159 
                           
                           + 
                           
                             5 
                             × 
                             255 
                           
                           + 
                           63 
                         
                         ) 
                       
                       8 
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     207. 
                   
                 
               
             
           
         
       
     
     Substituting MAX=255, γ=2, maxRGB=255, minRGB=63, Y[1]=207, and MAXw=127.5 in Formula (5) yields:
 
 fg (α×255+(1−α)×207,2)− fg (α×63+(1−α)×207,2)=127.5.
 
Solving this formula yields a color-saturation conversion rate α as follows:
 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             a 
                             × 
                             255 
                           
                           + 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 a 
                               
                               ) 
                             
                             × 
                             207 
                           
                         
                         ) 
                       
                       / 
                       255 
                     
                     } 
                   
                   2 
                 
                 × 
                 255 
               
               - 
               
                 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             a 
                             × 
                             63 
                           
                           + 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 a 
                               
                               ) 
                             
                             × 
                             207 
                           
                         
                         ) 
                       
                       / 
                       255 
                     
                     } 
                   
                   2 
                 
                 × 
                 255 
               
             
             = 
             127.5 
           
         
       
       
         
           
             
               
                 
                   { 
                   
                     ( 
                     
                       
                         a 
                         × 
                         255 
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             - 
                             a 
                           
                           ) 
                         
                         × 
                         207 
                       
                     
                     ) 
                   
                   } 
                 
                 2 
               
               - 
               
                 
                   { 
                   
                     ( 
                     
                       
                         a 
                         × 
                         63 
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             - 
                             a 
                           
                           ) 
                         
                         × 
                         207 
                       
                     
                     ) 
                   
                   } 
                 
                 2 
               
             
             = 
             32512.5 
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     
                       48 
                       × 
                       a 
                     
                     + 
                     207 
                   
                   ) 
                 
                 2 
               
               - 
               
                 
                   ( 
                   
                     
                       
                         - 
                         144 
                       
                       × 
                       a 
                     
                     + 
                     207 
                   
                   ) 
                 
                 2 
               
             
             = 
             32512.5 
           
         
       
       
         
           
             
               
                 2304 
                 × 
                 
                   a 
                   2 
                 
               
               + 
               
                 19872 
                 × 
                 a 
               
               + 
               42849 
               - 
               
                 20736 
                 × 
                 
                   a 
                   2 
                 
               
               + 
               
                 59616 
                 × 
                 a 
               
               - 
               42849 
             
             = 
             
               
                 32512.5 
                 ⁢ 
                 
                   
 
                 
                 - 
                 
                   18432 
                   × 
                   
                     a 
                     2 
                   
                 
                 + 
                 
                   2 
                   × 
                   39744 
                   × 
                   a 
                 
                 - 
                 32512.5 
               
               = 
               0. 
             
           
         
       
     
     Applying the formula for the solution of a quadratic equation to this formula yields: 
             α   =       √     -   39744       ±       √     {               (     -   39744     )     2     -       (     -   18432     )     ×                   (     -   32512.5     )     )           }       /       {     -   18432     )     .               
Therefore, α=0.457575 or α=3.854925.
 
     Since 0≦α&lt;1, α=0.457575. 
     Once the color-saturation conversion rate α is determined, a saturation-converted RGB signal (Rs[1], Gs[1], Bs[1]) is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               Rs 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   a 
                   × 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       a 
                     
                     ) 
                   
                   × 
                   
                     Y 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
               
               = 
               
                 
                   
                     0.457575 
                     × 
                     159 
                   
                   + 
                   
                     
                       ( 
                       
                         1 
                         - 
                         0.457575 
                       
                       ) 
                     
                     × 
                     207 
                   
                 
                 = 
                 185 
               
             
           
         
       
       
         
           
             
               Gs 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   a 
                   × 
                   
                     G 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       a 
                     
                     ) 
                   
                   × 
                   
                     Y 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
               
               = 
               
                 
                   
                     0.457575 
                     × 
                     255 
                   
                   + 
                   
                     
                       ( 
                       
                         1 
                         - 
                         0.457575 
                       
                       ) 
                     
                     × 
                     207 
                   
                 
                 = 
                 229 
               
             
           
         
       
       
         
           
             
               Bs 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   a 
                   × 
                   
                     B 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       a 
                     
                     ) 
                   
                   × 
                   
                     Y 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                 
               
               = 
               
                 
                   
                     0.457575 
                     × 
                     63 
                   
                   + 
                   
                     
                       ( 
                       
                         1 
                         - 
                         0.457575 
                       
                       ) 
                     
                     × 
                     207 
                   
                 
                 = 
                 141. 
               
             
           
         
       
     
     Performing gamma correction on the saturation-converted RGB signal yields a gamma-corrected RGB signal (Rsg[1], Gsg[1], Bsg[1]) as follows: 
     
       
         
           
             
               Rsg 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         Rs 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                       / 
                       MAX 
                     
                     ) 
                   
                   γ 
                 
                 × 
                 MAX 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         185 
                         / 
                         255 
                       
                       ) 
                     
                     2 
                   
                   × 
                   255 
                 
                 = 
                 
                   
                     134 
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       Gsg 
                       ⁡ 
                       
                         [ 
                         1 
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               Gs 
                               ⁡ 
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                             / 
                             MAX 
                           
                           ) 
                         
                         γ 
                       
                       × 
                       MAX 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               229 
                               / 
                               255 
                             
                             ) 
                           
                           2 
                         
                         × 
                         255 
                       
                       = 
                       
                         
                           206 
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             Bsg 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     Bs 
                                     ⁡ 
                                     
                                       [ 
                                       1 
                                       ] 
                                     
                                   
                                   / 
                                   MAX 
                                 
                                 ) 
                               
                               γ 
                             
                             × 
                             MAX 
                           
                           = 
                           
                             
                               
                                 
                                   ( 
                                   
                                     141 
                                     / 
                                     255 
                                   
                                   ) 
                                 
                                 2 
                               
                               × 
                               255 
                             
                             = 
                             78. 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Based on the gamma-corrected RGB signal, the W transmission amount Wtsg[1] is calculated as: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             RGBsg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           RGBsg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       min 
                       ⁡ 
                       
                         ( 
                         
                           
                             206 
                             / 
                             2 
                           
                           , 
                           78 
                         
                         ) 
                       
                     
                     = 
                     78. 
                   
                 
               
             
           
         
       
     
     Further, at this time, the RGB transmission amounts (Rtsg[1], Gtsg[1], Btsg[1]) are given by:
 
 Rtsg[ 1]= Rsg[ 1] −Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1] −Wtsg[ 1]=206−78=128
 
 Btsg[ 1 ]=Bsg[ 1 ]−Wtsg[ 1]=78−78=0.
 
     From the RGBW transmission amounts, a backlight value Wbsg is calculated as: 
                   Wbsg   =       ⁢     max   ⁡     (       Rtsg   ⁡     [   1   ]       ,     Gtsg   ⁡     [   1   ]       ,     Btsg   ⁡     [   1   ]       ,     Wtsg   ⁡     [   1   ]         )                   =       ⁢       max   ⁡     (     56   ,   128   ,   0   ,   78     )       =   128.                 
That is, the backlight value obtained in cases where the color-saturation conversion process is 128.
 
     Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     As described above, according to the process of the present embodiment, the gamma-correction section  12  is disposed behind the color-saturation conversion section  11  so that a gamma-correction process is performed after a color-saturation conversion process is performed. The reason for this is as follows: Although performing color-saturation conversion after gamma correction causes no difference in luminance between before and after the color-saturation conversion, there is a change in luminance when a human eye performs inverse gamma correction on light (an image) outputted from the display. The human-eye inverse gamma correction will be described below more in detail. 
     For example, in cases where a human observes a lamp of certain brightness and then observes the same lamp with its luminance value halved (physically), the eyes of the human do not recognize that the light obtained by halving the luminance value of the lamp is half as bright, but actually recognize the light as more than half as bright. That is, the visual property of a human (property of recognizing brightness) is nonlinear, and a graph whose abscissa represents the physical quantity of brightness and whose ordinate represents the quantity of light perceived by a human is an upwardly convex graph (i.e., a gamma curve) that is monotonically increasing but is not linear. 
     For this reason, the display device performs gamma correction (downwardly convex, monotonically increasing graph) on input image data so that the physical quantity of the luminance value of the gamma-corrected image data and the quantity of light perceived by a human are linear. In other words, the display device performs gamma correction and the human perception performs correction (inverse gamma correction) opposite to the gamma correction performed by the display. As a result, the gamma correction and the inverse gamma correction cancel each other out, and the characteristic of brightness finally perceived by a human becomes linear with respect to the physical quantity of the luminance value of the gamma-corrected image data. 
     Although performing gamma correction after color-saturation conversion as in the process according to the present embodiment causes a temporal change in luminance at the time of the gamma correction, a human eye performs inverse gamma correction, so that the change in luminance is cancelled. Finally, the luminance is retained. 
     For example, in the aforementioned example of calculation of a backlight value (where gamma correction is performed after color-saturation conversion), the luminance value Ys[1](=Y[1]) before the gamma correction (after the color-saturation conversion) is 207, and the luminance value Ysg[1] after the gamma correction is given by: 
                     Ysg   ⁡     [   1   ]       =       ⁢       (       2   ×     Rsg   ⁡     [   1   ]         +     5   ×     Gsg   ⁡     [   1   ]         +     Bsg   ⁡     [   1   ]         )     /   8                 =       ⁢       (       2   ×   134     +     5   ×   206     +   78     )     /   8                 =       ⁢   172.               
The gamma correction causes a temporal change in luminance.
 
     The RGB corresponding values of the amount of light that is outputted from (transmitted through) the display are identical to Rsg[1], Gsg[1], and Bsg[1], respectively. On the other hand, performing inverse gamma correction with a human eye yields RGB values (Rsgi[1], Gsgi[1], Bsgi[1]) as follows: 
               Rsgi   ⁡     [   1   ]       =           (       Rsg   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     134   /   255     )       1   /   2       ×   255     =       185   ⁢     
     ⁢     Gsgi   ⁡     [   1   ]         =           (       Gsg   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     206   /   255     )       1   /   2       ×   255     =   229                           Bsgi   ⁡     [   1   ]       =           (       Bsg   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     78   /   255     )       1   /   2       ×   255     =   141.             
Therefore, the RGB values that are obtained in cases where inverse gamma correction is performed with a human eye correspond to the following luminance Ysgi[1]:
 
                     Ysgi   ⁡     [   1   ]       =       ⁢       (       2   ×     Rsgi   ⁡     [   1   ]         +     5   ×     Gsgi   ⁡     [   1   ]         +     Bsgi   ⁡     [   1   ]         )     /   8                 =       ⁢       (       2   ×   185     +     5   ×   229     +   141     )     /   8                 =       ⁢   207.               
This shows that the luminance is retained.
 
     Meanwhile, a case where color-saturation conversion is preformed after gamma correction is as follows. 
     First, performing gamma correction on the pixel value (R[1], G[1]), B[1]) of the input image yields pixel values (Rg[1], Gg[1], Bg[1])=(99, 255, 16). 
     Next, such a relational expression of a that a backlight value that is obtained in cases where color-saturation conversion is preformed after gamma correction is not more than MAXw is represented by Formula (124) as follows:
 
MAX w =max RGBgs− min RGBgs   (124)
 
where
 
max RGBgs =α×max RGBg +(1−α)× Yg[ 1], and
 
min RGBgs =α×min RGBg +(1−α)× Yg[ 1].
 
     Transforming Formula (124) yields:
 
{α×max RGBg +(1−α)× Yg[ 1]}−{α×min RGBg +(1−α)× Yg[ 1]}=MAX w  
 
∵=MAX w /(max RGBg −min RGBg ).  (125)
 
     At this time, α is calculated from Formula (125) as follows: 
     
       
         
           
             
               
                 
                   a 
                   = 
                     
                   ⁢ 
                   
                     MAX 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       w 
                       / 
                       
                         ( 
                         
                           
                             max 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             RGBg 
                           
                           - 
                           
                             min 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             RGBg 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     127.5 
                     / 
                     
                       ( 
                       
                         255 
                         - 
                         16 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     0.533473 
                     . 
                   
                 
               
             
           
         
       
     
     Performing color-saturation conversion yields pixel values (Rgs[1], Ggs[1], Bgs[1]) as follows: 
               Rgs   ⁡     [   1   ]       =         a   ×     Rg   ⁡     [   1   ]         +       (     1   -   a     )     ×     Yg   ⁡     [   1   ]           =         0.533473   ×   99     +       (     1   -   0.533473     )     ×   186.125       =   140                     Ggs   ⁡     [   1   ]       =         a   ×     Gg   ⁡     [   1   ]         +       (     1   -   a     )     ×     Yg   ⁡     [   1   ]           =         0.533473   ×   223     +       (     1   -   0.533473     )     ×   186.125       =   223                     Bgs   ⁡     [   1   ]       =         a   ×     Bg   ⁡     [   1   ]         +       (     1   -   a     )     ×     Yg   ⁡     [   1   ]           =         0.533473   ×   95     +       (     1   -   0.533473     )     ×   186.125       =   95.             
The RGB corresponding values of the amounts of light that is outputted from (transmitted through) the display are identical to Rgs[1], Ggs[1], and Bgs[1], respectively. On the other hand, performing inverse gamma correction with a human eye yields RGB values (Rgs[i], Ggs[i], Bgs[i]) as follows:
 
               Rgsi   ⁡     [   1   ]       =           (       Rgs   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     140   /   255     )       1   /   2       ×   255     =   189                     Ggsi   ⁡     [   1   ]       =           (       Ggs   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     223   /   255     )       1   /   2       ×   255     =   238                     Bgsi   ⁡     [   1   ]       =           (       Bgs   ⁡     [   1   ]       /   MAX     )       1   /   γ       ×   MAX     =           (     95   /   255     )       1   /   2       ×   255     =   156.             
Therefore, the luminance Ygsi[1] is given by:
 
                     Ygsi   ⁡     [   1   ]       =       ⁢       (       2   ×     Rgsi   ⁡     [   1   ]         +     5   ×     Ggsi   ⁡     [   1   ]         +     Bgsi   ⁡     [   1   ]         )     /   8                 =       ⁢       (       2   ×   189     +     5   ×   238     +   156     )     /   8                 =       ⁢     215.5   .                 
That is, the original luminance Y[1]=207 is not retained. Therefore, in order to retain the luminance, it is necessary to perform gamma correction after performing color-saturation conversion.
 
     Embodiment 2 
     Embodiment 1 uses Formula (4) to judge for each target pixel whether or not it is necessary to reduce the color saturation. In this case, it is possible to perform a color-saturation conversion (color-saturation reduction) process only on a pixel that satisfies Formula (4), i.e., a pixel whose color saturation needs to be reduced, and to skip a color-saturation conversion process for a pixel that does not satisfy Formula (4), i.e., a pixel whose color saturation does not need to be reduced. 
     However, in consideration of realization of the color-saturation conversion section  11  by hardware, the hardware can be better simplified without such a skip process. Embodiment 2 describes a modified example where a color-saturation conversion process is performed on every pixel within an input image. It should be noted that processing sections identical to those of Embodiment 1 are given the same reference numerals as those of Embodiment 1 and will not be described below in detail. 
     In a process according to Embodiment 2, for a pixel that does not satisfy Formula (4), a deviates from a range of 0 to 1, depending on how α is calculated. In view of this, a limiter process is performed so that α is 0 or 1, with the same result as in the case where no color-saturation conversion is performed. 
       FIG. 10  shows an arrangement of a color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 2. The color-saturation conversion section  11  includes a backlight upper-limit calculation section  21 , a second color-saturation conversion rate calculation section  33 , and a saturation-converted RGB signal calculation section  25 . Further,  FIG. 11  is a flow chart that explains the operation of the color-saturation conversion section  11 . 
     First, the backlight upper-limit calculation section  21  calculates a backlight upper limit according to Formula (1) (S 11 ). 
     Next, the second color-saturation conversion rate calculation section  33  calculates, from the input KGB signal, the gamma coefficient, and the backlight upper limit, a color-saturation conversion rate α that satisfies Formula (5) (S 23 ). How to calculate a color-saturation conversion rate α will be fully described below in each of Embodiments 4 to 13. 
     Next, the saturation-converted RGB signal calculation section  25  calculates a saturation-converted RGB signal according to Formulae (6) to (8) (S 15 ). Steps S 23  and S 15  are repeated as many times as the number of pixels of the input RGB signal. 
       FIG. 12  shows another arrangement of the color-saturation conversion section  11  of the liquid crystal display device according to Embodiment 2. The color-saturation conversions section  11  of  FIG. 12  is arranged such that a color-saturation conversion rate limiter process section  34  is added so as to be positioned in front of the saturation-converted RGB signal calculation section  25  and behind the second color-saturation conversion rate calculation section  33 . The color-saturation conversion rate limiter process section  34  performs a limiter process so that a color-saturation conversion rate α calculated by the second color-saturation conversion rate calculation section  33  falls within a range of not less than 0 to not more than 1. 
       FIG. 13  is a flow chart for explaining the operation of the color-saturation conversion section  11  of  FIG. 12 . The flow chart of  FIG. 13  is obtained by adding a color-saturation conversion rate limiter process S 24  to the flow chart of  FIG. 11  so that Step S 24  immediately follows Step S 23 . 
     According to the flow chart of  FIG. 13 , after the calculation of a color-saturation conversion rate α by the second color-saturation conversion rate calculation section  33  (S 23 ), the color-saturation conversion rate limiter process section  34  makes such a correction that α=0 when α&lt;0 or α=1 when 1&lt;α (S 24 ). 
     However, in cases where it is obvious in the color-saturation conversion process (S 23 ) that α is not less than 0, it is possible to omit a process of judging whether α&lt;0 and the limiter process. Similarly, in cases where it is obvious in the color-saturation conversion process (S 23 ) that α is not greater than 1, it is possible to omit a process of judging whether 1&lt;α and the limiter process. 
     The following shows an example of calculation of a backlight value according to Embodiment 2. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63) (first example of calculation). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that a color-saturation conversion rate limiter process is performed, and that MAX=255, γ=2, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     First, the calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. Accordingly, the backlight upper limit MAXw is set as MAXw 127.5. 
     Since Embodiment 2 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 2 calculates a color-saturation conversion rate α for every pixel. In this example, the color-saturation conversion rate α is calculated in the same manner as in the example explained in Embodiment 1. Therefore, α=0.457575 or α=3.854925. However, the present example of calculation adopts the smaller value. Therefore, α=0.457575. 
     In this case, since 0≦α≦1, α after the limiter process is calculated as α=0.457575. The subsequent procedure up to the calculation of a backlight value is the same as in the example explained in Embodiment 1. Accordingly, the backlight value is 128. That is, the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The example of calculation thus explained is an example where the second color-saturation conversion rate calculation section  33  calculates a color-saturation conversion rate α as 0≦α≦1. The following shows an example (second example of calculation) where the second color-saturation conversion rate calculation section  33  does not calculate a color-saturation conversion rate α as 0≦α≦1. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). It should be noted that the following explanation assumes that a color-saturation conversion rate limiter process is performed, and that MAX=255, γ=2, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the following manner. Performing gamma correction on the pixels values of the input image yields gamma-corrected pixel values (Rg[1], Gg[1], Bg[1]) as follows: 
     
       
         
           
             
               Rg 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                       / 
                       MAX 
                     
                     ) 
                   
                   γ 
                 
                 × 
                 MAX 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         159 
                         / 
                         255 
                       
                       ) 
                     
                     2 
                   
                   × 
                   255 
                 
                 = 
                 99 
               
             
           
         
       
       
         
           
             
               Gg 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                       / 
                       MAX 
                     
                     ) 
                   
                   γ 
                 
                 × 
                 MAX 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         187 
                         / 
                         255 
                       
                       ) 
                     
                     2 
                   
                   × 
                   255 
                 
                 = 
                 137 
               
             
           
         
       
       
         
           
             
               Bg 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                       / 
                       MAX 
                     
                     ) 
                   
                   γ 
                 
                 × 
                 MAX 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         85 
                         / 
                         255 
                       
                       ) 
                     
                     2 
                   
                   × 
                   255 
                 
                 = 
                 28. 
               
             
           
         
       
     
     Based on the gamma-corrected pixel values, the W transmission amount Wtg[1] is calculated as: 
     
       
         
           
             
               
                 
                   
                     Wtg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             RGBg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           RGBg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       min 
                       ⁡ 
                       
                         ( 
                         
                           
                             137 
                             / 
                             2 
                           
                           , 
                           28 
                         
                         ) 
                       
                     
                     = 
                     28. 
                   
                 
               
             
           
         
       
     
     Further, at this time, the RGB transmission amounts (Rtg[1], Gtg[1], Btg[1]) are given by:
 
 Rtg[ 1]= Rg[ 1 ]−Wtg[ 1]=99−28=71
 
 Gtg[ 1]= Gg[ 1] −Wtg[ 1]=137−28=109
 
 Btg[ 1]= Bg[ 1] −Wtg[ 1]=28−28=0.
 
     From the RGBW transmission amounts, a backlight value Wbg is calculated as: 
                   Wbg   =     max   ⁡     (       Rtg   ⁡     [   1   ]       ,     Gtg   ⁡     [   1   ]       ,     Btg   ⁡     [   1   ]       ,     Wtg   ⁡     [   1   ]         )                   =       max   ⁡     (     17   ,   109   ,   0   ,   28     )       =   109.                 
That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109.
 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, a backlight upper limit is calculated. This calculation is performed in the same manner as in the example explained in Embodiment 1. Accordingly, the backlight upper limit MAXw is calculated as MAXw=127.5. 
     Since Embodiment 2 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 2 calculates a color-saturation conversion rate α for every pixel. At this time, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated as: 
     
       
         
           
             
               
                 
                   
                     Y 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           2 
                           × 
                           
                             R 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         + 
                         
                           5 
                           × 
                           
                             G 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         + 
                         
                           B 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                     / 
                     8 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             2 
                             × 
                             159 
                           
                           + 
                           
                             5 
                             × 
                             187 
                           
                           + 
                           85 
                         
                         ) 
                       
                       / 
                       8 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     167.25 
                     . 
                   
                 
               
             
           
         
       
     
     Substituting MAX=255, γ=2, maxRGB=187, minRGB 85, Y[1]=167.25, and MAXw=127.5 in Formula (5) yields:
 
 fg (α×187+(1−α)×167.25,2)− fg (α×85+(1−α)×167.25,2)=127.5.
 
Solving this formula yields a color-saturation conversion rate α as follows:
 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             α 
                             × 
                             187 
                           
                           + 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 α 
                               
                               ) 
                             
                             × 
                             167.25 
                           
                         
                         ) 
                       
                       / 
                       255 
                     
                     } 
                   
                   2 
                 
                 × 
                 255 
               
               - 
               
                 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             α 
                             × 
                             85 
                           
                           + 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 α 
                               
                               ) 
                             
                             × 
                             167.25 
                           
                         
                         ) 
                       
                       / 
                       255 
                     
                     } 
                   
                   2 
                 
                 × 
                 255 
               
             
             = 
             127.5 
           
         
       
       
         
           
             
               
                 
                   { 
                   
                     ( 
                     
                       
                         α 
                         × 
                         187 
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             - 
                             α 
                           
                           ) 
                         
                         × 
                         167.25 
                       
                     
                     ) 
                   
                   } 
                 
                 2 
               
               - 
               
                 
                   { 
                   
                     ( 
                     
                       
                         α 
                         × 
                         85 
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             - 
                             α 
                           
                           ) 
                         
                         × 
                         167.25 
                       
                     
                     ) 
                   
                   } 
                 
                 2 
               
             
             = 
             32512.5 
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     
                       19.75 
                       × 
                       α 
                     
                     + 
                     167.25 
                   
                   ) 
                 
                 2 
               
               - 
               
                 
                   ( 
                   
                     
                       
                         - 
                         82.25 
                       
                       × 
                       α 
                     
                     + 
                     167.25 
                   
                   ) 
                 
                 2 
               
             
             = 
             32512.5 
           
         
       
       
         
           
             
               
                 390.06 
                 × 
                 
                   α 
                   2 
                 
               
               + 
               
                 6606.4 
                 × 
                 α 
               
               + 
               27973 
               - 
               
                 6765.1 
                 × 
                 
                   α 
                   2 
                 
               
               + 
               
                 27512 
                 × 
                 α 
               
               - 
               27973 
             
             = 
             
               
                 32512.5 
                 ⁢ 
                 
                   
 
                 
                 - 
                 
                   6375.0 
                   × 
                   
                     α 
                     2 
                   
                 
                 + 
                 
                   2 
                   × 
                   17059 
                   × 
                   α 
                 
                 - 
                 32512.5 
               
               = 
               0. 
             
           
         
       
     
     Applying the formula for the solution of a quadratic equation to this formula yields: 
                 α   =       -   17059     ±     √     {         (     -   17059     )     2     -       (     -   6375.0     )     ×     (     -   32512.5     )         )           }     /     (     -   6375.0     )           
Therefore, α=1.240458 or α=4.11139. The present example of calculation adopts the smaller value. Therefore, α=1.240458. In this example, since a exceeds 1, a limiter process is performed so that α=1.
 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Embodiment 3 
     Embodiment 3 is a modified example of Embodiment 1.  FIG. 38  schematically shows an arrangement of a liquid crystal display device according to Embodiment 3. That is, the present liquid crystal display device includes a color-saturation conversion and gamma-correction section (color-saturation conversion section)  18  instead of the color saturation conversion section  11  and the gamma-correction section  12  of the liquid crystal display device (see  FIG. 1 ) according to Embodiment 1. 
     The color-saturation conversion and gamma-correction section  18  performs a color-saturation conversion process and a gamma-correction process on an input RGB signal (first input RGB signal), and then outputs, to the subsequent output signal generation section  13 , a saturation-converted and gamma-corrected RGB signal finished with the processes. It should be noted that the saturation-converted and gamma-corrected RGB signal outputted from the color-saturation conversion and gamma-correction section  18  in Embodiment 3 correspond to the gamma-corrected RGB signal (third input KGB signal) of Embodiment 1. 
       FIG. 39  shows an arrangement of the color-saturation conversion and gamma-correction section  18  of the liquid crystal display device according to Embodiment 3. The color-saturation conversion and gamma-correction section  18  includes a backlight upper-limit calculation section  21 , a gamma-correction section  26 , and a saturation-converted and gamma-corrected RGB signal calculation section  27 . 
     The backlight upper-limit calculation section  21  calculates a backlight upper limit from the upper limit of the input RGB signal and the backlight value determination ratio, and then outputs the backlight upper limit. The gamma-correction section  26  calculates a gamma-corrected RGB signal from the input RGB signal and the gamma coefficient, and then outputs the gamma-corrected RGB signal. The saturation-converted and gamma-corrected RGB signal calculation section  27  calculates a saturation-converted and gamma-corrected RGB signal from the input RGB signal, the gamma coefficient, the gamma-corrected RGB signal, and the backlight upper limit, and then outputs the saturation-converted and gamma-corrected RGB signal. 
       FIG. 40  is a flow chart for explaining the operation of the color-saturation conversion and gamma-correction section  18 . 
     First, the backlight upper-limit calculation section  21  calculates a backlight upper limit according to Formula (1) (S 11 ). 
     Once the backlight upper limit MAXw is determined, Steps S 16  to S 19  are repeated as many times as the number of pixels of the input RGB signal. 
     In Step  16 , the gamma-correction section  26  calculates an RGB signal (Rg[i], Gg[i], Bg[i]) that is obtained after gamma correction in a target pixel (to be processed). The calculation of the gamma-corrected RGB signal can be performed according to Formulae (101) to (103). 
     Then, in Step S 17 , the saturation-converted and gamma-corrected RGB signal calculation section  27  judges, according to Formula (4), whether or not the luminance and color saturation of the target pixel (to be processed) are both high. It should be noted here that although Embodiment 1 calculates the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal for use in Formula (4) from Formulae (2) and (3), Embodiment 3 can calculate maxRGBg and minRGBg with use of the gamma-corrected RGB signal as follows:
 
max RGBg =max( Rg[i],Gg[i],Bg[i ])
 
min RGBg =min( Rg[i],Gg[i],Bg[i ]).
 
     In cases where the luminance and the color saturation are both high, i.e., in cases where Formula (4) is satisfied, the same saturation-converted and gamma-corrected RGB signal calculation section  27  calculates a color-saturation conversion rate α that satisfies Formula (5), and then calculates a saturation-converted and gamma-corrected RGB signal with use of a thus calculated (S 18 ). It should be noted that the saturation-converted and gamma-corrected RGB signal is calculated by calculating a saturation-converted RGB signal (second input RGB signal) according to Formulae (6) to (8) and by further performing gamma correction on the saturation-converted RGB signal. 
     Alternatively, in cases where either the luminance or the color saturation is low, i.e., in cases where Formula (4) is not satisfied, the same saturation-converted and gamma-corrected RGB signal calculation section  27  uses the following formulae to cause the saturation-converted and gamma-corrected RGB signal (Rsg[i], Gsg[i], Bsg[i]) to be identical to the gamma-corrected RGB signal calculated in Step S 16  (no color-saturation conversion process is performed) (S 19 );
 
 Rsg[i]=Rg[i] 
 
 Gsg[i]=Gg[i] 
 
 Bsg[i]=Bg[i].  
 
     In comparison with Embodiment 1, the arrangement and processing procedure of Embodiment 3 bring about a merit of simplifying a process for a pixel that does not satisfy Formula (4), i.e., for a pixel on which no color-saturation conversion is performed. 
     That is, in the process of judging a pixel on which to perform a color-saturation conversion process, the color-saturation conversion section  11  according to Embodiment 1 calculates, for every pixel, an RGB signal (Rg[i], Gg[i] Bg[i]) that is obtained after gamma correction in cases where no color-saturation conversion process is performed. The gamma-corrected RGB signal is calculated in the process by which the calculation section  22  calculates the maximum and minimum values of the gamma-corrected RGB signal. 
     Then, for a pixel on which no color-saturation conversion process is performed, the input RGB signal is directly outputted to the gamma-correction section  12 , and the gamma-correction section  12  calculates a gamma-corrected RGB signal. Therefore, for a pixel on which no color-saturation conversion process is performed, the calculation section  22  and the gamma-correction section  12  perform exactly the same gamma-correction process once each. 
     On the other hand, for a pixel on which no color-saturation conversion process is performed, the color-saturation conversion and gamma-correction section  18  according to Embodiment 3 only causes the saturation-converted and gamma-corrected RGB signal to be identical to the gamma-corrected RGB signal calculated by the gamma-correction section  26 . This makes it necessary to perform a gamma-correction process only once. Therefore, the arrangement of Embodiment 3 makes it possible to simplify hardware configurations in comparison with Embodiment 1 in judging for each pixel whether or not it is necessary to perform a color-saturation conversion process. 
     As stated in Embodiment 1, the color-saturation conversion process explained in each of Embodiments 1 to 3 makes it necessary that, in the process, such a value of color-saturation conversion rate α that the backlight value becomes MAXw be derived by solving Formula (5). However, Formula (5) is a nonlinear equation. Accordingly, it is conceivable that Formula (5) may be solved with use of an approximation method. Each of Embodiments 4 to 13 below explains an example of how to calculate a color-saturation conversion rate α for use in a color-saturation conversion process. 
     Further, each of Embodiments 4 to 13 below is an exemplification of application of the arrangement of Embodiment 1 in an example where it is judged whether or not it is necessary to perform a color-saturation conversion process. However, each of Embodiments 4 to 13 below can also be applied as a method for calculating a color-saturation conversion rate α in the arrangement of Embodiment 3. 
     Embodiment 4 
     Embodiment 4 is a first example of means for approximately calculating a color-saturation conversion rate α in the first color-saturation conversion rate calculation section  23  (S 13 ) of Embodiment 1. 
     In Embodiment 1, in cases where Formula (4) is satisfied, it is necessary to calculate a color-saturation conversion rate α according to Formula (5). However, Formula (5) is a nonlinear equation, and therefore cannot by easily calculated. Accordingly, in Embodiment 4, α is calculated by performing piecewise linear approximation of a gamma correction curve. It should be noted that processing sections identical to those of Embodiment 1 are given the same reference numerals as those of Embodiment 1 and will not be described below in detail. 
       FIG. 14  shows an arrangement of a color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 4. The color-saturation conversion section  11  of  FIG. 14  includes a backlight upper-limit calculation section  21 , a calculation section  22  for calculating the maximum and minimum values of a gamma-corrected RGB signal, a third color-saturation conversion rate calculation section  53 , a gamma-curve normalization and piecewise-linear approximation section  61 , and a saturation-converted RGB signal calculation section  25 . 
     The gamma-curve normalization and piecewise-linear approximation section  61  serves as means for calculating normalized-gamma-curve polygonal-line information from the gamma coefficient and the number of normalized-gamma-curve polygonal-line segments, and for outputting the normalized-gamma-curve polygonal-line information. The normalized-gamma-curve polygonal-line information is a tabulation of data, obtained when piecewise linear approximation of a normalized gamma curve (i.e., a gamma curve that shows an output value of 1 with an input value of 1) is performed as many times as the number of normalized-gamma-curve polygonal-line segments, which indicate the x-y coordinates of an endpoint of each segment and the slope and intercept of each segment. 
       FIG. 15  is a flow chart that explains the operation of the gamma-curve normalization and piecewise-linear approximation section  61 . First, the gamma-curve normalization and piecewise-linear approximation section  61  calculates the coordinates of an endpoint of each normalized-gamma-curve polygonal-line segment according to Formulae (9) and (10) by repeating j from 0 to Ndg (S 51 ):
 
 x[j]=j/Ndg   (9)
 
 y[j]=x[j]   γ   (10)
 
     where 
     Ndg=the number of normalized-gamma-curve polygonal-line segments (&gt;0), 
     x[j]=the x coordinate of an endpoint of a normalized-gamma-curve polygonal-line segment (0≦j≦Ndg), and 
     y[j]=the y coordinate of an endpoint of a normalized-gamma-curve polygonal-line segment (0≦j≦Ndg). That is, Step S 51  is repeated as many times as the number of normalized-gamma-curve polygonal-line segments plus 1. 
     Next, the gamma-curve normalization and piecewise-linear approximation section  61  calculates the slope a[j] and intercept b[j] of each normalized-gamma-curve polygonal-line segment according to Formulae (11) and (12) by repeating j from 0 to Ndg−1 (S 52 ):
 
 a[j ]=( y[j+ 1 ]−y[j ])/ x[j− 1]− x[j ])  (11)
 
 b[j ]=( x[j+ 1 ]×y[j]−x[j]×y[j+ 1])/( x[j+ 1 ]−x[j ])  (12)
 
     where 
     a[j]=the slope of a normalized-gamma-curve polygonal-line segment j (0≦j&lt;Ndg), and 
     b[j]=the intercept of a normalized-gamma-curve polygonal-line segment j (0≦j&lt;Ndg). That is, Step S 52  is repeated as many times as the number of normalized-gamma-curve polygonal-line segments. 
     The third color-saturation conversion rate calculation section  53  calculates a color-saturation conversion rate α from an input RGB signal, a backlight upper limit, the maximum and minimum values of a gamma-corrected RGB signal, and normalized-gamma-curve polygonal-line information, and then outputs the color-saturation conversion rate α. 
       FIG. 16  is a flow chart that explains the operation of the third color-saturation conversion rate calculation section  53 . 
     According to the flow chart, Steps S 61  to S 65  are repeated from j=jMin to j=kMax, and further repeated from k=j to k=kMax. 
     It should be noted here that jMin is a normalized-gamma-curve polygonal-line segment index corresponding to min RGB, and is an array element number of x[j] that satisfies x[jMin]≦minRGB/MAX&lt;x[jMin+1]. Meanwhile, kMax is a normalized-gamma-curve polygonal-line segment index corresponding to maxRGB, and is an array element number of x[k] that satisfies x[kMax]&lt;maxRGB/MAX≦x[kMax+1]. Further, between jMin and kMax, the following relationship holds:
 
0≦ j Min≦ k Max&lt; Ndg.  
 
     First, in Step S 61 , the third color-saturation conversion rate calculation section  53  calculates an α candidate value (αt) according to Formula (13): 
     
       
         
           
             
               
                 
                   αt 
                   = 
                   
                     
                       { 
                       
                         
                           MAX 
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                       . 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Next, the third color-saturation conversion rate calculation section  53  judges whether or not the α candidate value αt falls within a range of 0≦αt&lt;1 (S 62 ). If the α candidate value αt falls within the range, the third color-saturation conversion rate calculation section  53  proceeds to Step S 63 . If the α candidate value αt goes out of the range, the third color-saturation conversion rate calculation section  53  skips Steps S 63  to S 65 . 
     In cases where the α candidate value αt falls within a range of 0≦αt&lt;1, the third color-saturation conversion rate calculation section  53  calculates an α calculation judgment value (judge) according to Formula (14) (S 63 ): 
     
       
         
           
             
               
                 
                   judge 
                   = 
                   
                     
                        
                       
                         
                           fg 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
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                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Next, the third color-saturation conversion rate calculation section  53  judges whether or not to update α (S 64 ). However, in cases where the third color-saturation conversion rate calculation section  53  judges for the first time whether or not to update α or in cases where Judge&lt;judgeMin, the third color-saturation conversion rate calculation section  53  proceeds to Step S 65 , and updates α and the minimum α calculation judgment value (judgeMin) according to Formulae (15) and (16):
 
α=αt  (15)
 
judgeMin=judge  (16)
 
     where judgeMin=the minimum a calculation judgment value. 
     The following explains how to derive a calculating formula in each step. 
     In order to calculate α from Formula (5), approximation of a gamma correction function fg(x,g) is performed with a polygonal line. The approximation formula is represented by Formula (126) as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                   126 
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     Solving Formula (5) to which Formula (126) has been applied yields α as follows: 
     
       
         
           
             
               
                 
                   
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                   ( 
                   
                     
                       b 
                       ⁡ 
                       
                         [ 
                         
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         ] 
                       
                     
                     - 
                     
                       b 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         ] 
                       
                     
                   
                   ) 
                 
                 × 
                 
                   MAX 
                   . 
                 
               
             
           
         
       
     
     Thus, α is given by Formula (127) as follows: 
     
       
         
           
             
               
                 
                   
                     
                       α 
                       = 
                       
                         
                           { 
                           
                             
                               MAX 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               w 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         k 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Max 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         s 
                                       
                                       ] 
                                     
                                   
                                   - 
                                   
                                     a 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Min 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         s 
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                               × 
                               
                                 Y 
                                 ⁡ 
                                 
                                   [ 
                                   i 
                                   ] 
                                 
                               
                             
                             - 
                             
                               ( 
                               
                                 b 
                                 ❘ 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Max 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   s 
                                 
                               
                               ] 
                             
                             - 
                             
                               b 
                               ⁡ 
                               
                                 [ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Min 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   s 
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                         × 
                         MAX 
                       
                     
                     } 
                   
                   / 
                   
                     { 
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Max 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                             
                             ] 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               max 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               RGB 
                             
                             - 
                             
                               Y 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Min 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                             
                             ] 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               min 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               RGB 
                             
                             - 
                             
                               Y 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   127 
                   ) 
                 
               
             
           
         
       
     
     where 
     jMins=the normalized-gamma-curve polygonal-line segment index corresponding to minRGBs, and 
     kMaxs=the normalized-gamma-curve polygonal-line segment index corresponding to maxRGBs. 
     Then, based on the magnitude relation between minRGBs and maxRGBs, Formula (128) holds as follows:
 
 j Min s≦k Max s.   (128)
 
     Furthermore, since the color-saturation conversion (reduction) process causes an increase in minRGB and a reduction in maxRGB, Formulae (129) and (130) hold as follows:
 
min RGB &lt;min RGBs   (129)
 
max RGBs &lt;max RGB.   (130)
 
     Based on Formulae (129) and (130), Formulae (131) and (132) hold as follows:
 
 j Min≦ j Min s   (131)
 
 k Max s≦k Max.  (132)
 
     The values of jMins and kMaxs are unknown. Therefore, based on Formulae (128), (131), and (132), the third color-saturation conversion rate calculation section  53  calculates, according to Formula (13) derived from Formula (127), an α candidate value for every j and k that satisfies Formula (133) (S 61 ):
 
 j Min≦ j≦k≦k Max.  (133)
 
     However, the third color-saturation conversion rate calculation section  53  excludes an α candidate value that deviates from a range of 0≦αt−1 (S 62 ). Among the α candidate values thus calculated, an α candidate value that satisfies Formula (5) becomes a desired value of α. However, since there occurs an error due to piecewise linear approximate, such an α candidate value that the α calculation judgment value of Formula (14) derived from Formula (5) is minimized is set as a desired value of α (S 63  to S 65 ). 
     The following shows an example of calculation a backlight value according to Embodiment 4. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information (j, x[j], y[j], a[j], b[j]) is calculated according to Formulae (9) to (12) to take on values shown below in Table 1. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 i 
                 x[j] 
                 y[j] 
                 a[j] 
                 b[j] 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 0 
                 0 
                 0 
                 0.0625 
                 0 
               
               
                 1 
                 0.0625 
                 0.003906 
                 0.1875 
                 −0.00781 
               
               
                 2 
                 0.125 
                 0.015625 
                 0.3125 
                 −0.02344 
               
               
                 3 
                 0.1875 
                 0.035156 
                 0.4375 
                 −0.04688 
               
               
                 4 
                 0.25 
                 0.0625 
                 0.5625 
                 −0.07813 
               
               
                 5 
                 0.3125 
                 0.097656 
                 0.6875 
                 −0.11719 
               
               
                 6 
                 0.375 
                 0.140625 
                 0.8125 
                 −0.16406 
               
               
                 7 
                 0.4375 
                 0.191406 
                 0.9375 
                 −0.21875 
               
               
                 8 
                 0.5 
                 0.25 
                 1.0625 
                 −0.28125 
               
               
                 9 
                 0.5625 
                 0.316406 
                 1.1875 
                 −0.35156 
               
               
                 10 
                 0.625 
                 0.390625 
                 1.3125 
                 −0.42969 
               
               
                 11 
                 0.6875 
                 0.472656 
                 1.4375 
                 −0.51563 
               
               
                 12 
                 0.75 
                 0.5625 
                 1.5625 
                 −0.60938 
               
               
                 13 
                 0.8125 
                 0.660156 
                 1.6875 
                 −0.71094 
               
               
                 14 
                 0.875 
                 0.765625 
                 1.8125 
                 −0.82031 
               
               
                 15 
                 0.9375 
                 0.878906 
                 1.9375 
                 −0.9375 
               
               
                 16 
                 1 
                 1 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     The calculation of a backlight upper limit and the calculation of the maximum and minimum values of a gamma-corrected RGB signal are performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5, and the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal are set as maxRGBg=255 and minRGBg=16, respectively. 
     Since the target pixel satisfies Formula (4), the target pixel is subjected to color-saturation conversion. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 207 as in Embodiment 1. 
     Further, minRGB/MAX=63/2505=0.2470558. At this time, it is found from Table 1 that jMin=3. Further, maxRGB/MAX=255/255=1. At this time, it is found from Table 1 that kMax=15. 
     Thus, since jMin=3 and kMax=15, the present example of calculation repeats Steps S 61  to S 65  of the flow chart of  FIG. 16  from j=3 to j=15 and from k=j to k=15 for each value of j. The repetition of the steps yields such variations in value of j, k, at, judge, α, and judgeMin as shown below in Table 2. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 j 
                 k 
                 αt 
                 judge 
                 α 
                 judgeMin 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 3 
                 3 
                 1.517867 
                 — 
                 — 
                 — 
               
               
                 3 
                 4 
                 1.217708 
                 — 
                 — 
                 — 
               
               
                 3 
                 5 
                 0.97583 
                 107.7788 
                 0.97583 
                 107.7788 
               
               
                 3 
                 6 
                 0.781939 
                 72.20267 
                 0.781939 
                 72.20267 
               
               
                 3 
                 7 
                 0.628038 
                 39.65104 
                 0.628038 
                 39.65104 
               
               
                 3 
                 8 
                 0.507813 
                 12.16406 
                 0.507813 
                 12.16406 
               
               
                 3 
                 9 
                 0.416211 
                 10.27383 
                 0.416211 
                 10.27383 
               
               
                 3 
                 10 
                 0.349144 
                 27.57701 
                 0.416211 
                 10.27383 
               
               
                 3 
                 11 
                 0.303267 
                 39.71165 
                 0.416211 
                 10.27383 
               
               
                 3 
                 12 
                 0.275815 
                 47.12364 
                 0.416211 
                 10.27383 
               
               
                 3 
                 13 
                 0.264486 
                 50.18262 
                 0.416211 
                 10.27383 
               
               
                 3 
                 14 
                 0.267344 
                 49.41094 
                 0.416211 
                 10.27383 
               
               
                 3 
                 15 
                 0.282752 
                 45.2506 
                 0.416211 
                 10.27383 
               
               
                 4 
                 4 
                 1.180556 
                 45.2506 
                 0.416211 
                 10.27383 
               
               
                 4 
                 5 
                 0.978824 
                 108.2998 
                 0.416211 
                 10.27383 
               
               
                 4 
                 6 
                 0.813867 
                 78.33281 
                 0.416211 
                 10.27383 
               
               
                 4 
                 7 
                 0.680432 
                 51.03906 
                 0.416211 
                 10.27383 
               
               
                 4 
                 8 
                 0.574219 
                 27.70313 
                 0.416211 
                 10.27383 
               
               
                 4 
                 9 
                 0.491678 
                 8.29178 
                 0.491678 
                 8.29178 
               
               
                 4 
                 10 
                 0.42985 
                 6.754883 
                 0.42985 
                 6.754883 
               
               
                 4 
                 11 
                 0.38625 
                 18.00375 
                 0.42985 
                 6.754883 
               
               
                 4 
                 12 
                 0.358774 
                 25.09255 
                 0.42985 
                 6.754883 
               
               
                 4 
                 13 
                 0.345631 
                 28.48351 
                 0.42985 
                 6.754883 
               
               
                 4 
                 14 
                 0.345285 
                 28.57282 
                 0.42985 
                 6.754883 
               
               
                 4 
                 15 
                 0.356412 
                 25.70205 
                 0.42985 
                 6.754883 
               
               
                 5 
                 5 
                 0.965909 
                 106.0526 
                 0.42985 
                 6.754883 
               
               
                 5 
                 6 
                 0.82303 
                 80.09205 
                 0.42985 
                 6.754883 
               
               
                 5 
                 7 
                 0.705892 
                 56.38574 
                 0.42985 
                 6.754883 
               
               
                 5 
                 8 
                 0.611406 
                 35.95875 
                 0.42985 
                 6.754883 
               
               
                 5 
                 9 
                 0.536959 
                 19.15926 
                 0.42985 
                 6.754883 
               
               
                 5 
                 10 
                 0.480324 
                 5.56684 
                 0.480324 
                 5.56684 
               
               
                 5 
                 11 
                 0.439593 
                 4.24135 
                 0.439593 
                 4.24135 
               
               
                 5 
                 12 
                 0.41312 
                 11.07139 
                 0.439593 
                 4.24135 
               
               
                 5 
                 13 
                 0.399479 
                 14.59063 
                 0.439593 
                 4.24135 
               
               
                 5 
                 14 
                 0.397429 
                 15.11946 
                 0.439593 
                 4.24135 
               
               
                 5 
                 15 
                 0.405884 
                 12.93823 
                 0.439593 
                 4.24135 
               
               
                 6 
                 6 
                 0.817308 
                 78.99339 
                 0.439593 
                 4.24135 
               
               
                 6 
                 7 
                 0.713397 
                 57.96181 
                 0.439593 
                 4.24135 
               
               
                 6 
                 8 
                 0.628767 
                 39.81278 
                 0.439593 
                 4.24135 
               
               
                 6 
                 9 
                 0.561422 
                 24.86234 
                 0.439593 
                 4.24135 
               
               
                 6 
                 10 
                 0.509635 
                 12.60156 
                 0.439593 
                 4.24135 
               
               
                 6 
                 11 
                 0.4719 
                 3.545111 
                 0.4719 
                 3.545111 
               
               
                 6 
                 12 
                 0.446899 
                 2.455078 
                 0.446899 
                 2.455078 
               
               
                 6 
                 13 
                 0.433475 
                 5.819602 
                 0.446899 
                 2.455078 
               
               
                 6 
                 14 
                 0.430607 
                 6.559743 
                 0.446899 
                 2.455078 
               
               
                 6 
                 15 
                 0.437388 
                 4.810045 
                 0.446899 
                 2.455078 
               
               
                 7 
                 7 
                 0.708338 
                 56.89844 
                 0.446899 
                 2.455078 
               
               
                 7 
                 8 
                 0.632056 
                 40.5431 
                 0.446899 
                 2.455078 
               
               
                 7 
                 9 
                 0.570923 
                 26.97144 
                 0.446899 
                 2.455078 
               
               
                 7 
                 10 
                 0.523556 
                 15.94247 
                 0.446899 
                 2.455078 
               
               
                 7 
                 11 
                 0.488741 
                 7.586857 
                 0.446899 
                 2.455078 
               
               
                 7 
                 12 
                 0.465402 
                 1.985491 
                 0.465402 
                 1.985491 
               
               
                 7 
                 13 
                 0.452582 
                 1.091146 
                 0.452582 
                 1.091146 
               
               
                 7 
                 14 
                 0.44943 
                 1.847762 
                 0.452582 
                 1.091146 
               
               
                 7 
                 15 
                 0.455181 
                 0.467516 
                 0.455181 
                 0.467516 
               
               
                 8 
                 8 
                 0.625 
                 38.97656 
                 0.455181 
                 0.467516 
               
               
                 8 
                 9 
                 0.569308 
                 26.61295 
                 0.455181 
                 0.467516 
               
               
                 8 
                 10 
                 0.525933 
                 16.51302 
                 0.455181 
                 0.467516 
               
               
                 8 
                 11 
                 0.493877 
                 8.819468 
                 0.455181 
                 0.467516 
               
               
                 8 
                 12 
                 0.472245 
                 3.627878 
                 0.455181 
                 0.467516 
               
               
                 8 
                 13 
                 0.460236 
                 0.745793 
                 0.455181 
                 0.467516 
               
               
                 8 
                 14 
                 0.457129 
                 0 
                 0.455181 
                 0 
               
               
                 8 
                 15 
                 0.462271 
                 1.234184 
                 0.457129 
                 0 
               
               
                 9 
                 9 
                 0.559211 
                 24.3713 
                 0.457129 
                 0 
               
               
                 9 
                 10 
                 0.519431 
                 14.95252 
                 0.457129 
                 0 
               
               
                 9 
                 11 
                 0.489941 
                 7.876 
                 0.457129 
                 0 
               
               
                 9 
                 12 
                 0.469989 
                 3.086319 
                 0.457129 
                 0 
               
               
                 9 
                 13 
                 0.458891 
                 0.422991 
                 0.457129 
                 0 
               
               
                 9 
                 14 
                 0.456032 
                 0.263263 
                 0.457129 
                 0 
               
               
                 9 
                 15 
                 0.460849 
                 0.892756 
                 0.457129 
                 0 
               
               
                 10 
                 10 
                 0.505952 
                 11.71763 
                 0.457129 
                 0 
               
               
                 10 
                 11 
                 0.478834 
                 5.209121 
                 0.457129 
                 0 
               
               
                 10 
                 12 
                 0.460494 
                 0.807528 
                 0.457129 
                 0 
               
               
                 10 
                 13 
                 0.450347 
                 1.627604 
                 0.457129 
                 0 
               
               
                 10 
                 14 
                 0.44786 
                 2.224524 
                 0.457129 
                 0 
               
               
                 10 
                 15 
                 0.452543 
                 1.100565 
                 0.457129 
                 0 
               
               
                 11 
                 11 
                 0.461957 
                 1.168628 
                 0.457129 
                 0 
               
               
                 11 
                 12 
                 0.445146 
                 2.875831 
                 0.457129 
                 0 
               
               
                 11 
                 13 
                 0.435954 
                 5.180176 
                 0.457129 
                 0 
               
               
                 11 
                 14 
                 0.433913 
                 5.706792 
                 0.457129 
                 0 
               
               
                 11 
                 15 
                 0.438594 
                 4.499063 
                 0.457129 
                 0 
               
               
                 12 
                 12 
                 0.425 
                 8.00625 
                 0.457129 
                 0 
               
               
                 12 
                 13 
                 0.416743 
                 10.13649 
                 0.457129 
                 0 
               
               
                 12 
                 14 
                 0.415189 
                 10.53741 
                 0.457129 
                 0 
               
               
                 12 
                 15 
                 0.419959 
                 9.306899 
                 0.457129 
                 0 
               
               
                 13 
                 13 
                 0.393519 
                 16.12847 
                 0.457129 
                 0 
               
               
                 13 
                 14 
                 0.392472 
                 16.39858 
                 0.457129 
                 0 
               
               
                 13 
                 15 
                 0.397391 
                 15.12933 
                 0.457129 
                 0 
               
               
                 14 
                 14 
                 0.366379 
                 23.13039 
                 0.457129 
                 0 
               
               
                 14 
                 15 
                 0.371491 
                 21.81157 
                 0.457129 
                 0 
               
               
                 15 
                 15 
                 0.342742 
                 29.22883 
                 0.457129 
                 0 
               
               
                   
               
            
           
         
       
     
     In Table 2, the final value of α is calculated as:
 
α=0.457129.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   185 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   229 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     v 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   134 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     v 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   206 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     v 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             RGBsg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           RGBsg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1]= Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1] −Wtsg[ 1]=206−78=128
 
 Btsg[ 1 ]=Bsg[ 1 ]−Wtsg[ 1]=78−78=0
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     Embodiment 5 
     Embodiment 5 is a first example of means for approximately calculating a color-saturation conversion rate α in the second color-saturation conversion rate calculation section  33  of Embodiment 2, and makes such a change to Embodiment 4 as to perform a color-saturation conversion on every pixel within the input image. It should be noted that processing sections identical to those of Embodiments 2 and 3 are given the same reference numerals as those of Embodiments 2 and 3 and will not be described below in detail. 
       FIG. 17  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 5. The color-saturation conversion section  11  of  FIG. 17  includes a backlight upper-limit calculation section  21 , a fourth color-saturation conversion rate calculation section  63 , a gamma-curve normalization and piecewise-linear approximation section  61 , a color-saturation conversion rate limiter process section  34 , and a saturation-converted RGB signal calculation section  25 . It should be noted, in the present embodiment, that since the color-saturation conversion rate α may exceed 1, the color-saturation conversion rate limiter process section  34  is mounted. 
     The fourth color-saturation conversion rate calculation section  63  calculates a color-saturation conversion rate α on the basis of an input RGB signal, a backlight upper limit, and normalized-gamma-curve polygonal-line information inputted from the gamma-curve normalization and piecewise-linear approximation section  61 , and then outputs the color-saturation conversion rate α. The gamma-curve normalization and piecewise-linear approximation section  61  calculates normalized-gamma-curve polygonal-line information as explained in Embodiment 4. 
       FIG. 18  is a flow chart that explains the operation of the fourth color-saturation conversion rate calculation section  63 . 
     According to the flow chart, Steps S 71  to S 73 , S 64 , and S 65  are repeated from j=0 to j=Ndg−1, and further repeated from k=j to k Ndg−1. 
     Embodiment 4 repeats j from jMin to kMax, and further repeats k from j to kMax. Meanwhile, Embodiment 5 calculates α in such a direction as to cause an increase in color saturation; therefore, Formulae (131) and (132) do not always hold. That is, Embodiment 5 repeats a different range from Embodiment 4 because the ranges of j and k are defined by Formula (134), which is derived exclusively from Formula (128):
 
0 ≦j≦k≦Ndg− 1.  (134)
 
     First, in Step S 71 , the fourth color-saturation conversion rate calculation section  63  calculates an α candidate value (αt) according to Formula (13). However, when the denominator of Formula (13) is 0, the target pixel has a color saturation of 0. Therefore, αt may take on any value. In Embodiment 5, when the denominator of Formula (13) is 0, αt is set, for example, as αt=1. 
     Next, the fourth color-saturation conversion rate calculation section  63  calculates the maximum and minimum values (maxRGBst, minRGBst) of the input RGB signal after color-saturation conversion according to Formulae (17) and (18) (S 72 ):
 
max RGBst =max(α t ×max RGB +(1 −αt )× Y[i], 0)  (17)
 
min RGBst =min(α t ×max RGB +(1− αt )× Y[i] ,MAX).  (18)
 
Depending on the value of αt, the maximum value after the color-saturation conversion may exceeds the upper limit MAX of the signal, or the minimum value after the color-saturation conversion may fall short of the lower limit 0 of the signal. Therefore, Formulae (17) and (18) are subjected to a limiter process.
 
     Next, the fourth color-saturation conversion rate calculation section  63  calculates an α calculation judgment value (judge) according to Formula (19) (S 73 ): 
     
       
         
           
             
               
                 
                   judge 
                   = 
                   
                     
                        
                       
                         
                           fg 
                           ⁡ 
                           
                             ( 
                             
                               
                                 max 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 st 
                               
                               , 
                               γ 
                             
                             ) 
                           
                         
                         - 
                         
                           fg 
                           ⁡ 
                           
                             ( 
                             
                               
                                 min 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 st 
                               
                               , 
                               γ 
                             
                             ) 
                           
                         
                         - 
                         
                           MAX 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           w 
                         
                       
                        
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Next, the fourth color-saturation conversion rate calculation section  63  judges whether or not to update α. However, in cases where the fourth color-saturation conversion rate calculation section  63  judges for the first time whether or not to update α or in cases where judge&lt;judgeMin, the fourth color-saturation conversion rate calculation section  63  proceeds to Step S 65  (S 64 ). Next, the fourth color-saturation conversion rate calculation section  63  updates α and the minimum a calculation judgment value (judgeMin) according to Formulae (15) and (16) (S 65 ). Steps S 64  and S 65  are identical to Steps S 64  and S 65  of  FIG. 16  explained in Embodiment 4. 
     The following shows a first example of calculation of a backlight value according to Embodiment 5. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information (j, x[j], y[j], a[j], b[j]) is calculated in the same manner as in Embodiment 4 to take on values shown above in Table 1. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 5 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 5 calculates a color-saturation conversion rate α for every pixel. In Embodiment 5, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 207 as in Embodiment 1. 
     Since Ndg=16, the present example of calculation repeats Steps S 71  to S 73 , S 64 , an S 65  of the flow chart of  FIG. 18  from j=0 to j=15 and from k=j to k=15 for each value of j. The repetition of the steps yields such variations in value of j, k, αt, judge, α, and judgeMin as shown below in Tables 3(a) and 3(b). 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 3(a) 
               
               
                   
               
               
                 j 
                 k 
                 αt 
                 judge 
                 α 
                 judgeMin 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0 
                 0 
                 10.625 
                 127.5 
                 10.625 
                 127.5 
               
               
                 0 
                 1 
                 5.75651 
                 127.5 
                 10.625 
                 127.5 
               
               
                 0 
                 2 
                 3.405273 
                 127.5 
                 10.625 
                 127.5 
               
               
                 0 
                 3 
                 2.060938 
                 127.5 
                 10.625 
                 127.5 
               
               
                 0 
                 4 
                 1.220052 
                 123.6211 
                 1.220052 
                 123.6211 
               
               
                 0 
                 5 
                 0.666853 
                 48.1942 
                 0.666853 
                 48.1942 
               
               
                 0 
                 6 
                 0.293457 
                 42.36035 
                 0.293457 
                 42.36035 
               
               
                 0 
                 7 
                 0.039931 
                 115.3047 
                 0.293457 
                 42.36035 
               
               
                 0 
                 8 
                 −0.12969 
                 169.0359 
                 0.293457 
                 42.36035 
               
               
                 0 
                 9 
                 −0.23828 
                 205.8047 
                 0.293457 
                 42.36035 
               
               
                 0 
                 10 
                 −0.30111 
                 228.0449 
                 0.293457 
                 42.36035 
               
               
                 0 
                 11 
                 −0.32873 
                 237.8185 
                 0.293457 
                 42.36035 
               
               
                 0 
                 12 
                 −0.32868 
                 237.8036 
                 0.293457 
                 42.36035 
               
               
                 0 
                 13 
                 −0.30651 
                 229.9578 
                 0.293457 
                 42.36035 
               
               
                 0 
                 14 
                 −0.26636 
                 215.7437 
                 0.293457 
                 42.36035 
               
               
                 0 
                 15 
                 −0.2114 
                 196.4903 
                 0.293457 
                 42.36035 
               
               
                 1 
                 1 
                 3.541667 
                 127.5 
                 0.293457 
                 42.36035 
               
               
                 1 
                 2 
                 2.514509 
                 127.5 
                 0.293457 
                 42.36035 
               
               
                 1 
                 3 
                 1.785645 
                 127.5 
                 0.293457 
                 42.36035 
               
               
                 1 
                 4 
                 1.255642 
                 124.582 
                 0.293457 
                 42.36035 
               
               
                 1 
                 5 
                 0.864844 
                 88.12031 
                 0.293457 
                 42.36035 
               
               
                 1 
                 6 
                 0.575284 
                 27.93963 
                 0.575284 
                 27.93963 
               
               
                 1 
                 7 
                 0.361654 
                 24.34961 
                 0.361654 
                 24.34961 
               
               
                 1 
                 8 
                 0.20643 
                 66.10276 
                 0.361654 
                 24.34961 
               
               
                 1 
                 9 
                 0.097098 
                 97.81138 
                 0.361654 
                 24.34961 
               
               
                 1 
                 10 
                 0.024479 
                 120.0328 
                 0.361654 
                 24.34961 
               
               
                 1 
                 11 
                 −0.01831 
                 133.2993 
                 0.361654 
                 24.34961 
               
               
                 1 
                 12 
                 −0.03654 
                 139.0947 
                 0.361654 
                 24.34961 
               
               
                 1 
                 13 
                 −0.03429 
                 138.3802 
                 0.361654 
                 24.34961 
               
               
                 1 
                 14 
                 −0.0148 
                 132.1838 
                 0.361654 
                 24.34961 
               
               
                 1 
                 15 
                 0.019336 
                 121.6066 
                 0.361654 
                 24.34961 
               
               
                 2 
                 2 
                 2.125 
                 127.5 
                 0.361654 
                 24.34961 
               
               
                 2 
                 3 
                 1.630327 
                 127.5 
                 0.361654 
                 24.34961 
               
               
                 2 
                 4 
                 1.245768 
                 124.3154 
                 0.361654 
                 24.34961 
               
               
                 2 
                 5 
                 0.945913 
                 102.5733 
                 0.361654 
                 24.34961 
               
               
                 2 
                 6 
                 0.712612 
                 57.79688 
                 0.361654 
                 24.34961 
               
               
                 2 
                 7 
                 0.532552 
                 18.10156 
                 0.532552 
                 18.10156 
               
               
                 2 
                 8 
                 0.395752 
                 15.55225 
                 0.395752 
                 15.55225 
               
               
                 2 
                 9 
                 0.294577 
                 42.0579 
                 0.395752 
                 15.55225 
               
               
                 2 
                 10 
                 0.22309 
                 61.35938 
                 0.395752 
                 15.55225 
               
               
                 2 
                 11 
                 0.176604 
                 74.69285 
                 0.395752 
                 15.55225 
               
               
                 2 
                 12 
                 0.151367 
                 81.96094 
                 0.395752 
                 15.55225 
               
               
                 2 
                 13 
                 0.144345 
                 83.98326 
                 0.395752 
                 15.55225 
               
               
                 2 
                 14 
                 0.153054 
                 81.47514 
                 0.395752 
                 15.55225 
               
               
                 2 
                 15 
                 0.175442 
                 75.02751 
                 0.395752 
                 15.55225 
               
               
                 3 
                 3 
                 1.517857 
                 127.5 
                 0.395752 
                 15.55225 
               
               
                 3 
                 4 
                 1.217708 
                 123.5678 
                 0.395752 
                 15.55225 
               
               
                 3 
                 5 
                 0.97583 
                 107.7788 
                 0.395752 
                 15.55225 
               
               
                 3 
                 6 
                 0.781939 
                 72.20267 
                 0.395752 
                 15.55225 
               
               
                 3 
                 7 
                 0.628038 
                 39.65104 
                 0.395752 
                 15.55225 
               
               
                 3 
                 8 
                 0.507813 
                 12.16406 
                 0.507813 
                 12.16406 
               
               
                 3 
                 9 
                 0.416211 
                 10.27383 
                 0.416211 
                 10.27383 
               
               
                 3 
                 10 
                 0.349144 
                 27.57701 
                 0.416211 
                 10.27383 
               
               
                 3 
                 11 
                 0.303267 
                 39.71165 
                 0.416211 
                 10.27383 
               
               
                 3 
                 12 
                 0.275815 
                 47.12364 
                 0.416211 
                 10.27383 
               
               
                 3 
                 13 
                 0.264486 
                 50.18262 
                 0.416211 
                 10.27383 
               
               
                 3 
                 14 
                 0.267344 
                 49.41094 
                 0.416211 
                 10.27383 
               
               
                 3 
                 15 
                 0.282752 
                 45.2506 
                 0.416211 
                 10.27383 
               
               
                 4 
                 4 
                 1.180556 
                 121.9141 
                 0.416211 
                 10.27383 
               
               
                 4 
                 5 
                 0.978824 
                 108.2998 
                 0.416211 
                 10.27383 
               
               
                 4 
                 6 
                 0.813867 
                 78.33281 
                 0.416211 
                 10.27383 
               
               
                 4 
                 7 
                 0.680432 
                 51.03906 
                 0.416211 
                 10.27383 
               
               
                 4 
                 8 
                 0.574219 
                 27.70313 
                 0.416211 
                 10.27383 
               
               
                 4 
                 9 
                 0.491678 
                 8.29178 
                 0.491678 
                 8.29178 
               
               
                 4 
                 10 
                 0.42985 
                 6.754883 
                 0.42985 
                 6.754883 
               
               
                 4 
                 11 
                 0.38625 
                 18.00375 
                 0.42985 
                 6.754883 
               
               
                 4 
                 12 
                 0.358774 
                 25.09255 
                 0.42985 
                 6.754883 
               
               
                 4 
                 13 
                 0.345631 
                 28.48351 
                 0.42985 
                 6.754883 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 3(b) 
               
               
                   
               
               
                 j 
                 k 
                 αt 
                 judge 
                 α 
                 judgeMin 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 4 
                 14 
                 0.345285 
                 28.57282 
                 0.42985 
                 6.754883 
               
               
                 4 
                 15 
                 0.356412 
                 25.70205 
                 0.42985 
                 6.754883 
               
               
                 5 
                 5 
                 0.965909 
                 106.0526 
                 0.42985 
                 6.754883 
               
               
                 5 
                 6 
                 0.82303 
                 80.09205 
                 0.42985 
                 6.754883 
               
               
                 5 
                 7 
                 0.705892 
                 56.38574 
                 0.42985 
                 6.754883 
               
               
                 5 
                 8 
                 0.611406 
                 35.95875 
                 0.42985 
                 6.754883 
               
               
                 5 
                 9 
                 0.536959 
                 19.15926 
                 0.42985 
                 6.754883 
               
               
                 5 
                 10 
                 0.480324 
                 5.56684 
                 0.480324 
                 5.56684 
               
               
                 5 
                 11 
                 0.439593 
                 4.24135 
                 0.439593 
                 4.24135 
               
               
                 5 
                 12 
                 0.41312 
                 11.07139 
                 0.439593 
                 4.24135 
               
               
                 5 
                 13 
                 0.399479 
                 14.59063 
                 0.439593 
                 4.24135 
               
               
                 5 
                 14 
                 0.397429 
                 15.11946 
                 0.439593 
                 4.24135 
               
               
                 5 
                 15 
                 0.405884 
                 12.93823 
                 0.439593 
                 4.24135 
               
               
                 6 
                 6 
                 0.817308 
                 78.99339 
                 0.439593 
                 4.24135 
               
               
                 6 
                 7 
                 0.713397 
                 57.96181 
                 0.439593 
                 4.24135 
               
               
                 6 
                 8 
                 0.628767 
                 39.81278 
                 0.439593 
                 4.24135 
               
               
                 6 
                 9 
                 0.561422 
                 24.86234 
                 0.439593 
                 4.24135 
               
               
                 6 
                 10 
                 0.509635 
                 12.60156 
                 0.439593 
                 4.24135 
               
               
                 6 
                 11 
                 0.4719 
                 3.545111 
                 0.4719 
                 3.545111 
               
               
                 6 
                 12 
                 0.446899 
                 2.455078 
                 0.446899 
                 2.455078 
               
               
                 6 
                 13 
                 0.433475 
                 5.819602 
                 0.446899 
                 2.455078 
               
               
                 6 
                 14 
                 0.430607 
                 6.559743 
                 0.446899 
                 2.455078 
               
               
                 6 
                 15 
                 0.437388 
                 4.810045 
                 0.446899 
                 2.455078 
               
               
                 7 
                 7 
                 0.708333 
                 56.89844 
                 0.446899 
                 2.455078 
               
               
                 7 
                 8 
                 0.632056 
                 40.5431 
                 0.446899 
                 2.455078 
               
               
                 7 
                 9 
                 0.570923 
                 26.97144 
                 0.446899 
                 2.455078 
               
               
                 7 
                 10 
                 0.523556 
                 15.94247 
                 0.446899 
                 2.455078 
               
               
                 7 
                 11 
                 0.488741 
                 7.586857 
                 0.446899 
                 2.455078 
               
               
                 7 
                 12 
                 0.465402 
                 1.985491 
                 0.465402 
                 1.985491 
               
               
                 7 
                 13 
                 0.452582 
                 1.091146 
                 0.452582 
                 1.091146 
               
               
                 7 
                 14 
                 0.44943 
                 1.847762 
                 0.452582 
                 1.091146 
               
               
                 7 
                 15 
                 0.455181 
                 0.467516 
                 0.455181 
                 0.467516 
               
               
                 8 
                 8 
                 0.625 
                 38.97656 
                 0.455181 
                 0.467516 
               
               
                 8 
                 9 
                 0.569308 
                 26.61295 
                 0.455181 
                 0.467516 
               
               
                 8 
                 10 
                 0.525933 
                 16.51302 
                 0.455181 
                 0.467516 
               
               
                 8 
                 11 
                 0.493877 
                 8.819468 
                 0.455181 
                 0.467516 
               
               
                 8 
                 12 
                 0.472245 
                 3.627878 
                 0.455181 
                 0.467516 
               
               
                 8 
                 13 
                 0.460236 
                 0.745793 
                 0.455181 
                 0.467516 
               
               
                 8 
                 14 
                 0.457129 
                 0 
                 0.457129 
                 0 
               
               
                 8 
                 15 
                 0.462271 
                 1.234184 
                 0.457129 
                 0 
               
               
                 9 
                 9 
                 0.559211 
                 24.3713 
                 0.457129 
                 0 
               
               
                 9 
                 10 
                 0.519431 
                 14.95252 
                 0.457129 
                 0 
               
               
                 9 
                 11 
                 0.489941 
                 7.875 
                 0.457129 
                 0 
               
               
                 9 
                 12 
                 0.469989 
                 3.086319 
                 0.457129 
                 0 
               
               
                 9 
                 13 
                 0.458891 
                 0.422991 
                 0.457129 
                 0 
               
               
                 9 
                 14 
                 0.456032 
                 0.263263 
                 0.457129 
                 0 
               
               
                 9 
                 15 
                 0.460849 
                 0.892756 
                 0.457129 
                 0 
               
               
                 10 
                 10 
                 0.505952 
                 11.71763 
                 0.457129 
                 0 
               
               
                 10 
                 11 
                 0.478834 
                 5.209121 
                 0.457129 
                 0 
               
               
                 10 
                 12 
                 0.460494 
                 0.807528 
                 0.457129 
                 0 
               
               
                 10 
                 13 
                 0.450347 
                 1.627604 
                 0.457129 
                 0 
               
               
                 10 
                 14 
                 0.44786 
                 2.224524 
                 0.457129 
                 0 
               
               
                 10 
                 15 
                 0.452543 
                 1.100565 
                 0.457129 
                 0 
               
               
                 11 
                 11 
                 0.461957 
                 1.158628 
                 0.457129 
                 0 
               
               
                 11 
                 12 
                 0.445146 
                 2.875831 
                 0.457129 
                 0 
               
               
                 11 
                 13 
                 0.435954 
                 5.180176 
                 0.457129 
                 0 
               
               
                 11 
                 14 
                 0.433913 
                 5.706792 
                 0.457129 
                 0 
               
               
                 11 
                 15 
                 0.438594 
                 4.499063 
                 0.457129 
                 0 
               
               
                 12 
                 12 
                 0.425 
                 8.00625 
                 0.457129 
                 0 
               
               
                 12 
                 13 
                 0.416743 
                 10.13649 
                 0.457129 
                 0 
               
               
                 12 
                 14 
                 0.415189 
                 10.53741 
                 0.457129 
                 0 
               
               
                 12 
                 15 
                 0.419959 
                 9.306899 
                 0.457129 
                 0 
               
               
                 13 
                 13 
                 0.393519 
                 16.12847 
                 0.457129 
                 0 
               
               
                 13 
                 14 
                 0.392472 
                 16.39858 
                 0.457129 
                 0 
               
               
                 13 
                 15 
                 0.397391 
                 15.12933 
                 0.457129 
                 0 
               
               
                 14 
                 14 
                 0.366379 
                 23.13039 
                 0.457129 
                 0 
               
               
                 14 
                 15 
                 0.371491 
                 21.81157 
                 0.457129 
                 0 
               
               
                 15 
                 15 
                 0.342742 
                 29.22883 
                 0.457129 
                 0 
               
               
                   
               
            
           
         
       
     
     In Tables 3(a) and 3(b), the final value of α is calculated as:
 
α=0.457129.
 
     Since the color-saturation conversion rate α thus calculated falls within a range of 0≦α≦1, α is set as α=0.457129 even after the limiter process. 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.459129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   185 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.459129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   229 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.457129 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457129 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             185 
                             / 
                             255 
                           
                           ) 
                         
                         2 
                       
                       × 
                       255 
                     
                     = 
                     134 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   206 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             RGBsg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           RGBsg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1]= Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1]− Wtsg[ 1]=206−78=128
 
 Btsg[ 1]= Bsg[ 1] −Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       max 
                       ⁡ 
                       
                         ( 
                         
                           56 
                           , 
                           128 
                           , 
                           0 
                           , 
                           56 
                         
                         ) 
                       
                     
                     = 
                     128. 
                   
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The first example of calculation thus explained is an example where the fourth color-saturation conversion rate calculation section  63  calculates a color-saturation conversion rate α as 0≦α≦1. The following shows an example (second example of calculation) where the fourth color-saturation conversion rate calculation section  63  does not calculate a color-saturation conversion rate α as 0≦α≦1. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). That is, the following shows an example where the same pixel values are used as in the second example of calculation of a backlight value according to Embodiment 2. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the second example of calculation explained in Embodiment 2. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information j, x[j], y[j], a[j], b[j]) is calculated in the same manner as in Embodiment 4 to take on values shown above in Table 1. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 5 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 5 calculates a color-saturation conversion rate α for every pixel. In Embodiment 5, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 167.25 as in the second example of calculation of Embodiment 2. 
     Since Ndg=16, the present example of calculation repeats Steps S 71  to S 73 , S 64 , and S 65  of the flow chart of  FIG. 18  from j=0 to j=15 and from k=j to k=15 for each value of j. The repetition of the steps yields such variations in value of j, k, at, judge, α, and judgeMin as shown below in Tables 4(a) and 4(b). 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 4(a) 
               
               
                   
               
               
                 j 
                 k 
                 αt 
                 judge 
                 α 
                 judgeMin 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0 
                 0 
                 20 
                 127.5 
                 20 
                 127.5 
               
               
                 0 
                 1 
                 12.27827 
                 127.5 
                 20 
                 127.5 
               
               
                 0 
                 2 
                 8.102901 
                 127.5 
                 20 
                 127.5 
               
               
                 0 
                 3 
                 5.568027 
                 127.5 
                 20 
                 127.5 
               
               
                 0 
                 4 
                 3.925962 
                 107.7137 
                 3.925962 
                 107.7137 
               
               
                 0 
                 5 
                 2.823456 
                 67.5458 
                 2.823456 
                 67.5458 
               
               
                 0 
                 6 
                 2.071903 
                 42.49794 
                 2.071903 
                 42.49794 
               
               
                 0 
                 7 
                 1.561427 
                 20.46663 
                 1.561427 
                 20.46663 
               
               
                 0 
                 8 
                 1.223684 
                 1.342105 
                 1.223684 
                 1.342105 
               
               
                 0 
                 9 
                 1.013934 
                 17.54301 
                 1.223684 
                 1.342105 
               
               
                 0 
                 10 
                 0.90166 
                 27.07932 
                 1.223684 
                 1.342105 
               
               
                 0 
                 11 
                 0.865331 
                 30.22147 
                 1.223684 
                 1.342105 
               
               
                 0 
                 12 
                 0.889323 
                 28.1272 
                 1.223684 
                 1.342105 
               
               
                 0 
                 13 
                 0.962023 
                 21.95226 
                 1.223684 
                 1.342105 
               
               
                 0 
                 14 
                 1.074618 
                 12.49178 
                 1.223684 
                 1.342105 
               
               
                 0 
                 15 
                 1.220302 
                 1.602929 
                 1.223684 
                 1.342105 
               
               
                 1 
                 1 
                 6.666667 
                 127.5 
                 1.223684 
                 1.342105 
               
               
                 1 
                 2 
                 5.120639 
                 127.5 
                 1.223684 
                 1.342105 
               
               
                 1 
                 3 
                 3.975 
                 109.5902 
                 1.223684 
                 1.342105 
               
               
                 1 
                 4 
                 3.117491 
                 78.05738 
                 1.223684 
                 1.342105 
               
               
                 1 
                 5 
                 2.474677 
                 55.92165 
                 1.223684 
                 1.342105 
               
               
                 1 
                 6 
                 1.996028 
                 39.84138 
                 1.223684 
                 1.342105 
               
               
                 1 
                 7 
                 1.645718 
                 25.23438 
                 1.223684 
                 1.342105 
               
               
                 1 
                 8 
                 1.397639 
                 10.64221 
                 1.223684 
                 1.342105 
               
               
                 1 
                 9 
                 1.232315 
                 0.676447 
                 1.232315 
                 0.676447 
               
               
                 1 
                 10 
                 1.134921 
                 7.989831 
                 1.232315 
                 0.676447 
               
               
                 1 
                 11 
                 1.093973 
                 11.04684 
                 1.232315 
                 0.676447 
               
               
                 1 
                 12 
                 1.100439 
                 10.56411 
                 1.232315 
                 0.676447 
               
               
                 1 
                 13 
                 1.147115 
                 7.079417 
                 1.232315 
                 0.676447 
               
               
                 1 
                 14 
                 1.228188 
                 0.994757 
                 1.232315 
                 0.676447 
               
               
                 1 
                 15 
                 1.338912 
                 6.716617 
                 1.232315 
                 0.676447 
               
               
                 2 
                 2 
                 4 
                 110.5469 
                 1.232315 
                 0.676447 
               
               
                 2 
                 3 
                 3.277753 
                 83.79423 
                 1.232315 
                 0.676447 
               
               
                 2 
                 4 
                 2.706494 
                 63.64769 
                 1.232315 
                 0.676447 
               
               
                 2 
                 5 
                 2.257757 
                 48.69211 
                 1.232315 
                 0.676447 
               
               
                 2 
                 6 
                 1.909805 
                 36.73737 
                 1.232315 
                 0.676447 
               
               
                 2 
                 7 
                 1.64576 
                 25.23672 
                 1.232315 
                 0.676447 
               
               
                 2 
                 8 
                 1.452309 
                 14.29468 
                 1.232315 
                 0.676447 
               
               
                 2 
                 9 
                 1.318818 
                 5.37346 
                 1.232315 
                 0.676447 
               
               
                 2 
                 10 
                 1.236683 
                 0.339588 
                 1.236683 
                 0.339588 
               
               
                 2 
                 11 
                 1.198873 
                 3.215351 
                 1.236683 
                 0.339588 
               
               
                 2 
                 12 
                 1.199586 
                 3.162185 
                 1.236683 
                 0.339588 
               
               
                 2 
                 13 
                 1.233986 
                 0.547562 
                 1.236683 
                 0.339588 
               
               
                 2 
                 14 
                 1.298018 
                 3.98316 
                 1.236683 
                 0.339588 
               
               
                 2 
                 15 
                 1.388251 
                 10.01466 
                 1.236683 
                 0.339588 
               
               
                 3 
                 3 
                 2.857143 
                 68.73772 
                 1.236683 
                 0.339588 
               
               
                 3 
                 4 
                 2.432648 
                 54.5209 
                 1.236683 
                 0.339588 
               
               
                 3 
                 5 
                 2.090637 
                 43.12232 
                 1.236683 
                 0.339588 
               
               
                 3 
                 6 
                 1.819369 
                 33.273 
                 1.236683 
                 0.339588 
               
               
                 3 
                 7 
                 1.609232 
                 23.1706 
                 1.236683 
                 0.339588 
               
               
                 3 
                 8 
                 1.452276 
                 14.29283 
                 1.236683 
                 0.339588 
               
               
                 3 
                 9 
                 1.341877 
                 6.914839 
                 1.236683 
                 0.339588 
               
               
                 3 
                 10 
                 1.272463 
                 2.274976 
                 1.236683 
                 0.339588 
               
               
                 3 
                 11 
                 1.23932 
                 0.136165 
                 1.23932 
                 0.136165 
               
               
                 3 
                 12 
                 1.238429 
                 0.2049 
                 1.23932 
                 0.136165 
               
               
                 3 
                 13 
                 1.266344 
                 1.865902 
                 1.23932 
                 0.136165 
               
               
                 3 
                 14 
                 1.320091 
                 5.458611 
                 1.23932 
                 0.136165 
               
               
                 3 
                 15 
                 1.397096 
                 10.60538 
                 1.23932 
                 0.136165 
               
               
                 4 
                 4 
                 2.222222 
                 47.60781 
                 1.23932 
                 0.136165 
               
               
                 4 
                 5 
                 1.94765 
                 38.09978 
                 1.23932 
                 0.136165 
               
               
                 4 
                 6 
                 1.726805 
                 28.98903 
                 1.23932 
                 0.136165 
               
               
                 4 
                 7 
                 1.553546 
                 20.02086 
                 1.23932 
                 0.136165 
               
               
                 4 
                 8 
                 1.42263 
                 12.31268 
                 1.23932 
                 0.136165 
               
               
                 4 
                 9 
                 1.329561 
                 6.091575 
                 1.23932 
                 0.136165 
               
               
                 4 
                 10 
                 1.270455 
                 2.140696 
                 1.23932 
                 0.136165 
               
               
                 4 
                 11 
                 1.241942 
                 0.066045 
                 1.241942 
                 0.066045 
               
               
                 4 
                 12 
                 1.241086 
                 0 
                 1.241086 
                 0 
               
               
                 4 
                 13 
                 1.265312 
                 1.796958 
                 1.241086 
                 0 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 4(b) 
               
               
                   
               
               
                 j 
                 k 
                 αt 
                 judge 
                 α 
                 judgeMin 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 4 
                 14 
                 1.312357 
                 4.941626 
                 1.241086 
                 0 
               
               
                 4 
                 15 
                 1.380222 
                 9.477952 
                 1.241086 
                 0 
               
               
                 5 
                 5 
                 1.818182 
                 33.21804 
                 1.241086 
                 0 
               
               
                 5 
                 6 
                 1.633018 
                 24.516 
                 1.241086 
                 0 
               
               
                 5 
                 7 
                 1.486574 
                 16.23276 
                 1.241086 
                 0 
               
               
                 5 
                 8 
                 1.375151 
                 9.13901 
                 1.241086 
                 0 
               
               
                 5 
                 9 
                 1.295508 
                 3.81535 
                 1.241086 
                 0 
               
               
                 5 
                 10 
                 1.24479 
                 0.285655 
                 1.241086 
                 0 
               
               
                 5 
                 11 
                 1.220475 
                 1.589645 
                 1.241086 
                 0 
               
               
                 5 
                 12 
                 1.220325 
                 1.601157 
                 1.241086 
                 0 
               
               
                 5 
                 13 
                 1.24235 
                 0.097531 
                 1.241086 
                 0 
               
               
                 5 
                 14 
                 1.284772 
                 3.0977 
                 1.241086 
                 0 
               
               
                 5 
                 15 
                 1.345995 
                 7.190129 
                 1.241086 
                 0 
               
               
                 6 
                 6 
                 1.538462 
                 19.16767 
                 1.241086 
                 0 
               
               
                 6 
                 7 
                 1.412395 
                 11.62851 
                 1.241086 
                 0 
               
               
                 6 
                 8 
                 1.316103 
                 5.192023 
                 1.241086 
                 0 
               
               
                 6 
                 9 
                 1.247144 
                 0.467257 
                 1.241086 
                 0 
               
               
                 6 
                 10 
                 1.203336 
                 2.882228 
                 1.241086 
                 0 
               
               
                 6 
                 11 
                 1.182721 
                 4.421257 
                 1.241086 
                 0 
               
               
                 6 
                 12 
                 1.183541 
                 4.359997 
                 1.241086 
                 0 
               
               
                 6 
                 13 
                 1.204212 
                 2.816785 
                 1.241086 
                 0 
               
               
                 6 
                 14 
                 1.243301 
                 0.170828 
                 1.241086 
                 0 
               
               
                 6 
                 15 
                 1.299509 
                 4.082829 
                 1.241086 
                 0 
               
               
                 7 
                 7 
                 1.333333 
                 6.34375 
                 1.241086 
                 0 
               
               
                 7 
                 8 
                 1.249124 
                 0.619933 
                 1.241086 
                 0 
               
               
                 7 
                 9 
                 1.18886 
                 3.962955 
                 1.241086 
                 0 
               
               
                 7 
                 10 
                 1.150819 
                 6.802925 
                 1.241086 
                 0 
               
               
                 7 
                 11 
                 1.133442 
                 8.100225 
                 1.241086 
                 0 
               
               
                 7 
                 12 
                 1.135311 
                 7.960677 
                 1.241086 
                 0 
               
               
                 7 
                 13 
                 1.155136 
                 6.480641 
                 1.241086 
                 0 
               
               
                 7 
                 14 
                 1.191738 
                 3.748047 
                 1.241086 
                 0 
               
               
                 7 
                 15 
                 1.244041 
                 0.227925 
                 1.241086 
                 0 
               
               
                 8 
                 8 
                 1.176471 
                 4.887868 
                 1.241086 
                 0 
               
               
                 8 
                 9 
                 1.123414 
                 8.848862 
                 1.241086 
                 0 
               
               
                 8 
                 10 
                 1.090251 
                 11.3247 
                 1.241086 
                 0 
               
               
                 8 
                 11 
                 1.075709 
                 12.41039 
                 1.241086 
                 0 
               
               
                 8 
                 12 
                 1.078621 
                 12.19299 
                 1.241086 
                 0 
               
               
                 8 
                 13 
                 1.097916 
                 10.75245 
                 1.241086 
                 0 
               
               
                 8 
                 14 
                 1.13261 
                 8.162309 
                 1.241086 
                 0 
               
               
                 8 
                 15 
                 1.181796 
                 4.490324 
                 1.241086 
                 0 
               
               
                 9 
                 9 
                 1.052632 
                 14.25617 
                 1.241086 
                 0 
               
               
                 9 
                 10 
                 1.023641 
                 16.71856 
                 1.241086 
                 0 
               
               
                 9 
                 11 
                 1.011589 
                 17.74222 
                 1.241086 
                 0 
               
               
                 9 
                 12 
                 1.0155 
                 17.41006 
                 1.241086 
                 0 
               
               
                 9 
                 13 
                 1.03447 
                 15.79873 
                 1.241086 
                 0 
               
               
                 9 
                 14 
                 1.067666 
                 13.01084 
                 1.241086 
                 0 
               
               
                 9 
                 15 
                 1.11431 
                 9.528518 
                 1.241086 
                 0 
               
               
                 10 
                 10 
                 0.952381 
                 22.77121 
                 1.241086 
                 0 
               
               
                 10 
                 11 
                 0.942526 
                 23.60808 
                 1.241086 
                 0 
               
               
                 10 
                 12 
                 0.947377 
                 23.1962 
                 1.241086 
                 0 
               
               
                 10 
                 13 
                 0.966158 
                 21.60102 
                 1.241086 
                 0 
               
               
                 10 
                 14 
                 0.998152 
                 18.88351 
                 1.241086 
                 0 
               
               
                 10 
                 15 
                 1.042691 
                 15.10052 
                 1.241086 
                 0 
               
               
                 11 
                 11 
                 0.869565 
                 29.81827 
                 1.241086 
                 0 
               
               
                 11 
                 12 
                 0.875288 
                 29.31927 
                 1.241086 
                 0 
               
               
                 11 
                 13 
                 0.893969 
                 27.73256 
                 1.241086 
                 0 
               
               
                 11 
                 14 
                 0.924965 
                 25.09817 
                 1.241086 
                 0 
               
               
                 11 
                 15 
                 0.967752 
                 21.46566 
                 1.241086 
                 0 
               
               
                 12 
                 12 
                 0.8 
                 36.44219 
                 1.241086 
                 0 
               
               
                 12 
                 13 
                 0.818643 
                 34.66706 
                 1.241086 
                 0 
               
               
                 12 
                 14 
                 0.848049 
                 31.79081 
                 1.241086 
                 0 
               
               
                 12 
                 15 
                 0.890107 
                 28.06062 
                 1.241086 
                 0 
               
               
                 13 
                 13 
                 0.740741 
                 42.08478 
                 1.241086 
                 0 
               
               
                 13 
                 14 
                 0.77027 
                 39.27302 
                 1.241086 
                 0 
               
               
                 13 
                 15 
                 0.810228 
                 35.46832 
                 1.241086 
                 0 
               
               
                 14 
                 14 
                 0.689655 
                 46.94908 
                 1.241086 
                 0 
               
               
                 14 
                 15 
                 0.728482 
                 43.25204 
                 1.241086 
                 0 
               
               
                 15 
                 15 
                 0.645161 
                 51.61361 
                 1.241086 
                 0 
               
               
                   
               
            
           
         
       
     
     In Tables 4(a) and 4(b), the final value of α is calculated as:
 
α=1.241086.
 
In this example, α exceeds 1. Therefore, a limiter process is performed so that α=1.
 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Embodiment 6 
     Embodiment 6 is a second example of means for approximately calculating a color-saturation conversion rate α in the first color-saturation conversion rate calculation section  23  of Embodiment 1, and makes such a change to Embodiment 4 as to speed up the color-saturation converting means. It should be noted that processing sections identical to those of Embodiments 1 to 3 are given the same reference numerals as those of Embodiments 1 to 3 and will not be described below in detail. 
     In Embodiment 4, all the α candidate values are calculated, and an α candidate value from which the smallest α calculation judgment value is calculated is selected as α from among the α candidate values. Meanwhile, Embodiment 6 performs a loop process, finishes the loop process at a point of time where the α calculation judgment value is not more than a threshold, and outputs a obtained at the point of time, thereby achieving speeding up. 
       FIG. 19  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 6. The color-saturation conversion section  11  of  FIG. 19  is obtained by replacing the third color-saturation conversion rate calculation section  53  of the color-saturation conversion section  11  (see  FIG. 14 ) of Embodiment 4 with a fifth color-saturation conversion rate calculation section  73 . 
     The fifth color-saturation conversion rate calculation section  73  calculates a color-saturation conversion rate α from an input RGB signal, a backlight upper limit, the maximum and minimum values of a gamma-corrected KGB signal, and normalized-gamma-curve polygonal-line information according to a different method from the third color-saturation conversion rate calculation section  53 , and then outputs the color-saturation conversion rate α. 
       FIG. 20  is a flow chart that explains the operation of the fifth color-saturation conversion rate calculation section  73 . 
     According to the flow chart, Steps S 81  to S 84  are repeated from j=jMin to j=kMax, and further repeated from k=kMax to k=j or repeated from k=j to k=kMax. 
     First, in Step S 81 , the fifth color-saturation conversion rate calculation section  73  calculates the current value of α according to Formula (20): 
     
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   MAX 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   w 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         ⁡ 
                                         
                                           [ 
                                           k 
                                           ] 
                                         
                                       
                                       - 
                                       
                                         a 
                                         ⁡ 
                                         
                                           [ 
                                           j 
                                           ] 
                                         
                                       
                                     
                                     ) 
                                   
                                   × 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   Y 
                                   ⁡ 
                                   
                                     [ 
                                     i 
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         b 
                                         ⁡ 
                                         
                                           [ 
                                           k 
                                           ] 
                                         
                                       
                                       - 
                                       
                                         b 
                                         ⁡ 
                                         
                                           [ 
                                           j 
                                           ] 
                                         
                                       
                                     
                                     ) 
                                   
                                   × 
                                   MAX 
                                 
                               
                             
                           
                         
                         } 
                       
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     ⁡ 
                                     
                                       [ 
                                       k 
                                       ] 
                                     
                                   
                                   × 
                                   
                                     ( 
                                     
                                       
                                         max 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         G 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                       
                                       - 
                                       
                                         Y 
                                         ⁡ 
                                         
                                           [ 
                                           i 
                                           ] 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   a 
                                   ⁡ 
                                   
                                     [ 
                                     j 
                                     ] 
                                   
                                 
                                 × 
                                 
                                   ( 
                                   
                                     
                                       min 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       B 
                                     
                                     - 
                                     
                                       Y 
                                       ⁡ 
                                       
                                         [ 
                                         i 
                                         ] 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Next, the fifth color-saturation conversion rate calculation section  73  judges whether or not a falls within a range of 0≦α&lt;1 (S 82 ). If a falls within the range, the fifth color-saturation conversion rate calculation section  73  proceeds to Step S 83 . If α is out of the range, the fifth color-saturation conversion rate calculation section  73  skips Steps S 83  and S 84 . 
     In cases where the color-saturation conversion rate α falls within a range of 0≦α&lt;1, the fifth color-saturation conversion rate calculation section  73  calculates an α calculation judgment value (judge) according to Formula (21) (S 83 ): 
     
       
         
           
             
               
                 
                   judge 
                   = 
                   
                     
                        
                       
                         
                           
                             
                               
                                 fg 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                             α 
                                             × 
                                             max 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             R 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             G 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             B 
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 1 
                                                 - 
                                                 α 
                                               
                                               ) 
                                             
                                             × 
                                             
                                               Y 
                                               ⁡ 
                                               
                                                 [ 
                                                 i 
                                                 ] 
                                               
                                             
                                           
                                           , 
                                           γ 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 fg 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                             α 
                                             × 
                                             min 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             R 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             G 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             B 
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 1 
                                                 - 
                                                 α 
                                               
                                               ) 
                                             
                                             × 
                                             
                                               Y 
                                               ⁡ 
                                               
                                                 [ 
                                                 i 
                                                 ] 
                                               
                                             
                                           
                                           , 
                                           γ 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 MAX 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 w 
                               
                             
                           
                         
                       
                        
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Next, if the α calculation judgment value judge is not more than a predetermined threshold (judgeTol) for use in calculation judgment of α, the fifth color-saturation conversion rate calculation section  73  goes out of the loop to finish the process. If not, the fifth color-saturation conversion rate calculation section  73  proceeds with the loop process (S 84 ). In cases where the fifth color-saturation conversion rate calculation section  73  finishes the loop process without satisfying the judgment condition of S 84 , the fifth color-saturation conversion rate calculation section  73  cannot calculate the desired value of α. Therefore, the fifth color-saturation conversion rate calculation section  73  performs some sort of error handling (S 85 ). 
     The following shows an example of calculation of a backlight value according to Embodiment 6. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, B1 Ratio=0.5, and judgeTol=1. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information (j, x[j], y[j], a[j], b[j]) is calculated according to Formulae (9) to (12) in the same manner as in Embodiment 4 to take on values shown above in Table 1. 
     The calculation of a backlight upper limit and the calculation of the maximum and minimum values of a gamma-corrected RGB signal are performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5, and the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal are set as maxRGBg=255 and minRGBg=16, respectively. 
     Since the target pixel satisfies Formula (4), the target pixel is subjected to color-saturation conversion. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1], jMin, and kMax each corresponding to the input pixel values are calculated to take on the same values of 207, 3, and 15 as in Embodiment 4, respectively. 
     Thus, since jMin=3 and kMax=15, the present example of calculation repeats Steps S 81  to S 84  of the flow chart of  FIG. 20  from j=3 to j=15 and from k=15 to k=j for each value of j. The repetition of the steps yields such variations in value of j, k, αt, judge, α, and judgeMin as shown below in Table 5. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 5 
               
               
                   
                   
               
               
                   
                 j 
                 k 
                 α 
                 judge 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 3 
                 15 
                 0.282752 
                 45.2506 
               
               
                   
                 3 
                 14 
                 0.267344 
                 49.41094 
               
               
                   
                 3 
                 13 
                 0.264486 
                 50.18262 
               
               
                   
                 3 
                 12 
                 0.275815 
                 47.12364 
               
               
                   
                 3 
                 11 
                 0.303267 
                 39.71165 
               
               
                   
                 3 
                 10 
                 0.349144 
                 27.57701 
               
               
                   
                 3 
                 9 
                 0.416211 
                 10.27383 
               
               
                   
                 3 
                 8 
                 0.507813 
                 12.16406 
               
               
                   
                 3 
                 7 
                 0.628038 
                 39.65104 
               
               
                   
                 3 
                 6 
                 0.781939 
                 72.20267 
               
               
                   
                 3 
                 5 
                 0.97583 
                 107.7788 
               
               
                   
                 3 
                 4 
                 1.217708 
                 — 
               
               
                   
                 3 
                 3 
                 1.517867 
                 — 
               
               
                   
                 4 
                 15 
                 0.356412 
                 25.70205 
               
               
                   
                 4 
                 14 
                 0.345285 
                 28.67282 
               
               
                   
                 4 
                 13 
                 0.345631 
                 28.48351 
               
               
                   
                 4 
                 12 
                 0.358774 
                 25.09255 
               
               
                   
                 4 
                 11 
                 0.38625 
                 18.00375 
               
               
                   
                 4 
                 10 
                 0.42985 
                 6.754883 
               
               
                   
                 4 
                 9 
                 0.491678 
                 8.29178 
               
               
                   
                 4 
                 8 
                 0.574219 
                 27.70313 
               
               
                   
                 4 
                 7 
                 0.680482 
                 51.03906 
               
               
                   
                 4 
                 6 
                 0.813867 
                 78.33281 
               
               
                   
                 4 
                 5 
                 0.978824 
                 108.2998 
               
               
                   
                 4 
                 4 
                 1.180556 
                 — 
               
               
                   
                 5 
                 15 
                 0.405884 
                 12.93823 
               
               
                   
                 5 
                 14 
                 0.397429 
                 15.11946 
               
               
                   
                 5 
                 13 
                 0.399479 
                 14.59063 
               
               
                   
                 5 
                 12 
                 0.41312 
                 11.07139 
               
               
                   
                 5 
                 11 
                 0.439593 
                 4.24135 
               
               
                   
                 5 
                 10 
                 0.480324 
                 5.56684 
               
               
                   
                 5 
                 9 
                 0.436959 
                 19.15926 
               
               
                   
                 5 
                 8 
                 0.611406 
                 35.95675 
               
               
                   
                 5 
                 7 
                 0.705892 
                 56.38574 
               
               
                   
                 5 
                 6 
                 0.82303 
                 80.09205 
               
               
                   
                 5 
                 5 
                 0.965909 
                 106.0526 
               
               
                   
                 6 
                 15 
                 0.437388 
                 4.810045 
               
               
                   
                 6 
                 14 
                 0.430607 
                 6.559743 
               
               
                   
                 6 
                 13 
                 0.433475 
                 5.819602 
               
               
                   
                 6 
                 12 
                 0.446899 
                 2.455078 
               
               
                   
                 6 
                 11 
                 0.4719 
                 3.545111 
               
               
                   
                 6 
                 10 
                 0.509635 
                 12.60156 
               
               
                   
                 6 
                 9 
                 0.561422 
                 24.86234 
               
               
                   
                 6 
                 8 
                 0.628797 
                 39.81278 
               
               
                   
                 6 
                 7 
                 0.713397 
                 57.96181 
               
               
                   
                 6 
                 6 
                 0.817308 
                 78.99339 
               
               
                   
                 7 
                 15 
                 0.455181 
                 0.467516 
               
               
                   
                   
               
            
           
         
       
     
     In Table 5, the final value of α is calculated as:
 
α=0.455181.
 
In Embodiment 4, the number of times the loop is repeated is 91. Meanwhile, in Embodiment 6, the number of times the loop is repeated is 47, which is only about half as many as in Embodiment 4.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   185 
                 
               
             
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   229 
                 
               
             
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   134 
                 
               
             
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   206 
                 
               
             
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1] =Rsg[ 1] Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1]− Wtsg[ 1]=206−78=128
 
 Btsg[ 1]= Bsg[ 1]− Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     Embodiment 7 
     Embodiment 7 is a second example of means for approximately calculating a color-saturation conversion rate α in the second color-saturation conversion rate section  33  of Embodiment 2, makes such a change to Embodiment 5 as to speed up the color-saturation converting means, and makes such a change to Embodiment 6 as to perform a color-saturation conversion process on every pixel within the input image. It should be noted that processing sections identical to those of Embodiments 2 to 5 are given the same reference numerals as those of Embodiments 2 to 5 and will not be described below in detail. 
       FIG. 21  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 7. The color-saturation conversion section  11  of  FIG. 21  is obtained by replacing the fourth color-saturation conversion rate calculation section  63  of the color-saturation conversion section  11  (see  FIG. 17 ) of Embodiment 5 with a sixth color-saturation conversion rate calculation section  83 . 
     The sixth color-saturation conversion rate calculation section  83  calculates a color-saturation conversion rate α from an input RGB signal, a backlight upper limit, and normalized-gamma-curve polygonal-line information according to a different method from the fourth color-saturation conversion rate calculation section  63 , and then outputs the color-saturation conversion rate α. 
       FIG. 22  is a flow chart that explains the operation of the sixth color-saturation conversion rate calculation section  83 . 
     According to the flow chart, Steps S 91  to S 93  and S 84  are repeated from j=0 to j=Ndg−1, and further repeated from k=Ndg−1 to k=j or repeated from k=j to k=Ndg−1. 
     First, the sixth color-saturation conversion rate calculation section  83  calculates the current value of α according to Formula (20) (S 91 ). However, when the denominator of Formula (20) is 0, the target pixel has a color saturation of 0. Therefore, a may take on any value. In Embodiment 7, when the denominator of Formula (20) is 0, α is set, for example, as α=1. 
     Next, the sixth color-saturation conversion rate calculation section  83  calculates the maximum and minimum values (maxRGBs, minRGBs) of the input RGB signal after color-saturation conversion according to Formulae (22) and (23) (S 92 ):
 
max RGBs =max(α×max RGB +(1−α)× Y[i], 0)  (22)
 
min RGBs =min(α×max RGB +(1−α)× Y[i] ,MAX).  (23)
 
Depending on the value of α, the maximum value after the color-saturation conversion may exceeds the upper limit MAX of the signal, or the minimum value after the color-saturation conversion may fall short of the lower limit 0 of the signal. Therefore, Formulae (22) and (23) are subjected to a limiter process.
 
     Next, the sixth color-saturation conversion rate calculation section  83  calculates an α calculation judgment value (judge) according to Formula (24) (S 93 ):
 
judge=| fg (max RGBs ,γ)− fg (min RGBs,γ )−MAX w|.   (24)
 
     Next, if the α calculation judgment value judge is not more than a predetermined threshold (judgeTol) for use in calculation judgment of α, the sixth color-saturation conversion rate calculation section  83  goes out of the loop to finish the process. If not, the sixth color-saturation conversion rate calculation section  83  proceeds with the loop process (S 84 ). In cases where the sixth color-saturation conversion rate calculation section  83  finishes the loop process without satisfying the judgment condition of S 84 , the sixth color-saturation conversion rate calculation section  83  sets the color-saturation conversion rate αas α=1 (S 95 ). 
     The following shows a first example of calculation of a backlight value according to Embodiment 7. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, B1 Ratio=0.5, and judgeTol=1. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information (j, x[j], y[j], a[j], b[j]) is calculated in the same manner as in Embodiment 5 to take on values shown above in Table 1. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 7 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 7 calculates a color-saturation conversion rate α for every pixel. The calculation of a color-saturation conversion rate α in this example is performed in a way similar to the example explained in Embodiment 6 (differs only in range of repetition of j and k). 
     That is, since Ndg=16, the present example of calculation repeats Steps S 91  to S 93  and S 84  of the flow chart of  FIG. 22  from j=0 to j=15 and from k=15 to k=j for each value of j. The repetition of the steps yields such variations in value of j, k, α, and judge as shown below in Table 6. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 6 
               
               
                   
                   
               
               
                   
                 j 
                 k 
                 α 
                 judge 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0 
                 15 
                 −0.2114 
                 196.4903 
               
               
                   
                 0 
                 14 
                 −0.26636 
                 215.7437 
               
               
                   
                 0 
                 13 
                 −0.30651 
                 229.9678 
               
               
                   
                 0 
                 12 
                 −0.32868 
                 237.8036 
               
               
                   
                 0 
                 11 
                 −0.32873 
                 237.8185 
               
               
                   
                 0 
                 10 
                 −0.30111 
                 228.0449 
               
               
                   
                 0 
                 9 
                 −0.23828 
                 205.8047 
               
               
                   
                 0 
                 8 
                 −0.12969 
                 169.0359 
               
               
                   
                 0 
                 7 
                 0.039931 
                 115.3047 
               
               
                   
                 0 
                 6 
                 0.293457 
                 42.36035 
               
               
                   
                 0 
                 5 
                 0.666853 
                 48.1942 
               
               
                   
                 0 
                 4 
                 1.220052 
                 123.6211 
               
               
                   
                 0 
                 3 
                 2.060938 
                 127.5 
               
               
                   
                 0 
                 2 
                 3.405273 
                 127.5 
               
               
                   
                 0 
                 1 
                 5.75651 
                 127.5 
               
               
                   
                 0 
                 0 
                 10.625 
                 127.5 
               
               
                   
                 1 
                 15 
                 0.019336 
                 121.6066 
               
               
                   
                 1 
                 14 
                 −0.0148 
                 132.1838 
               
               
                   
                 1 
                 13 
                 −0.03429 
                 138.3802 
               
               
                   
                 1 
                 12 
                 −0.03654 
                 139.0947 
               
               
                   
                 1 
                 11 
                 −0.01831 
                 133.2993 
               
               
                   
                 1 
                 10 
                 0.024479 
                 120.0328 
               
               
                   
                 1 
                 9 
                 0.097098 
                 97.81133 
               
               
                   
                 1 
                 8 
                 0.20643 
                 66.10276 
               
               
                   
                 1 
                 7 
                 0.361654 
                 24.34961 
               
               
                   
                 1 
                 6 
                 0.575284 
                 27.93963 
               
               
                   
                 1 
                 5 
                 0.864844 
                 88.12031 
               
               
                   
                 1 
                 4 
                 1.255642 
                 124.582 
               
               
                   
                 1 
                 3 
                 1.785645 
                 127.5 
               
               
                   
                 1 
                 2 
                 2.514509 
                 127.5 
               
               
                   
                 1 
                 1 
                 3.541667 
                 127.5 
               
               
                   
                 2 
                 15 
                 0.175442 
                 75.02751 
               
               
                   
                 2 
                 14 
                 0.153054 
                 81.47514 
               
               
                   
                 2 
                 13 
                 0.144345 
                 83.96326 
               
               
                   
                 2 
                 12 
                 0.151367 
                 81.96094 
               
               
                   
                 2 
                 11 
                 0.176604 
                 74.69285 
               
               
                   
                 2 
                 10 
                 0.22309 
                 61.35938 
               
               
                   
                 2 
                 9 
                 0.294577 
                 42.0579 
               
               
                   
                 2 
                 8 
                 0.395752 
                 15.55225 
               
               
                   
                 2 
                 7 
                 0.532552 
                 18.10156 
               
               
                   
                 2 
                 6 
                 0.712612 
                 57.79688 
               
               
                   
                 2 
                 5 
                 0.945913 
                 102.5733 
               
               
                   
                 2 
                 4 
                 1.245768 
                 124.3154 
               
               
                   
                 2 
                 3 
                 1.630327 
                 127.5 
               
               
                   
                 2 
                 2 
                 2.125 
                 127.5 
               
               
                   
                 3 
                 15 
                 0.282752 
                 45.2506 
               
               
                   
                 3 
                 14 
                 0.287344 
                 49.41094 
               
               
                   
                 3 
                 13 
                 0.264486 
                 50.18262 
               
               
                   
                 3 
                 12 
                 0.275815 
                 47.12364 
               
               
                   
                 3 
                 11 
                 0.303267 
                 39.71165 
               
               
                   
                 3 
                 10 
                 0.349144 
                 27.57701 
               
               
                   
                 3 
                 9 
                 0.416211 
                 10.27383 
               
               
                   
                 3 
                 8 
                 0.507813 
                 12.16406 
               
               
                   
                 3 
                 7 
                 0.628038 
                 39.65104 
               
               
                   
                 3 
                 6 
                 0.781939 
                 72.20267 
               
               
                   
                 3 
                 5 
                 0.97583 
                 107.7788 
               
               
                   
                 3 
                 4 
                 1.217708 
                 123.5578 
               
               
                   
                 3 
                 3 
                 1.517867 
                 127.5 
               
               
                   
                 4 
                 15 
                 0.356412 
                 25.70205 
               
               
                   
                 4 
                 14 
                 0.345285 
                 28.57282 
               
               
                   
                 4 
                 13 
                 0.345631 
                 28.48351 
               
               
                   
                 4 
                 12 
                 0.358774 
                 25.09255 
               
               
                   
                 4 
                 11 
                 0.38625 
                 18.00375 
               
               
                   
                 4 
                 10 
                 0.42965 
                 6.754883 
               
               
                   
                 4 
                 9 
                 0.491678 
                 8.29178 
               
               
                   
                 4 
                 8 
                 0.574219 
                 27.70313 
               
               
                   
                 4 
                 7 
                 0.680432 
                 51.03906 
               
               
                   
                 4 
                 6 
                 0.813867 
                 78.33281 
               
               
                   
                 4 
                 5 
                 0.978824 
                 108.2998 
               
               
                   
                 4 
                 4 
                 1.180556 
                 121.9141 
               
               
                   
                 5 
                 15 
                 0.405884 
                 12.93823 
               
               
                   
                 5 
                 14 
                 0.397429 
                 15.11946 
               
               
                   
                 5 
                 13 
                 0.399479 
                 14.59063 
               
               
                   
                 5 
                 12 
                 0.41312 
                 11.07139 
               
               
                   
                 5 
                 11 
                 0.439593 
                 4.24135 
               
               
                   
                 5 
                 10 
                 0.480324 
                 5.56684 
               
               
                   
                 5 
                 9 
                 0.536959 
                 19.15926 
               
               
                   
                 5 
                 8 
                 0.611406 
                 35.96875 
               
               
                   
                 5 
                 7 
                 0.705892 
                 56.38574 
               
               
                   
                 5 
                 6 
                 0.82303 
                 80.09205 
               
               
                   
                 5 
                 5 
                 0.965909 
                 106.0526 
               
               
                   
                 6 
                 15 
                 0.437388 
                 4.810045 
               
               
                   
                 6 
                 14 
                 0.430607 
                 6.559743 
               
               
                   
                 6 
                 13 
                 0.433475 
                 5.819602 
               
               
                   
                 6 
                 12 
                 0.446699 
                 2.455078 
               
               
                   
                 6 
                 11 
                 0.4719 
                 3.545111 
               
               
                   
                 6 
                 10 
                 0.509635 
                 12.60158 
               
               
                   
                 6 
                 9 
                 0.561422 
                 24.86234 
               
               
                   
                 6 
                 8 
                 0.628767 
                 39.81278 
               
               
                   
                 6 
                 7 
                 0.713397 
                 57.96181 
               
               
                   
                 6 
                 6 
                 0.817308 
                 78.99339 
               
               
                   
                 7 
                 15 
                 0.455181 
                 0.467516 
               
               
                   
                   
               
            
           
         
       
     
     In Table 6, the final value of α is calculated as:
 
α=0.455181.
 
In Embodiment 5, the number of times the loop is repeated is 136. Meanwhile, in Embodiment 7, the number of times the loop is repeated is 92, which means a reduction in number of times the loop is repeated.
 
     Since the color-saturation conversion rate α thus calculated falls within a range of 0≦α≦1, α is set as α=0.455181 even after the limiter process. 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   185 
                 
               
             
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   229 
                 
               
             
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.455181 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.455181 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   134 
                 
               
             
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   206 
                 
               
             
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1] =Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1] −Wtsg[ 1]=206−78=128
 
 Btsg[ 1]= Bsg[ 1] −Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The first example of calculation thus explained is an example where the sixth color-saturation conversion rate calculation section  83  calculates a color-saturation conversion rate α as 0≦α≦1. The following shows an example (second example of calculation) where the sixth color-saturation conversion rate calculation section  83  does not calculate a color-saturation conversion rate α as 0≦α≦1. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). That is, the following shows an example where the same pixel values are used as in the second example of calculation of a backlight value according to Embodiment 2. It should be noted that the following explanation assumes that MAX=255, γ=2, Ndg=16, B1 Ratio=0.5, and judgeTol=1. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the second example of calculation explained in Embodiment 2. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. First, normalized-gamma-curve polygonal-line information (j, x[j], y[j], a[j], b[j]) is calculated in the same manner as in Embodiment 5 to take on values shown above in Table 1. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 7 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 7 calculates a color-saturation conversion rate α for every pixel. In Embodiment 7, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 167.25 as in the second example of calculation of Embodiment 2. 
     Since Ndg=16, the present example of calculation repeats Steps S 91  to S 93  and S 84  of the flow chart of  FIG. 22  from j=0 to j=15 and from k=15 to k=j for each value of j. The repetition of the steps yields such variations in value of j, k, α, and judge as shown below in Table 7. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 7 
               
               
                   
                   
               
               
                   
                 i 
                 k 
                 α 
                 judge 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0 
                 15 
                 1.220302 
                 1.602929 
               
               
                   
                 0 
                 14 
                 1.074618 
                 12.49178 
               
               
                   
                 0 
                 13 
                 0.962023 
                 21.95226 
               
               
                   
                 0 
                 12 
                 0.889323 
                 28.1272 
               
               
                   
                 0 
                 11 
                 0.865331 
                 30.22147 
               
               
                   
                 0 
                 10 
                 0.90166 
                 27.07932 
               
               
                   
                 0 
                 9 
                 1.013934 
                 17.54301 
               
               
                   
                 0 
                 8 
                 1.223684 
                 1.342105 
               
               
                   
                 0 
                 7 
                 1.561427 
                 20.46663 
               
               
                   
                 0 
                 6 
                 2.071903 
                 42.49794 
               
               
                   
                 0 
                 5 
                 2.823456 
                 67.5458 
               
               
                   
                 0 
                 4 
                 3.925962 
                 107.7137 
               
               
                   
                 0 
                 3 
                 5.568027 
                 127.5 
               
               
                   
                 0 
                 2 
                 8.102901 
                 127.5 
               
               
                   
                 0 
                 1 
                 12.27827 
                 127.5 
               
               
                   
                 0 
                 0 
                 20 
                 127.5 
               
               
                   
                 1 
                 15 
                 1.338912 
                 6.716617 
               
               
                   
                 1 
                 14 
                 1.228188 
                 0.994757 
               
               
                   
                   
               
            
           
         
       
     
     In Table 7, the final value of α is calculated as:
 
α=1.228188.
 
In Embodiment 5, the number of times the loop is repeated is 116. Meanwhile, in Embodiment 7, the number of times the loop is repeated is 18, which means a reduction in number of times the loop is repeated. In this example, since α exceeds 1, a limiter process is performed so that α=1.
 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Embodiment 8 
     Embodiment 8 is a third example of means for approximately calculating a color-saturation conversion rate α in the first color-saturation conversion rate calculation section  23  of Embodiment 1. Embodiment 8 calculates a color-saturation conversion rate α by using a dichotomizing search. It should be noted that processing sections identical to those of Embodiment 1 are given the same reference numerals as those of Embodiment 1 and will not be described below in detail. 
       FIG. 23  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 8. The color-saturation conversion section  11  of  FIG. 23  is obtained by replacing the first color-saturation conversion rate calculation section  23  of the color-saturation conversion section  11  (see  FIG. 6 ) of Embodiment 1 with a seventh color-saturation conversion rate calculation section  93 . 
     The seventh color-saturation conversion rate calculation section  93  calculates a color-saturation conversion rate α from an input RGB signal, a gamma coefficient, a backlight upper limit, and the maximum and minimum values of a gamma-corrected RGB signal according to a dichotomizing search, and then outputs the color-saturation conversion rate α. 
       FIG. 24  is a flow chart that explains the operation of the seventh color-saturation conversion rate calculation section  93 . 
     First, in Step S 101 , the seventh color-saturation conversion rate calculation section  93  sets the lower and upper limits low and high of a color-saturation conversion rate α for use in a dichotomizing search as low=0 and high=1, respectively. 
     Next, the seventh color-saturation conversion rate calculation section  93  repeats Steps S 102  to S 107  while low+alphaTol≦high, where alphaTol is a threshold for use in calculation loop judgment of a color-saturation conversion rate α. Further, Steps S 102  to S 107  may be repeated a predetermined number of times. 
     In Step S 102 , the seventh color-saturation conversion rate calculation section  93  calculates the current value of a according to Formula (25).
 
α=(low+high)/2.  (25)
 
     Next, in Step S 103 , the seventh color-saturation conversion rate calculation section  93  calculates an α calculation judgment value (judgeSi) according to Formula (26): 
     
       
         
           
             
               
                 
                   judgeSi 
                   = 
                   
                     
                       fg 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   α 
                                   × 
                                   max 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   B 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       α 
                                     
                                     ) 
                                   
                                   × 
                                   
                                     Y 
                                     ⁡ 
                                     
                                       [ 
                                       i 
                                       ] 
                                     
                                   
                                 
                                 , 
                                 γ 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       fg 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   α 
                                   × 
                                   min 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   B 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       α 
                                     
                                     ) 
                                   
                                   × 
                                   
                                     Y 
                                     ⁡ 
                                     
                                       [ 
                                       i 
                                       ] 
                                     
                                   
                                 
                                 , 
                                 γ 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       MAX 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         w 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The closer to 0 judgeSi is, the closer to the desired value of α the current value of α is. Further, the current value of α is smaller than the desired value in cases where judgeSi takes on a negative value, and the current value of α is larger than the desired value in cases where judgeSi takes on a positive value. 
     Next, if judgeSi is smaller than −judgeTol in Step S 104 , the current value of α is smaller than the desired value. Therefore, the seventh color-saturation conversion rate calculation section  93  updates low so that low takes on the current value of α (S 105 ). Alternatively, if judgeSi is not smaller than −judgeTol in Step S 104 , the seventh color-saturation conversion rate calculation section  93  proceeds to Step S 106 . It should be noted here that judgeTol is a threshold for use in calculation judgment of a color-saturation conversion rate. 
     Next, if judgeSi is larger than judgeTol in Step S 106 , the current value of α is larger than the desired value. Therefore, the seventh color-saturation conversion rate calculation section  93  updates high so that high takes on the current value of α (S 107 ). Alternatively, if judgeSi is not larger than judgeTol in Step S 106 , the seventh color-saturation conversion rate calculation section  93  judges that the current value of α is the desired value of α, and goes out of the loop to finish the process. 
     Alternatively, in cases where the seventh color-saturation conversion rate calculation section  93  finishes the loop process without saying No to the judgment condition of S 106 , the seventh color-saturation conversion rate calculation section  93  cannot calculate the desired value of α. Therefore, the seventh color-saturation conversion rate calculation section  93  performs some sort of error handling (S 108 ). 
     The following shows an example of calculation of a backlight value according to Embodiment 8. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, B1 Ratio=0.5, judgeTol=1, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit and the calculation of the maximum and minimum values of a gamma-corrected RGB signal are performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5, and the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal are set as maxRGBg=255 and minRGBg=16, respectively. 
     Since the target pixel satisfies Formula (4), the target pixel is subjected to color-saturation conversion. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in Embodiment 1 to take on a value of 207. 
     In the present example of calculation, the seventh color-saturation conversion rate calculation section  93  repeats Steps S 101  to S 107  of the flow chart of  FIG. 24  while low+alphatol≦high, until the seventh color-saturation conversion rate calculation section  93  goes out of the loop by satisfying the judgment condition of S 106 . The values of low, high, α, and judgeSi that are calculated based on the flow chart vary as shown below in Table 8. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 8 
               
               
                   
                   
               
               
                   
                 low 
                 high 
                 α 
                 judgeSi 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0 
                 1 
                 0.5 
                 10.28824 
               
               
                   
                 0 
                 0.5 
                 0.25 
                 −54.0882 
               
               
                   
                 0.25 
                 0.5 
                 0.375 
                 −20.7706 
               
               
                   
                 0.375 
                 0.5 
                 0.4375 
                 −4.95882 
               
               
                   
                 0.4375 
                 0.5 
                 0.46875 
                 2.735294 
               
               
                   
                 0.4375 
                 0.46875 
                 0.453125 
                 −1.09412 
               
               
                   
                 0.453125 
                 0.46875 
                 0.460938 
                 0.825 
               
               
                   
                   
               
            
           
         
       
     
     In Table 8, the final value of α is calculated as:
 
α=0.460938.
 
In Embodiments 4 and 6, the numbers of times the loops are repeated is 91 and 47, respectively. Meanwhile, in the present embodiment, the number of times the loop is repeated is only 7.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.460938 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.460938 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   185 
                 
               
             
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.40938 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.460938 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   229 
                 
               
             
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       0.460938 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.460938 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   134 
                 
               
             
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   206 
                 
               
             
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wrsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1]= Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1 ]−Wtsg[ 1]=206−78=128
 
 Btsg[ 1 ]=Bsg[ 1 ]−Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in eases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     Embodiment 9 
     Embodiment 9 is a third example of means for approximately calculating a color-saturation conversion rate α in the second color-saturation conversion rate section  33  of Embodiment 2, and makes such a change to Embodiment 8 as to perform a color-saturation conversion process on every pixel within the input image. It should be noted that processing sections identical to those of Embodiments 2 and 7 are given the same reference numerals as those of Embodiments 2 and 7 and will not be described below in detail. 
       FIG. 25  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 9. The color-saturation conversion section  11  of  FIG. 25  is obtained by replacing the calculation section  22  and the seventh color-saturation conversion rate calculation section  93  of the color-saturation conversion section  11  (see  FIG. 23 ) of Embodiment 8 with an eighth color-saturation conversion rate calculation section  103 . It should be noted, in the present embodiment, that since the color-saturation conversion rate α does not deviate from a range of 0 to 1, no color-saturation conversion rate limiter process section  34  is mounted. 
     The eighth color-saturation conversion rate calculation section  103  calculates a color-saturation conversion rate α from an input RGB signal, a gamma coefficient, and a backlight upper limit according to a dichotomizing search, and then outputs the color-saturation conversion rate α. 
       FIG. 26  is a flow char that explains the operation of the eighth color-saturation conversion rate calculation section  103 . 
     First, in Step S 101 , the eighth color-saturation conversion rate calculation section  103  sets the lower and upper limits low and high of a color-saturation conversion rate α for use in a dichotomizing search as low=0 and high=1, respectively. Step S 101  of  FIG. 26  is identical to Step S 101  of  FIG. 24 . 
     Next, the eighth color-saturation conversion rate calculation section  103  repeats Steps S 102  to S 107  while low+alphaTol≦high. Steps S 102  to S 107  of  FIG. 26  are also identical to Steps S 102  to S 107  of  FIG. 24 . 
     Further, in cases where the eighth color-saturation conversion rate calculation section  103  finishes the loop process without saying No to the judgment condition of S 106 , the eighth color-saturation conversion rate calculation section  103  performs such a process that the current value of α is the desired value of α. 
     The following shows a first example of calculation of a backlight value according to Embodiment 9. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=62, B1 Ratio=0.5, judgeTol=1, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw 127.5. 
     Since Embodiment 9 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 9 calculates a color-saturation conversion rate α for every pixel. In Embodiment 9, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 207 as in Embodiment 1. 
     In the present example of calculation, the eighth color-saturation conversion rate calculation section  103  repeats Steps S 101  to S 107  of the flow chart of  FIG. 26  until the eighth color-saturation conversion rate calculation section  103  goes out of the loop by satisfying the judgment condition of S 106 . The values of low, high, α, and judgeSi that are calculated based on the flow chart vary as shown below in Table 8, as with Embodiment 8. 
     In Table 8, the final value of α is calculated as:
 
α=0.460938.
 
In Embodiments 5 and 7, the numbers of times the loops are repeated is 136 and 92, respectively. Meanwhile, in the present embodiment, the number of times the loop is repeated is only 7.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. The subsequent procedure is the same as in the example of calculation of a backlight value according to Embodiment 8. That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The following shows a second example of calculation of a backlight value according to Embodiment 9. For convenience of explanation. The following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). That is, the following shows an example where the same pixel values are used as in the second example of calculation of a backlight value according to Embodiment 2. It should be noted that the following explanation assumes that MAX=255, γ=2, B1 Ratio=0.5, judgeTol=1, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the second example of calculation explained in Embodiment 2. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. Further, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in the second example of calculation of Embodiment 2 to take on a value of 167.25. 
     In the present example of calculation, the seventh color-saturation conversion rate calculation section  93  repeats Steps S 101  to S 107  of the flow chart of  FIG. 26  while low+alphaTol≦high. The values of low, high, α, and judgeSi that are calculated based on the flow chart vary as shown below in Table 9. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 9 
               
               
                   
                   
               
               
                   
                 low 
                 high 
                 α 
                 judgeSi 
               
               
                   
                   
               
             
            
               
                   
                 0.00000000 
                 1.00000000 
                 0.50000000 
                 −66.85000000 
               
               
                   
                 0.50000000 
                 1.00000000 
                 0.75000000 
                 −41.21250000 
               
               
                   
                 0.75000000 
                 1.00000000 
                 0.87500000 
                 −29.56562500 
               
               
                   
                 0.87500000 
                 1.00000000 
                 0.93750000 
                 −24.03515625 
               
               
                   
                 0.93750000 
                 1.00000000 
                 0.96875000 
                 −21.34316406 
               
               
                   
                 0.96875000 
                 1.00000000 
                 0.98437500 
                 −20.01547852 
               
               
                   
                 0.98437500 
                 1.00000000 
                 0.99218750 
                 −19.35621338 
               
               
                   
                 0.99218750 
                 1.00000000 
                 0.99609375 
                 −19.02772522 
               
               
                   
                 0.99609375 
                 1.00000000 
                 0.99804688 
                 −18.86376724 
               
               
                   
                 0.99804688 
                 1.00000000 
                 0.99902344 
                 −18.78185978 
               
               
                   
                 0.99902344 
                 1.00000000 
                 0.99951172 
                 −18.74092393 
               
               
                   
                 0.99951172 
                 1.00000000 
                 0.99975586 
                 −18.72046047 
               
               
                   
                 0.99975586 
                 1.00000000 
                 0.99987793 
                 −18.71022986 
               
               
                   
                 0.99987793 
                 1.00000000 
                 0.99993896 
                 −18.70511484 
               
               
                   
                 0.99993896 
                 1.00000000 
                 0.99996948 
                 −18.70255740 
               
               
                   
                 0.99996948 
                 1.00000000 
                 0.99998474 
                 −18.70127869 
               
               
                   
                 0.99998474 
                 1.00000000 
                 0.99999237 
                 −18.70063934 
               
               
                   
                 0.99999237 
                 1.00000000 
                 0.99999619 
                 −18.70031967 
               
               
                   
                 0.99999619 
                 1.00000000 
                 0.99999809 
                 −18.70015984 
               
               
                   
                 0.99999809 
                 1.00000000 
                 0.99999905 
                 −18.70007992 
               
               
                   
                 0.99999905 
                 1.00000000 
                 0.99999952 
                 −18.70003996 
               
               
                   
                 0.99999952 
                 1.00000000 
                 0.99999976 
                 −18.70001998 
               
               
                   
                 0.99999976 
                 1.00000000 
                 0.99999988 
                 −18.70000999 
               
               
                   
                 0.99999988 
                 1.00000000 
                 0.99999994 
                 −18.70000499 
               
               
                   
                   
               
            
           
         
       
     
     In Table 9, the final value of α is calculated as:
 
α=0.99999994≈1.
 
In the second examples of calculation of Embodiments 5 and 7, the numbers of times the loops are repeated is 136 and 18, respectively. Meanwhile, in the present embodiment, the number of times the loop is repeated is 24.
 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Embodiment 10 
     Embodiment 10 is a fourth example of means for approximately calculating a color-saturation conversion rate α in the color-saturation conversion rate calculation section  23  of Embodiment 1. Embodiment 10 calculates a color-saturation conversion rate α by using Newton&#39;s method. It should be noted that processing sections identical to those of Embodiment 1 are given the same reference numerals as those of Embodiment 1 and will not be described below in detail. 
       FIG. 27  shows another example of a color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 10. The color-saturation conversion section  11  of  FIG. 27  is obtained by replacing the first color-saturation conversion section  23  (see  FIG. 6 ) of Embodiment 1 with a ninth color-saturation conversion rate calculation section  113 . 
     The ninth color-saturation conversion rate calculation section  113  calculates a color-saturation conversion rate α from an input RGB signal, a gamma coefficient, a backlight upper limit, and the maximum and minimum values of a gamma-corrected RGB signal according to Newton&#39;s method, and then outputs the color-saturation conversion rate α. 
       FIG. 28  is a flow chart that explains the operation of the ninth color-saturation conversion rate calculation section  113 . 
     First, in Step S 111 , the ninth color-saturation conversion rate calculation section  113  initializes variables for use in processing. Specifically, the ninth color-saturation conversion rate calculation section  113  initializes variables as follows: 
     αprev=a predetermined constant (e.g., 0.5) 
     maxRGBn=maxRGB/MAX 
     minRGBn=minRGB/MAX 
     Yn=Y[i]/MAX 
     MAXwn=MAXw/MAX 
     overFlag=FALSE 
     underFlag=FALSE, 
     where 
     αprev=a previous color-saturation conversion rate, 
     maxRGBn=the maximum value of a normalized KGB signal, 
     minRGBn=the minimum value of the normalized RGB signal, 
     Yn=a normalized luminance signal, 
     overFlag=an indicator of whether or not an out-of-range correction for α&gt;1 has been made, 
     underflag=an indicator of whether or not an out-of-range correction for α&lt;0 has been made, and 
     FALSE=a logic value “FALSE”. 
     Next, the ninth color-saturation conversion rate calculation section  113  repeats Steps S 112  to S 122  a predetermined number of times. It should be noted that it is in order to avoid an endless loop that the ninth color-saturation conversion rate calculation section  113  repeats the steps a predetermined number of times. 
     In Step S 112 , the ninth color-saturation conversion rate calculation section  113  calculates the current value of α according to Formula (27): 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       α 
                       = 
                       
                         
                           α 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           prev 
                         
                         - 
                         
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 α 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 prev 
                               
                               ) 
                             
                           
                           / 
                           
                             
                               g 
                               ′ 
                             
                             ⁡ 
                             
                               ( 
                               
                                 α 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 prev 
                               
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     where 
                     ⁢ 
                     
                        
                     
                     ⁢ 
                     
                       
                         
                           g 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         x 
                                         × 
                                         max 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         G 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           x 
                                         
                                         ) 
                                       
                                       × 
                                       Yn 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             γ 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         x 
                                         × 
                                         min 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         G 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           x 
                                         
                                         ) 
                                       
                                       × 
                                       Yn 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             γ 
                           
                           - 
                           
                             MAX 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             wn 
                           
                         
                       
                       , 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           g 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           w 
                           ) 
                         
                       
                       = 
                       
                         
                           γ 
                           × 
                           
                             ( 
                             
                               
                                 max 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               Yn 
                             
                             ) 
                           
                           × 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         x 
                                         × 
                                         max 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         G 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           x 
                                         
                                         ) 
                                       
                                       × 
                                       Yn 
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               ( 
                               
                                 γ 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           γ 
                           × 
                           
                             ( 
                             
                               
                                 min 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               Yn 
                             
                             ) 
                           
                           × 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           x 
                                           × 
                                           min 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           R 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           G 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           B 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           n 
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             x 
                                           
                                           ) 
                                         
                                         × 
                                         Yn 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   γ 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     It should be noted here that g(x) in Formula (27) is a normal function obtained by transforming Formula (5) so that the right-hand side is 0, and that g′(x) is the first differential of g(x). That is, Formula (27) serves as a formula for calculating the next solution according to Newton&#39;s method. 
     Next, in Step S 113 , the ninth color-saturation conversion rate calculation section  113  judges whether or not Formula (28) below is satisfied as follows:
 
|α−αprev|&lt;alphaTol  (28)
 
     where alphaTol=a threshold for calculating a (alphaTol&gt;0). That is, the ninth color-saturation conversion rate calculation section  113  judges whether or not the current value of α is the desired value of α. If Formula (28) is satisfied, the ninth color-saturation conversion rate calculation section  113  finishes the process of calculating α. 
     If Formula (28) is not satisfied, the ninth color-saturation conversion rate calculation section  113  shifts to Step S 114 , and then judges whether or not α is negative. If α is negative, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 115 . If not, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 118 . 
     In cases where α is negative in Step S 114 , the ninth color-saturation conversion rate calculation section  113  makes an out-of-range correction for α&lt;0 by setting α to 0 in Step S 115 , and further judges whether or not underFlag is TRUE, i.e., whether or not an out-of-range correction for α&lt;0 has been made (S 116 ). If an out-of-range correction for α&lt;0 has already been made, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 123 . If not, the ninth color-saturation conversion rate calculation section  113  sets underFlag to TRUE (S 117 ). 
     Further, in cases where α is not negative in Step S 114 , the ninth color-saturation conversion rate calculation section  113  judges whether or not a exceeds 1. If a exceeds 1, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 119 . If not so, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 122 . 
     In cases where a exceeds 1 in Step S 118 , the ninth color-saturation conversion rate calculation section  113  makes an out-of-range correction for α&gt;1 by setting α to 1 in Step S 119 , and further judges whether or not overflag is TRUE, i.e., whether or not an out-of-range correction for α&gt;1 has been made (S 120 ). If an out-of-range correction for α&gt;1 has already been made, the ninth color-saturation conversion rate calculation section  113  proceeds to Step S 123 . If not, the ninth color-saturation conversion rate calculation section  113  sets overFlag to TRUE (S 121 ). 
     In cases where the ninth color-saturation conversion rate calculation section  113  has proceeded to Step S 117  or S 121 , i.e., in cases where an out-of-range correction for α&lt;0 or an out-of-range correction for α&gt;1 has been made, or in cases where the ninth color-saturation conversion rate calculation section  113  says No in both Steps S 114  and S 118 , i.e., in cases where 0≦α≦1, the ninth color-saturation conversion rate calculation section  113  updates the previous color-saturation conversion rate αprev according to Formula (29)
 
αprev=α  (29)
 
so that the previous color-saturation conversion rate αprev takes on the current value of α(0 or 1) (S 122 ).
 
     In cases where the ninth color-saturation conversion rate calculation section  113  says Yes to the judgment condition of S 116  or S 120  or finishes the loop process, the ninth color-saturation conversion rate calculation section  113  cannot calculate the desired value of α. Therefore, the ninth color-saturation conversion rate calculation section  113  performs some sort of error handling (S 123 ). 
     In comparison with Embodiment 8, which uses a dichotomizing search, Embodiment 10, which uses Newton&#39;s method, has a larger amount of calculation per loop, but can be expected to have a smaller number of loops. 
     The following shows an example of calculation of a backlight value according to Embodiment 10. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, B1 Ratio=0.5, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit and the calculation of the maximum and minimum values of a gamma-corrected RGB signal are performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5, and the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal are set as maxRGBg=255 and minRGBg=16, respectively. 
     Since the target pixel satisfies Formula (4), the target pixel is subjected to color-saturation conversion. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in Embodiment 1 to take on a value of 207. 
     In the present example of calculation, the ninth color-saturation conversion rate calculation section  113  repeats Steps S 112  to S 122  of the flow chart of  FIG. 28  until the ninth color-saturation conversion rate calculation section  113  goes out of the loop by satisfying the judgment condition of S 113 . The values of αprev, g(αprev), g′(αprev), and α that are calculated based on the flow chart vary as shown below in Table 10. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 10 
               
               
                   
                   
               
               
                   
                 αprev 
                 g(αprev) 
                 g′(αprev) 
                 α 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0.5 
                 0.040346 
                 0.938962 
                 0.457031 
               
               
                   
                 0.457031 
                 −0.00052 
                 0.963322 
                 0.457575 
               
               
                   
                 0.457575 
                 0 
                 0.963014 
                 0.457575 
               
               
                   
                   
               
            
           
         
       
     
     In Table 10, the final value of α is calculated as:
 
α=0.457575.
 
In Embodiments 4, 6, and 8, the numbers of times the loops are repeated is 91, 47, and 7, respectively. Meanwhile, in the present embodiment, the number of times the loop is repeated is only 3.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   185 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   229 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           Rs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   134 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           Gs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           229 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   206 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           Bs 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         / 
                         MAX 
                       
                       ) 
                     
                     ⁢ 
                     γ 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1]= Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1 ]−Wtsg[ 1]=206−78=128
 
 Btsg[ 1 ]=Bsg[ 1 ]−Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows; 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     Embodiment 11 
     Embodiment 11 is a fourth example of means for approximately calculating a color-saturation conversion rate α in the second color-saturation conversion rate section  33  of Embodiment 2, and makes such a change to Embodiment 10 as to perform a color-saturation conversion process on every pixel within the input image. It should be noted that processing sections identical to those of Embodiments 2 and 9 are given the same reference numerals as those of Embodiments 2 and 9 and will not be described below in detail. 
       FIG. 29  shows another arrangement of the color-saturation conversion section  11  of Embodiment 11. The color-saturation conversion section  11  of  FIG. 29  is obtained by replacing the second color-saturation conversion rate calculation section  33  of the color-saturation conversion section  11  (see  FIG. 10 ) of Embodiment 2 with a tenth color-saturation conversion rate calculation section  123 . It should be noted, in the present embodiment, that since the color-saturation conversion rate α does not deviate from a range of 0 to 1, no color-saturation conversion rate limiter process section  34  is mounted. 
     The tenth color-saturation conversion rate calculation section  123  calculates a color-saturation conversion rate α from an input RGB signal, a gamma coefficient, and a backlight upper limit according to Newton&#39;s method, and then outputs the color-saturation conversion rate α. 
       FIG. 30  is a flow chart for explaining the operation of the tenth color-saturation conversion rate calculation section  123 . 
     First, in Step S 111 , the tenth color-saturation conversion rate calculation section  123  initializes variables for use in processing. Step S 111  of  FIG. 30  is identical to Step S 111  of  FIG. 28 . 
     Next, the tenth color-saturation conversion rate calculation section  123  repeats Steps S 132  and S 113  to S 122  a predetermined number of times. 
     In Step S 132 , the tenth color-saturation conversion rate calculation section  123  calculates the current value of α according to Formula (27). However, it is meant in this step that when g′(αprev) is 0, the target pixel has a color saturation of 0. Therefore, α may take on any value. In the present embodiment, when g′(αprev) is 0, α is set, for example, as α=1. Steps S 113  to S 122  that are repeated thereafter are identical to the steps of the flow chart of  FIG. 28 . However, in cases where the tenth color-saturation conversion rate calculation section  123  finishes the loop process without saying No in Step S 116  or S 120 , the tenth color-saturation conversion rate calculation section  123  performs such a process that the current value of α is the desired value of α. 
     The following shows a first example of calculation of a backlight value according to Embodiment 11. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (K[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value is calculated according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255, γ=2, B1 Ratio=0.5, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 11 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 11 calculates a color-saturation conversion rate α for every pixel. In Embodiment 11, the color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated to take on the same value of 207 as in Embodiment 1. 
     In the present example of calculation, the tenth color-saturation conversion rate calculation section  123  repeats Steps S 132  and S 113  to S 122  of the flow chart of  FIG. 30  until the tenth color-saturation conversion rate calculation section  123  goes out of the loop by satisfying the judgment condition of S 113 . The values of αprev, g(αprev), g′(αprev), and a that are calculated based on the flow chart vary as shown below in Table 10, as with Embodiment 10. 
     In Table 10, the final value of α is calculated as:
 
α=0.457575.
 
In Embodiments 5, 7, and 9, the numbers of times the loops are repeated is 136, 92, and 7, respectively. Meanwhile, in the present embodiment, the number of times the loop is repeated is only 3.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. The subsequent procedure is the same as in the example of calculation of a backlight value according to Embodiment 10. That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The following shows a second example of calculation of a backlight value according to Embodiment 11. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). That is, the following shows an example where the same pixel values are used as in the second example of calculation of a backlight value according to Embodiment 2. It should be noted that the following explanation assumes that MAX=255, γ=2, B1 Ratio=0.5, and alphaTol=10 −7 . 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the second example of calculation explained in Embodiment 2. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. Further, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in the second example of calculation of Embodiment 2 to take on a value of 167.25. 
     In the present example of calculation, the tenth color-saturation conversion rate calculation section  123  goes out of the loop by saying Yes in Step S 120  of the flow chart of  FIG. 30 . The values of αprev, g(αprev), g′(αprev), and a that are calculated based on the flow chart vary as shown below in Table 11. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 11 
               
               
                   
                   
               
               
                   
                 αprev 
                 g(αprev) 
                 g′(αprev) 
                 α 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0.5 
                 −0.26216 
                 0.426667 
                 1.11443 
               
               
                   
                 1 
                 −0.07333 
                 0.328627 
                 1.22315 
               
               
                   
                   
               
            
           
         
       
     
     Further, although the final value of α in Table 11 is α=1.22315, α is set as α=1 by Step S 119  in this case. Therefore, the final value of α is calculated as:
 
α=1.
 
     The numbers of times the loops are repeated in the second examples of calculation of Embodiments 5, 7, and 9 are 136, 18, and 24, respectively. Meanwhile, the number of times the loop is repeated in the present embodiment is only 2. 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Embodiment 12 
     Embodiment 12 is a fifth example of means for approximately calculating a color-saturation conversion rate α in the first color-saturation conversion section  23  of Embodiment 1. Embodiment 12 directly calculates a according to Formula (5) by fixing the gamma coefficient to 2. 
     Therefore, Embodiment 12 always gives a gamma coefficient of 2 to the calculation section  22  for calculating the maximum and minimum values of a gamma-corrected RGB signal. The fixation of the gamma coefficient eliminates the degree of freedom of the gamma coefficient. However, since the gamma coefficient usually takes on a value of approximately 1.8 to 2.3, it does not matter if the gamma coefficient is fixed to 2. 
     It should be noted that processing sections identical to those of Embodiment 1 are given the same reference numerals as those of Embodiment 1 and will not be described below in detail. 
       FIG. 31  shows another arrangement of the color-saturation conversion section  11  of a liquid crystal display device according to Embodiment 12. The color-saturation conversion section  11  of  FIG. 31  is obtained by replacing the first color-saturation conversion rate calculation section  23  of the color-saturation conversion section  11  (see  FIG. 6 ) of Embodiment 1 with an eleventh color-saturation conversion rate calculation section  133 . 
     The eleventh color-saturation conversion rate calculation section  133  calculates a color-saturation conversion rate α from an input RGB signal, a backlight upper limit, and the maximum and minimum values of a gamma-corrected RGB signal according to a direct solving method, and then outputs the color-saturation conversion rate α. 
       FIG. 32  is a flow chart for explaining the operation of the eleventh color-saturation conversion rate calculation section  133 . 
     First, in Step S 141 , the eleventh color-saturation conversion rate calculation section  133  calculates a coefficient for use in calculation of α according to Formulae (30) to (32): 
                   a   =       (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B       )     ×     (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     2   ×     Y   ⁡     [   i   ]           )               (   30   )                 b=Y[i]× (max RGB− min RGB )  (31)
 
 c =−MAX w× MAX.  (32)
 
     Next, if the coefficient a for use in calculation of α is 0, the eleventh color-saturation conversion rate calculation section  133  calculates α according to Formula (33). If not so, the eleventh color-saturation conversion rate calculation section  133  calculates α according to Formula (34) (S 142 ):
 
α=− c /(2 ×b )  (33)
 
α=(− b+√d )/ a   (34)
 
     where d=b 2 −a×c. 
     The following explains how to derive the above calculating formulae. 
     First, substituting γ=2 in Formula (5) transforms Formula (5) as follows: 
                           fg   ⁡     (             α   ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                   (     1   -   α     )     ×     Y   ⁡     [   i   ]         ,   2           )       -     fg   ⁡     (             α   ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                   (     1   -   α     )     ×     Y   ⁡     [   i   ]         ,   2           )         =     MAX   ⁢           ⁢   w       ⁢     
     ⁢             {       (             α   ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                 (     1   -   α     )     ×     Y   ⁡     [   i   ]               )     MAX     }     2     ×   MAX     -         {       (             α   ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                 (     1   -   α     )     ×     Y   ⁡     [   i   ]               )     MAX     }     2     ×   MAX       =     MAX   ⁢           ⁢   w       ⁢     
     ⁢           {     (             α   ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                 (     1   -   α     )     ×     Y   ⁡     [   i   ]               )     }     2     -       {     (             α   ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +                 (     1   -   α     )     ×     Y   ⁡     [   i   ]               )     }     2       =       MAX   ⁢           ⁢   w   ×   MAX   ⁢     
     ⁢         α   2     ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       +     2   ×   α   ×     (     1   -   α     )     ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         -       α   2     ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       -     2   ×   α   ×     (     1   -   α     )     ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         -     MAW   ⁢           ⁢   w   ×   MAX         =       0   ⁢     
     ⁢         α   2     ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       +     2   ×   α   ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         -     2   ×     α   2     ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         -       α   2     ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       -     2   ×   α   ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         +     2   ×     α   2     ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         -     MAW   ⁢           ⁢   w   ×   MAX         =       0   ⁢     
     ⁢         (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       -     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢     B   2       -     2   ×   max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]         +     2   ×   min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B   ×     Y   ⁡     [   i   ]           )     ×     α   2       +     2   ×     (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B       )     ×     Y   ⁡     [   i   ]       ×   α     -     MAW   ⁢           ⁢   w   ×   MAX         =       0   ⁢     
     ⁢     (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B       )     ×     (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     +     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     2   ×     Y   ⁡     [   i   ]           )     ×     α   2       +     2   ×     (       max   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B     -     min   ⁢           ⁢   R   ⁢           ⁢   G   ⁢           ⁢   B       )     ×     Y   ⁡     [   i   ]                     )     ×   α     -     MAW   ⁢           ⁢   w   ×   MAX       =   0         
It should be noted, however, that fg(x,g)=(x/MAX) g ×MAX.
 
     Denoting the second term by a, the first term by 2×b, and the constant term by c in the final formula of the transformation yields Formulae (30) to (32), which are expressed by Formula (135):
 
 a×α   2 +2 ×b×α+c= 0.  (135)
 
     First, in cases where a=0, Formula (33) is obtained from Formula (135). 
     On the other hand, in cases where a≠0, the formula for the solution of a quadratic equation yields:
 
 a={−−b±√b   2   −a×c )}/ a.  
 
Denoting the square-root term by d yields:
 
α=(− b±√d )/ a.  
 
Although the above formula indicates that there two solutions of α, it is judged which one of the solutions is the right solution.
 
     First, denoting the left-hand side of Formula (135) by f(α) yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         ⁡ 
                         
                           ( 
                           α 
                           ) 
                         
                       
                       = 
                       
                         
                           a 
                           × 
                           
                             α 
                             2 
                           
                         
                         + 
                         
                           2 
                           × 
                           b 
                           × 
                           α 
                         
                         + 
                         c 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       in 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       which 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       case 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       = 
                       
                         c 
                         &lt; 
                         0 
                       
                     
                     , 
                     and 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             a 
                             + 
                             
                               2 
                               × 
                               b 
                             
                             + 
                             c 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     max 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   - 
                                   
                                     min 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                 
                                 ) 
                               
                               × 
                               
                                 ( 
                                 
                                   
                                     max 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   + 
                                   
                                     min 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   - 
                                   
                                     2 
                                     × 
                                     
                                       Y 
                                       ⁡ 
                                       
                                         [ 
                                         i 
                                         ] 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               2 
                               × 
                               
                                 Y 
                                 ⁡ 
                                 
                                   [ 
                                   i 
                                   ] 
                                 
                               
                               × 
                               
                                 ( 
                                 
                                   
                                     max 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   - 
                                   
                                     min 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               MAX 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               w 
                               × 
                               MAX 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               max 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                             - 
                             
                               min 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                             - 
                             
                               MAW 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               w 
                               × 
                               MAX 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     max 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       / 
                                       MAX 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               × 
                               
                                 MAX 
                                 2 
                               
                             
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     min 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     G 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       / 
                                       MAX 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               × 
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               MAX 
                               2 
                             
                             - 
                             
                               MAX 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               w 
                               × 
                               MAX 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       max 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         B 
                                         / 
                                         MAX 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 × 
                                 MAX 
                               
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       min 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         B 
                                         / 
                                         MAX 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 × 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               MAX 
                               - 
                               
                                 MAX 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 w 
                               
                             
                             } 
                           
                           × 
                           MAX 
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   max 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   B 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                                 - 
                                 
                                   min 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   B 
                                 
                                 - 
                                 
                                   MAX 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   w 
                                 
                               
                               ) 
                             
                             × 
                             
                               MAX 
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   136 
                   ) 
                 
               
             
           
         
       
     
     Since it is found from Formula (4) that f(1)&gt;0 and f(α) is a quadratic equation, there surely exists a solution between 0 and 1 according to f(0)&lt;0 and f(1)&gt;0. 
     When a&gt;0, Formula (136) gives a downwardly convex parabola. Therefore, according to f(0)&lt;0 and f(1)&gt;0, the bigger one of the solutions of α=(−b±√d)/a is a solution between 0 and 1. Accordingly, the bigger solution obtained when a&gt;0 is expressed as α=(−b+√d)/a, i.e., Formula (34). 
     On the other hand, when a&lt;0, Formula (136) gives an upwardly convex parabola. Therefore, according to f(0)&lt;0 and f(1)&gt;0, the smaller one of the solutions of α=(—b±·d)/a is a solution between 0 and 1. Accordingly, the smaller solution obtained when a&lt;0 is similarly expressed as α=(−b+√d)/a, i.e., Formula (34). 
     That is, when a≠0, α is expressed by Formula (34). 
     The following shows an example of calculation of a backlight value according to Embodiment 12. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255 and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit and the calculation of the maximum and minimum values of a gamma-corrected RGB signal are performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5, and the maximum and minimum values maxRGBg and minRGBg of the gamma-corrected RGB signal are set as maxRGBg=255 and minRGBg=16, respectively. 
     Since the target pixel satisfies Formula (4), the target pixel is subjected to color-saturation conversion. At this time, a color-saturation conversion rate α is calculated in the following manner. First, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in Embodiment 1 to take on a value of 207. 
     Substituting MAX=255, maxRGB=255, minRGB=63, Y[i]=207, and MAXw=127.5 in Formulae (30) to (32) yields coefficients (a, b, c) for use in calculation of α as follows: 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         + 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           2 
                           × 
                           
                             Y 
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         255 
                         - 
                         63 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         255 
                         + 
                         63 
                         - 
                         
                           2 
                           × 
                           207 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     - 
                     18432 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   b 
                   = 
                   
                     
                       Y 
                       ⁡ 
                       
                         [ 
                         i 
                         ] 
                       
                     
                     × 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     207 
                     × 
                     
                       ( 
                       
                         255 
                         - 
                         63 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   39744 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       - 
                       MAX 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     w 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       - 
                       127.5 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   
                     - 
                     
                       32512.5 
                       . 
                     
                   
                 
               
             
           
         
       
     
     Since a≠0, it is found from Formula (34) that: 
                   α   =       (       -   b     ±     √   d       )     /   a                 =       (       -   39744     +     √   980315136       )     /     (     -   18432     )                   =     0.457575   .                 
Thus, the present embodiment can calculate α without a loop process.
 
     Once the color-saturation conversion rate α is thus calculated, the RGBW transmission amounts and the backlight value are calculated according to the same procedure as in Embodiment 1. 
     That is, the saturation-converted input RGB signal is calculated according to Formulae (6) to (8) as follows: 
     
       
         
           
             
               
                 
                   
                     Rs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         R 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       159 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   185 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         G 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       255 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   229 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bs 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       α 
                       × 
                       
                         B 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                       × 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       0.457575 
                       × 
                       63 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           0.457575 
                         
                         ) 
                       
                       × 
                       207 
                     
                   
                 
               
             
             
               
                 
                   = 
                   141. 
                 
               
             
           
         
       
     
     Next, the gamma-corrected input RGB signal is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     Rsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             Rs 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           / 
                           Max 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           185 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   134 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Gsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             Gs 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           / 
                           Max 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           299 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   206 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Bsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             Bs 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                           / 
                           Max 
                         
                         ) 
                       
                       γ 
                     
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ( 
                         
                           141 
                           / 
                           255 
                         
                         ) 
                       
                       2 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   78. 
                 
               
             
           
         
       
     
     Next, the W transmission amount is calculated according to Formula (55) as follows: 
     
       
         
           
             
               
                 
                   
                     Wtsg 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sg 
                             / 
                             2 
                           
                         
                         , 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sg 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           206 
                           / 
                           2 
                         
                         , 
                         78 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   78. 
                 
               
             
           
         
       
     
     Next, the RGB transmission amounts are calculated according to Formulae (56) to (58) as follows:
 
 Rtsg[ 1]= Rsg[ 1]− Wtsg[ 1]=134−78=56
 
 Gtsg[ 1]= Gsg[ 1 ]−Wtsg[ 1]=206−78=128
 
 Btsg[ 1 ]=Bsg[ 1 ]−Wtsg[ 1]=78−78=0.
 
     Finally, the backlight value is calculated according to Formula (59) as follows: 
     
       
         
           
             
               
                 
                   Wbsg 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         
                           Rtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Gtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Btsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         , 
                         
                           Wtsg 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     max 
                     ⁡ 
                     
                       ( 
                       
                         56 
                         , 
                         128 
                         , 
                         0 
                         , 
                         56 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   128. 
                 
               
             
           
         
       
     
     That is, the backlight value obtained in cases where the color-saturation conversion process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     Embodiment 13 
     Embodiment 13 is a fifth example of means for approximately calculating a color-saturation conversion rate α in the second color-saturation conversion section  33  of Embodiment 2, and makes such a change to Embodiment 12 as to perform a color-saturation conversion process on every pixel within the input image. It should be noted that processing sections identical to those of Embodiments 2 and 11 are given the same reference numerals as those of Embodiments 2 and 11 and will not be described below in detail. 
       FIG. 33  shows another arrangement of the color-saturation conversion section  11  of Embodiment 13. The color-saturation conversion section  11  of  FIG. 33  is obtained by replacing the second color-saturation conversion rate calculation section  33  of the color-saturation conversion section  11  (see  FIG. 12 ) of Embodiment 2 with a twelfth color-saturation conversion rate calculation section  143 . 
     The twelfth color-saturation conversion rate calculation section  143  calculates a color-saturation conversion rate α from an input RGB signal and a backlight upper limit according a direct solving method, and then outputs the color-saturation conversion rate α. 
       FIG. 34  is a flow chart for explaining the operation of the twelfth color-saturation conversion rate calculation section  143 . 
     First, in Step S 141 , the twelfth color-saturation conversion rate calculation section  143  calculates a coefficient for use in calculation of α according to Formulae (30) to (32). Step S 141  of  FIG. 34  is identical to Step S 141  of  FIG. 32 . 
     Next, if the coefficient a for use in calculation of α is 0, the twelfth color-saturation conversion rate calculation section  143  calculates a according to Formula (33). If not so, the twelfth color-saturation conversion rate calculation section  143  calculates a according to Formula (34) (S 152 ). However, it is meant in the case of use of Formula (33) that when b is 0, the target pixel has a color saturation of 0. Therefore, a may take on any value. In the present embodiment, when b is 0, α is set, for example, as α=1. 
     The following explains how to derive the above calculating formulae. 
     First, transforming Formula in the same manner as in Embodiment 12 by substituting γ=2 in Formula (5) yields Formulae (30) to (32) and Formula (135). Further, in cases where a=0, Formula (33) is obtained from Formula (135). On the other hand, in cases where a≠0, the formula for the solution of a quadratic equation yields:
 
α=(− b±√d )/ a.  
 
The calculating formulae have been derived so far in the same manner as in Embodiment 12.
 
     At this point of time, it is indicated that there are two solutions of α. Either of the solutions is chosen according to the following way of thinking. 
     First, denoting the left-hand side of Formula (135) by f(α) yields Formula (136), in which case the following relationship always holds:
 
 f (0)= c&lt; 0.
 
     On the other hand, f(1) is expressed as:
 
 f (1)=(max RGBg −min RGBg −MAX w )×MAX.
 
     In cases where MAXw&lt;maxRGBg−minRGBg, f(1)&gt;0. This is the same case as in Formula (4), and the twelfth color-saturation conversion rate calculation section  143  performs a process in such a direction as to cause a reduction in color saturation. Therefore, the desired value of α falls within a range of 0&lt;α&lt;1. 
     Meanwhile, since f(α) is a quadratic equation, there surely exists a solution between 0 and 1 according to f(0)=c&lt;0 and f(1)&gt;0. 
     When a&gt;0, Formula (136) gives a downwardly convex parabola. Therefore, according to f(0)&lt;0 and f(1)&gt;0, the bigger one of the solutions of α=(−b±√d)/a is a solution between 0 and 1. Accordingly, the bigger solution obtained when a&gt;0 is expressed as α=(−b+√d)/a, i.e., Formula (34). 
     On the other hand, when a&lt;0, Formula (136) gives an upwardly convex parabola. Therefore, according to f(0)&lt;0 and f(1)&gt;0, the smaller one of the solutions of α=(−b±√d)/a is a solution between 0 and 1. Accordingly, the smaller solution obtained when a&lt;0 is similarly expressed as α=(−b+√d)/a, i.e., Formula (34). 
     That is, in cases where MAXw&lt;maxRGBg−minRGBg, α is expressed by Formula (34) when a≠0. 
     On the other hand, in cases where maxRGBg−minRGBg≦MAXw, f(1)≦0. This is a case opposite to Formula (4), and the twelfth color-saturation conversion rate calculation section  143  performs a process in such a direction as to cause no change or an increase in color saturation. Therefore, the desired value of α falls within a range of 1≦α. Furthermore, in this case, f(0)&lt;f(1). 
     When a&gt;0, Formula (136) gives a downwardly convex parabola. Therefore, according to f(0)&lt;0 and f(0)&lt;f(1), one of the solutions of α=(−b±√d)/a is negative and the other is a value of not less than 1. In this case, the value of not less than 1, i.e., the bigger solution is the desired value of α. Accordingly, the bigger solution obtained when a&gt;0 is expressed by Formula (34). 
     On the other hand, when a&lt;0, Formula (136) gives an upwardly convex parabola. Therefore, according to f(0)&lt;0 and f(0)&lt;f(1), both the solutions of α=(−b±√d)/a are solutions of not less than 1. In this case, the solution closer to 1, i.e., the smaller solution is adopted. Accordingly, the smaller solution obtained when a&lt;0 is similarly expressed by Formula (34). 
     That is, also in cases where maxRGBg−minRGBg≦MAXw, α is expressed by Formula (34) when a≠0. 
     The following shows a first example of calculation of a backlight value according to Embodiment 13. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 255, 63). That is, the following shows an example where the same pixel values are used as in the example of calculation of a backlight value according to Embodiment 1. It should be noted that the following explanation assumes that MAX=255 and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the example explained in Embodiment 1. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 239. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     First, the calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. 
     Since Embodiment 13 does not judge whether or not it is necessary to reduce the color saturation, Embodiment 13 calculates a color-saturation conversion rate α for every pixel. In Embodiment 13, the color-saturation conversion rate α is calculated in the same manner as in Embodiment 12. In this case, since 0≦α≦1, α after the limiter process is also calculated as α=0.457575 as in Embodiment 12. 
     Subsequently, the backlight value is calculated under the same procedure as in Embodiment 12. That is, the backlight value obtained in cases where the color-saturation process is performed is 128. Thus, the above example shows that the color-saturation conversion process can reduce the backlight value from 239 to 128 (reduction of approximately 46%). 
     The following shows a second example of calculation of a backlight value according to Embodiment 13. For convenience of explanation, the following shows an example where an image constituted by a single pixel is inputted and the input image has pixel values (R[1], G[1], B[1])=(159, 187, 85). That is, the following shows an example where the same pixel values are used as in the second example of calculation of a backlight value according to Embodiment 2. It should be noted that the following explanation assumes that MAX=255 and B1 Ratio=0.5. 
     First, in cases where no color-saturation conversion process is performed, a backlight value is calculated in the same manner as in the second example of calculation explained in Embodiment 2. That is, the backlight value obtained in cases where no color-saturation conversion process is performed is 109. 
     On the other hand, in cases where a color-saturation conversion process is performed, a backlight value is calculated in the following manner. 
     The calculation of a backlight upper limit is performed in the same manner as in the example explained in Embodiment 1. That is, the backlight upper limit MAXw is set as MAXw=127.5. Further, the luminance Y[1] corresponding to the input pixel values is calculated in the same manner as in the second example of calculation of Embodiment 2 to take on a value of 167.25. 
     Substituting MAX=255, maxRGB=187, minRGB=85, Y[i]=167.25, and MAXw=127.5 in Formulae (30) to (32) yields coefficients (a, b, c) for use in calculation of α as follows: 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         + 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           2 
                           × 
                           
                             Y 
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         187 
                         - 
                         85 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         187 
                         + 
                         85 
                         - 
                         
                           2 
                           × 
                           167.25 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     - 
                     6375 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   b 
                   = 
                   
                     
                       Y 
                       ⁡ 
                       
                         [ 
                         i 
                         ] 
                       
                     
                     × 
                     
                       ( 
                       
                         
                           max 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                         - 
                         
                           min 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     167.25 
                     × 
                     
                       ( 
                       
                         187 
                         - 
                         85 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   17059.5 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       - 
                       MAX 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     w 
                     × 
                     MAX 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       - 
                       127.5 
                     
                     × 
                     255 
                   
                 
               
             
             
               
                 
                   = 
                   
                     - 
                     
                       32512.5 
                       . 
                     
                   
                 
               
             
           
         
       
     
     Since a≠0, it is found from Formula (34) that: 
                   α   =       (       -   b     ±     √   d       )     a                 =       (       -   17059.5     +     √   83759352.75       )       (     -   6375     )                   =     1.24039   .                 
Thus, the present embodiment can calculate a without a loop process.
 
     In the present example of calculation, since a exceeds 1, a limiter process is performed so that α=1. 
     Thus, in cases where α=1, there are no differences in RGB values between before and after color-saturation conversion according to Formulae (6) to (8), with the same result as in the case where no color-saturation conversion (color-saturation reduction) is performed. 
     That is, in cases where a backlight value obtained in cases where no color-saturation conversion is performed falls short of a backlight upper limit, no color-saturation conversion is needed. Also in the present example of calculation, no color-saturation conversion is performed. Accordingly, the backlight value is 109 as in the case where no color-saturation conversion is performed. 
     Basically, the present liquid crystal display device has a single backlight  17  provided for a plurality of pixels. For this reason, for example, the liquid crystal display device of  FIG. 1  is arranged such that a single white backlight  17  corresponds to the entire display screen of the liquid crystal panel  15 . However, the present invention is not limited to this. The display screen of the liquid crystal panel  15  may be divided into a plurality of areas, and a plurality of backlights may be provided so that it becomes possible to adjust the luminance of the backlight of each area. 
       FIG. 35  shows an example where a single display area has two white backlights. It should be noted, however, that the number of backlights is not limited. 
     The liquid crystal display device of  FIG. 35  includes a color-saturation conversion section  11 , a gamma-correction section  12 , an input signal division section  151 , output signal generation sections  13   a  and  13   b , liquid crystal panel control sections  14   a  and  14   b , a liquid crystal panel  15 , backlight control sections  16   a  and  16   b , and white backlights  17   a  and  17   b.    
     The input signal division section  151  splits, into signals corresponding to two areas, a gamma-corrected input RGB signal, inputted from the gamma-correction section  12 , which corresponds to a single screen, and inputs the RGB input signals to the output signal generation sections  13   a  and  13   b  for the respective areas. Each of the output signal generation sections  13   a  and  13   b  performs, on a corresponding one of the areas, a process equivalent to the process performed by the output signal generation section  13  of  FIG. 1 . 
     Each of the liquid crystal panel control sections  14   a  and  14   b  performs, on a corresponding one of the areas, a process equivalent to the process performed by the liquid crystal panel control section  14  of  FIG. 1 . Bach of the control sections controls the transmittance of a pixel positioned so as to correspond to that area of the liquid crystal panel  15  which corresponds to the control section. 
     Each of the backlight control sections  16   a  and  16   b  performs, on a corresponding one of the areas, a process equivalent to the process performed by the backlight control section  16  of  FIG. 1 . Each of the white backlights  17   a  and  17   b  is identical in structure to the backlight  17 . Each of the backlights illuminates a corresponding one of the areas. 
     By thus dividing a single screen into a plurality of areas and controlling each area, the backlight value can be further reduced. It should be noted that although the present embodiment divides a single screen into two areas, it is also possible to divide a single screen into three or more areas and control each area. 
     In a general image, similar colors tend to be contiguous in a neighborhood area. Thus, dividing the backlight area as shown in  FIG. 35  makes it possible to further darken the backlight for an area where dark pixels gather. Accordingly, the amount of electricity that is used by the entire backlight is better reduced when the backlight is divided than when the backlight is not divided. 
     The processes that are performed by the color-saturation conversion section  11 , the gamma-correction section  12 , and the output signal generation section  13  can be realized by software operable on personal computers. The following describes a procedure for realizing the processes by software. 
       FIG. 36  is a system configuration in which the processes are realized by software. The system is constituted by a PC main unit  161  and an input-output device  165 . Further, the PC main unit  161  includes a CPU  162 , a memory  163 , and an input-output interface  164 . The input-output device  165  includes a storage medium  166 . 
     The CPU  162  controls the input-output device  165  via the input-output interface  164 . The CPU  162  reads out, from the storage medium  166 , programs for color-saturation reduction, gamma correction, and output signal generation, parameter files (e.g., the upper limit of an input RGB signal, the backlight value determination ratio, area information for use in division of a single screen into a plurality of areas), and input image data, and stores them in the memory  163 . 
     Further, the CPU  162  reads out, from the memory  163 , the programs for color-saturation reduction, gamma correction, and output signal generation, the parameter files, and the input image data. In accordance with each command of the programs for color-saturation reduction, gamma correction, and output signal generation, the CPU  162  performs color-saturation reduction, gamma correction, and output signal generation on the input image data thus inputted, and then controls, via the input-output interface  164 , the input-output device  165  to output, to the storage medium  166 , the backlight value and the RGBW transmittances obtained after the generation of the output signal. 
     Alternatively, as shown in  FIG. 37 , the CPU  162  outputs the backlight value obtained after the generation of the output signal to the backlight control section  16  and the RGBW transmittances to the liquid crystal panel control section  14  via the input-output interface  164 , thereby controlling the white backlight  17  and the liquid crystal panel  15  so that an image is actually displayed. 
     Thus, the system makes it possible to perform the aforementioned color-saturation reduction, the aforementioned gamma correction, and the aforementioned output signal generation on a personal computer. This makes it possible to confirm the validity of a method for reducing color saturation and a method for generating an output signal and the effect of reduction in backlight value before actually manufacturing a color-saturation conversion section and an output signal generation section by way of trial. 
     In order to solve the foregoing problems, a transmissive liquid crystal device according to the present invention includes: a liquid crystal panel whose pixels are each divided into four subpixels, namely red (K), green (G), blue (B), and white (W) subpixels; a white active backlight whose light emission luminance is controllable; a color-saturation conversion section that performs a color-saturation reduction process on pixel data high in luminance and color saturation, from among pixel data contained in a first input RGB signal serving as an input image, and thereby converts the first input RGB signal into a second input RGB signal; a gamma-correction section that converts the second input RGB signal into a third input RGB signal by performing a gamma-correction process on the second input RGB signal; an output signal generation section that generates, from the third input RGB signal, respective transmittance signals of the R, G, B, and W subpixels of each pixel of the liquid crystal panel and calculates a backlight value of the active backlight; a liquid crystal panel control section that drives and controls the liquid crystal panel in accordance with the transmittance signals generated by the output signal generation section; and a backlight control section that controls the light emission luminance of the backlight in accordance with the backlight value calculated by the output signal generation section. 
     Alternatively, in order to solve the foregoing problems, another transmissive liquid crystal device according to the present invention includes: a liquid crystal panel whose pixels are each divided into four subpixels, namely red (R), green (G), blue (B), and white (W) sub pixels; a white active backlight whose light emission luminance is controllable; a color-saturation conversion and gamma-correction section that performs a color-saturation reduction process and a gamma-correction section in this order on pixel data high in luminance and color saturation, from among pixel data contained in a first input RGB signal serving as an input image, and converts the first input RGB signal into a third input RGB signal by performing a gamma-correction process on pixel data low in at least either luminance or color saturation; an output signal generation section that generates, from the third input RGB signal, respective transmittance signals of the R, G, B, and W subpixels of each pixel of the liquid crystal panel and calculates a backlight value of the active backlight; a liquid crystal panel control section that drives and controls the liquid crystal panel in accordance with the transmittance signals generated by the output signal generation section; and a backlight control section that controls the light emission luminance of the backlight in accordance with the backlight value calculated by the output signal generation section. 
     According to the foregoing arrangement, the use of a liquid crystal panel whose pixels are each divided into four subpixels, namely R, G, B, and W subpixels, makes it possible to transfer a part of each of the R, G, and B components to the subpixel W, which suffers no loss (or little loss) of light due to absorption by a filter. This makes it possible to reduce the amount of light that is absorbed by the color filter and to reduce the backlight value accordingly, thereby enabling a reduction in the amount of electricity that is used by the transmissive liquid crystal display device. 
     Furthermore, the backlight value can be more surely reduced by calculating the backlight value and the RGBW transmittances after performing the color-saturation reduction process on the first input RGB signal, which is an original input. 
     Further, as for the color-saturation reduction process and the gamma-correction process, the gamma-correction process is performed after the color-saturation reduction process. In cases where the color-saturation reduction process and the gamma-correction process are performed in an inverse order, there is a difference in luminance between before the processes and after the processes. However, since the gamma-correction process is performed after the color-saturation reduction process, the luminance after the processes stays the same as the luminance before the processes. 
     Further, the transmissive liquid crystal display device is preferably arranged such that the color-saturation conversion section reduces only the color saturation of the pixel data on which the color-saturation reduction process is performed, without causing a difference in luminance and hue between before and after the color-saturation reduction process. 
     The foregoing arrangement does not cause a change in luminance and hue, both of which exert a great influence on the visual property of a human, but causes a reduction only in saturation, which exerts only a minor influence on the visual property, thereby enabling the suppression of deterioration in image quality due to the color-saturation reduction process. 
     Further, the transmissive liquid crystal display device is preferably arranged such that the color-saturation conversion section is able to change a degree of the color-saturation reduction process. 
     The foregoing arrangement allows a user to selectively set a balance between a power consumption reduction effect that is brought about by the color-saturation reduction process and deterioration in image quality due to the color-saturation reduction process. 
     Further, the transmissive liquid crystal display device may be arranged so as to include a plurality of active backlights with respect to the liquid crystal panel, wherein the transmissive liquid crystal display device controls a transmittance of the liquid crystal panel and a backlight value of a backlight for each of those areas respectively corresponding to the active backlights. 
     According to the foregoing arrangement, the backlight is divided into areas so that the backlight value can be suitably set for each of the areas of the backlight thus divided, and that the entire amount of electricity that is used by the backlight can be reduced. 
     The embodiments and concrete examples of implementation discussed in the foregoing detailed explanation serve solely to illustrate the technical details of the present invention, which should not be narrowly interpreted within the limits of such embodiments and concrete examples, but rather may be applied in many variations within the spirit of the present invention, provided such variations do not exceed the scope of the patent claims set forth below.