Abstract:
Methods and apparatus are provided for performing reversible color conversion on digital image data using integer computer arithmetic. This color conversion provides an approximation of the luminance component to the luminance as perceived by the human visual system with any necessary precision (Kd) and without multiplication and division operations other than shifts. The color conversion can be used in conjunction with a lossless or lossy compression/decompression process. Other embodiments, including non-reversible and non-integer color conversion, are also provided.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   The present application is a division of U.S. patent application Ser. No. 10/060,606 filed on Jan. 29, 2002, now U.S. Pat. No. 6,934,411 incorporated herein by reference. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to image conversion between the RGB (Red, Green Blue) color representation and another color representation. Some embodiments of the invention are suitable for use with lossless and lossy data compression/decompression systems for image data. 
   2. Related Art 
   In many applications that carry out color image processing, it is necessary to convert data between two color spaces. Many modern devices that deal with digital color images, such as computer monitors and TV equipment, use RGB (Red, Green, Blue) color representation. However, if we need to use compression for storing or transmitting large amounts of color image data, the RGB representation is not optimal. Usually data is converted from the RGB color coordinate system to a TV standard (YC b C r , YUV, YIQ) prior to compression. All of these TV standard systems provide luminance and chrominance separation. The change of chrominance is less perceptible to the human eye than the change of luminance, and some lossy compression techniques take advantage of this fact. 
   For example, the JPEG2000 standard uses the YC b C r  color coordinate system for lossy compression. JPEG2000 is a shorthand for the Joint Photographic Experts Group JPEG 2000 standard for coding of still pictures, described in “JPEG 2000 Final Committee Draft version 1.0,” 16 Mar. 2000, incorporated herein by reference. The YC b C r  system is defined in terms of R, G, and B by the following matrix equation: 
             [         Y             C   b               C   r           ]     =       [         0.299       0.587       0.114             -   0.1687           -   0.3313         0.5           0.5         -   0.4187           -   0.0813           ]     ⁡     [         R           G           B         ]             
Y is the luminance component, and C b  and C r  are chrominance components. The matrix equation implies that the luminance Y is given by the following equation:
   Y= 0.299 R+ 0.587 G+ 0.114 B   (1) 
According to studies of the human visual perception, this equation is a good representation of the luminance as perceived by the human eye. Below, this representation is referred to as “the model of human visual system” and used as a standard against which other luminance representations are measured.
 
   In digital image processing, the color at each pixel is usually represented by integer numbers. Thus, the data R, G, B, and the luminance and chrominance components are usually formatted as integers. However, the coefficients of the YC b C r  transformation are not integers and actual computer implementations of the RGB to YC b C r  conversion are not reversible because of numeric precision problems and an accumulation of errors. 
   When color conversion is used in conjunction with lossless compression, it is typically expected that the R, G, and B data can be recovered precisely (through decompression and reverse conversion), so the color conversion should be reversible. This is why the JPEG2000 standard uses a different color representation for lossless compression, namely: 
                   [         Y           U           V         ]     =       [         0.25       0.5       0.25           1         -   1         0           0         -   1         1         ]     ⁡     [         R           G           B         ]               (   2.1   )               
The luminance component in this representation is:
 
                 Y   =         1   4     ⁢   R     +       1   2     ⁢   G     +       1   4     ⁢   B               (   2.2   )               
The chrominance components are:
   U=R−G   (2.3)   V=B−G   (2.4) 
   It has been shown in U.S. Pat. No. 5,731,988 issued Mar. 24, 1998 to Zandi et al. that even though the Y coefficients in equation (2.2) are not integers, the transformation (2.1) is reversible. 
   Therefore, to perform both lossy and lossless compression according to the JPEG2000 standard, the compression system has to contain color converters of both types, i.e. both a YC b C r  converter and a converter for the transformation (2.1). 
   U.S. Pat. No. 6,044,172, issued Mar. 28, 2000 to Allen, discloses another reversible color conversion with a luminance that is closer to the model (1) than the luminance (2.2), namely: 
   
     
       
         
           
             
               
                 Y 
                 = 
                 
                   ⌊ 
                   
                     
                       
                         3 
                         ⁢ 
                         R 
                       
                       + 
                       
                         6 
                         ⁢ 
                         G 
                       
                       + 
                       B 
                     
                     10 
                   
                   ⌋ 
                 
               
             
             
               
                 ( 
                 3.1 
                 ) 
               
             
           
           
             
               
                 U 
                 = 
                 
                   R 
                   - 
                   G 
                 
               
             
             
               
                 ( 
                 3.2 
                 ) 
               
             
           
           
             
               
                 V 
                 = 
                 
                   B 
                   - 
                   G 
                 
               
             
             
               
                 ( 
                 3.3 
                 ) 
               
             
           
         
       
     
   
   Here └x┘ symbolizes a floor function and indicates the largest integer not exceeding “x”. 
   The transformation (3.1) is however more computationally expensive than (2.1). 
   SUMMARY 
   The invention is defined by the appended claims, which are incorporated into this section by reference. The following is a summary of some features of the invention. 
   The invention provide methods and apparatus for color conversion. In some embodiments, the color conversion is reversible, the luminance is close to the model (1), and the conversion is computationally inexpensive, it can be performed without any division or multiplication operations other than shifts. Some embodiments are suitable for use with both lossless and lossy compression/decompression processes. (The invention is not limited to such embodiments however. Also, the color conversion provided herein can be used not in conjunction with compression, but for any other purpose, whether known or not at this time.) 
   As discussed above, if a color conversion is to be used with lossy compression, the luminance coefficients should be close to the model representation (1). These coefficients are typically chosen so that their sum is equal to 1 in order for the number of bits in the Y representation be about the same as the number of bits in the R, G and B representation. For the chrominance components, the equations (2.3), (2.4) are suitable. Given these equations and the condition that the sum of the Y coefficients is 1, the color conversion can be defined by the following matrix: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         α 
                       
                       
                         
                           1 
                           - 
                           α 
                           - 
                           β 
                         
                       
                       
                         β 
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   0 
                   ≤ 
                   α 
                   ≤ 
                   1 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   0 
                   ≤ 
                   β 
                   ≤ 
                   1 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   0 
                   ≤ 
                   
                     α 
                     + 
                     β 
                   
                   ≤ 
                   1 
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   If α=0.299 and β=0.114, the model equation (1) is obtained for the luminance. If α=0.25 and β=0.25, the JPEG2000 transformation (2.1) is obtained. 
   For any given α and β, the degree to which the luminance differs from the model luminance (1) can be measured as: 
   
     
       
         
           
             
               
                 
                   K 
                   d 
                 
                 = 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       α 
                                       - 
                                       0.299 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       α 
                                       - 
                                       β 
                                       - 
                                       0.587 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     β 
                                     - 
                                     0.114 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                             0.299 
                             2 
                           
                           + 
                           
                             0.587 
                             2 
                           
                           + 
                           
                             0.114 
                             2 
                           
                         
                       
                     
                     · 
                     100 
                   
                   ⁢ 
                   % 
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   K d  is a scaled magnitude of the difference between the vector (α, 1−α−β, β) and the corresponding vector (0.299, 0.587, 0.114) for the model luminance (1). 
   Transformation (2.1) has K d =25.24%. The K d  value for transformation (3.1, 3.2, 3.3) is only 2.86%, but, as mentioned, this transformation is more computationally expensive. 
   Denoting the chrominance components as R g , B g , the transformation (4) can be written as follows:
 
 Y=α*R +(1−α−β) *G+β*B   (6.1)
 
 R   g   =R−G   (6.2)
 
 B   g   =B−G   (6.3)
 
   The reverse transformation is:
 
 R=Y +(1−α) *R   g   −β*B   g   (7.1)
 
 G=Y−α*R   g   −β*B   g   (7.2)
 
 B=Y−α*R   g +(1−β) *B   g   (7.3)
 
   It follows from equations (6.1), (6.2), and (6.3) that:
 
 Y=G +(α *R   g   +β*B   g )  (8)
 
   Let us assume that the components R, G, B, Y, and the chrominance components are formatted as integers. From the purely mathematical standpoint, the system of equations (6.1, 6.2, 6.3) and the system of equations (8, 6.2, 6.3) are equivalent to each other. However, if the color data are integers, modification of the order of operations can affect reversibility as explained below. Also note that the equation (8) has only two multiplications versus three multiplications in equation (6.1). 
   The forward transformation (6), i.e. the transformation (6.1)–(6.3), can be written as follows:
 
 R   g   =R−G;   (9.1)
 
 B   g   =B−G;   (9.2)
 
 P=αR   g   +βB   g ;  (9.3)
 
 Y=G+P.   (9.4)
 
Equations (9.1) through (9.4) involve only two multiplications and four additions, compared to three multiplications and four additions in equations (6).
 
   The reverse transformation (7), i.e. the transformation (7.1)–(7.3), can be written as follows:
 
 P=αR   g   +βB   g ;  (10.1)
 
 G=Y−P;   (10.2)
 
 R=R   g   +G;   (10.3)
 
 B=B   g   +G.   (10.4)
 
   The reverse transformation equations also have only two multiplications and four additions. 
   In some embodiments of the present invention, the forward and reverse transformations are implemented by performing a computation for each of the equations (9.1) through (9.4) and (10.1) through (10.4). In other words, to perform the forward transformation, the image processing system calculates R g  and B g  according to equations (9.1) and (9.2). Each of these values is calculated by an adder or a subtractor. From these values, the value for P is calculated according to equation (9.3), which requires two multiplications and an addition. Y is calculated according to equation (9.4) by an adder. Likewise, the reverse transformations are implemented by performing a computation for each of the equations (10.1) through (10.4). 
   Implemented this way, the forward transformation is reversible for any values of a and β whether or not the values P in equations (9.3), (10.1) are computed with a rounding error, as long as the rounding error is the same in both of these computations and the remaining forward and reverse computations are performed without any new rounding errors. (It should be noted that a transformation may be reversible or non-reversible depending on its implementation.) Indeed, suppose that the computation (9.3) involves some rounding error E, that is, instead of P the system generates PE=P+E. Then Y will be computed, according to equation (9.4), with the same error E, that is, instead of Y the system will generate YE=Y+E. 
   The reverse computation will start with YE, R g , and B g . P will be calculated according to equation (10.1), with the same rounding error E, so the system will generate PE=P+E. G will be generated as YE−PE according to equation (10.2). YE−PE=Y−P, so G will be recovered without an error. Therefore, R and B will also be recovered without an error in computations (10.3) and (10.4). 
   This reasoning applies when the values R, G, B, P, Y, R g , B g  are formatted as integers, and all of the computations are performed by integer computer arithmetic devices. This reasoning is also valid for non-integer values and computations. 
   Similar reasoning shows that the transformation (9), i.e., the transformation defined by equations (9.1) through (9.4), when implemented with a computation for each of the equations, is reversible when P is any function f(R g , B g ), whether or not the values R, G, B, P, Y, R g , B g  are integers. In other words, the following computations define a reversible color transformation:
 
 R   g   =R−G;   (11.1)
 
 B   g   =B−G;   (11.2)
 
 P=f ( R   g   ,B   g );  (11.3)
 
 Y=G+P.   (11.4)
 
The reverse transformation can be performed with the following computations:
 
 P=f ( R   g   ,B   g );  (12.1)
 
 G=Y−P;   (12.2)
 
 R=R   g   +G;   (12.3)
 
 B=B   g   +G.   (12.4)
 
   The reversibility is obtained even if the values P in equations (11.3), (12.1) are computed with a rounding error, as long as the rounding error is the same in both of these computations and the remaining forward and reverse computations are performed without any new rounding errors. 
   In the linear case (equations (9), (10)), computationally inexpensive color conversion, with a good approximation of the luminance component to the model (1), can be found by minimizing the value Kd for all of the possible values of α and β in the range [0,1 ] such that α+β is in the same range and each of the values of α and β is in the form k/2 n , where k and n are integers and n is some fixed value. Multiplications and divisions by k/2 n  can be implemented as shifts and additions, and often are computationally inexpensive. 
   The invention is not limited to the computations (9), (10), (11), (12), or reversible implementations. Other embodiments of the invention are described below. The invention is defined by the appended claims. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a flowchart of a color transformation process used in an embodiment of the present invention. 
       FIG. 2  is a block diagram of a circuit implementing a forward color conversion in accordance with an embodiment of the present invention. 
       FIG. 3  is a block diagram of a circuit implementing a reverse color conversion in accordance with an embodiment of the present invention. 
       FIGS. 4–9  are block diagrams of circuits, in accordance with embodiments of the present invention. 
   

   DESCRIPTION OF PREFERRED EMBODIMENTS 
     FIG. 1  is a color data processing flowchart. At step  104 , RGB data  102  is converted to the YR g B g  format. This step implements the transformation (11). The YR g B g  data are then compressed (step  106 ). The compressed data are stored in memory (step  108 ) or transmitted over a network (step  110 ) or both (the data can be stored in memory before and/or after transmission). The stored or transmitted data are decompressed (step  112 ) and converted back to the RGB format (step  114 ). Step  114  implements the transformation (12). The reconstructed RGB data are shown at step  116 . The data are then displayed by a suitable display system (not shown). 
   Process  100  can be performed by circuits specially designed for that purpose, or by one or more general purpose computers, or by a combination of the above. Both hardwired and software implementations are possible, as well as a combination of these implementation types. The software can be stored on a computer readable medium (a semiconductor memory, a disk, or some other medium) known or to be invented. 
   The signals R, G, B, R g , B g , P, Y are digital data signals. Each signal represents the corresponding value R, G, B, R g , B g , P, or Y in 8 bits or some other number of bits. In some embodiments, more bits are used for the values R g , B g  than for the values R, G, B in order to calculate the values R g , B g  without error. For example, if n bits are used for the values R, G, B, then n+1 bits can be used for the values R g , B g . The signal (−G) in equations (11.1), (11.2) can also be represented with n+1 bits. Signals R, G and B can be formatted as unsigned integers. The loss of precision in calculating the values P and Y is recoverable in the reverse transformation (12) as explained above. 
   In some integer embodiments, the data −G, R g , B g  are represented using only n bits. These data are calculated modulo 2 n . The transformation (9) is reversible since the data R, G, B are between 0 and 2 n −1 (inclusive). However, in some of such embodiments, the luminance Y could deviate more from the model (1). 
     FIG. 2  illustrates a circuit  200  implementing the forward transformation (11). Block  202  (an adder or a subtractor, for example) receives the n-bit digital data signals R and G and generates the n+1 bit digital data signals R g =R−G (equation (11.1)). Block  204  (an adder or a subtractor, for example) receives the n-bit data signals B and G and generates the n+1 bit data signal B g =B−G (equation (11.2)). Block  206  receives the data R g , B g  from blocks  202 ,  204  and generates the n+1 bit data signal P=f(R g ,B g ) (equation (11.3)). Block  208  (an adder, for example) receives the data P from block  206  and also receives the data G, and generates the n bit data signal Y=G+P (equation (11.4)). Signal (−G) is produced from unsigned integer signal G outside or inside of blocks  202 ,  204  depending on implementation. 
   Exemplary implementations of block  206  are described below with reference to  FIGS. 4 through 9 . 
     FIG. 3  shows a circuit  300  implementing the inverse transformation (12). Block  302  receives the n+1 bit data signals R g , B g  and generates the signal P=f(R g ,B g ) (equation (12.1)). The function f implemented by block  302  is the same function as implemented by block  206 . The same precision is used in blocks  206 ,  302  if reversibility is desired. Blocks  206 ,  302  can be identical to each other. 
   Block  304  (an adder or a subtractor, for example) receives the data signal P from block  302  and also receives the n-bit data Y, and generates the n-bit data signal G=Y−P (equation (12.2)). Block  306  (an adder, for example) receives the data G from block  304  and also receives the signal R g , and generates the n-bit data signal R=R g +G (equation (12.3)). Block  308  (an adder, for example) receives the signal G from block  304  and also receives the signal B g , and generates the n-bit signal B=B g +G (equation (12.4)). 
     FIG. 4  shows a circuit  400  which can implement either block  206  or block  302  ( FIGS. 2 and 3 , respectively) to generate the P signal when f(R g ,B g ) is a linear function. See equations (9.3), (10.1). Block  402  receives the n+1 bit data signal R g  and generates the signal α*R g . Block  404  receives the n+1 bit data signal B g  and generates the signal β*B g . Blocks  402 ,  404  can include fixed or floating point multipliers or dividers, shifters, or other circuits. Signals αR g , βB g  can be in integer, fixed or floating point format. For example, in some embodiments, each of these signals has a signed integer format. In another example, signals R g  and B g  have a signed integer format, signals α and β have a floating point format, and the blocks  402 ,  404  perform floating point computations. Block  406 , described below, may also perform a floating point computation. In some embodiments, the function of each of blocks  402 ,  404 ,  406  is performed by a computer processor, and the number of bits at the output of these blocks can be a standard number of bits for the particular processor (64 bits for example). 
   Block  406  (a floating-point adder, for example) receives the signals α*R g  and β*B g  from blocks  402 ,  404  and generates the sum signal α*R g +β*B g . Block  408  receives the signal α*R g +β*B g  from block  406 , rounds the value α*R g +β*B g  to an n+1 bit signed integer, and provides the n+1 bit signal P. 
   Table 1 below shows the values Kd, defined above, for the values of α and β chosen to minimize Kd given a certain number n of bits for representing each of the α and β parameters. Thus, each of α and β can be written as k/2 n . The number n, shown in the last column, is recommended as the minimum number of bits to represent R, G and B for obtaining a luminance Y close to the model (1). The rounding errors in the Y computation should be small for a good approximation of the model luminance (1). 
   
     
       
             
             
             
             
             
           
             
             
             
             
             
           
         
             
               TABLE 1 
             
             
                 
             
             
                 
                 
                 
                 
               Number of bits 
             
             
               α 
               1 − α − β 
               β 
               Kd 
               for α and β 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
                2/8 = 0.2500 
               ⅝ = 0.6250 
                ⅛ = 0.1250 
               9.42% 
               3 
             
             
                5/16 = 0.3125  
                9/16 = 0.5625  
                2/16 = 0.1250 
               4.50% 
               4 or 5 
             
             
                19/64 = 0.2969  
                38/64 = 0.5938  
                7/64 = 0.1094 
               1.26% 
               6 or 7 
             
             
                77/256 = 0.3008   
                150/256 = 0.5859   
                29/256 = 0.1133  
               0.33% 
               8 
             
             
                306/1024 = 0.2988    
                601/1024 = 0.5869    
                117/1024 = 0.1143   
               0.04% 
               10 
             
             
                 
             
           
        
       
     
   
     FIGS. 5 through 9  show circuits that can implement the blocks  206 ,  302  to generate the P signal for the embodiments of Table 1.  FIG. 5  shows a circuit  500  that generates the P signal for α= 2/8 and β=⅛. Shifter  502  receives the n+1 bit signal R g  and shifts R g  left by 1 bit, thus generating an n+2 bit signal R g *2. Adder  504  adds together the output of shifter  502  and the n+1 bit value B g  and a number 4 to generate an n+3 bits signal R g *2+B g +4. Shifter  506  shifts the output of adder  504  right by 3 bits, thus generating the signal P=R g /4+B g /8+½. The three least significant bits (LSBs) are dropped (the shift can be implemented simply by dropping the three LSBs). This corresponds to P being αR g +βB g  rounded to the nearest integer (as is well know, such rounding can be accomplished by adding ½ and dropping the fractional bits). 
     FIG. 6  shows a circuit  600  that generates the signal P for α= 5/16, β= 2/16. Shifter  602  shifts the n+1 bit data signal R g  left by 2 bits, thus generating an n+3 bit signal R g *4. Shifter  604  shift n+1 bit signal B g  left by 1 bit, thus generating an n+2 bit signal B g *2. Adder  606  adds together the outputs of shifters  602 ,  604 , number 8 and the data R g , thus generating an n+4 bit signal R g *5+B g *2+8. Number 8 is added for rounding to the nearest integer to improve luminance approximation as explained above in reference to  FIG. 5 . Shifter  608  shifts the output of adder  606  right by 4 bits dropping the four LSBs, generating an n bits signal P=R g * 5/16+B g * 2/16 rounded to the nearest integer. The shifting can be accomplished simply by dropping the four LSBs. 
     FIG. 7  shows a circuit  700  that generates the signal P for α= 19/64 and β= 7/64. Shifter  702  shifts the n+1 bit signal R g  left by 1 bit, to generate an n+2 bit signal R g *2. Shifter  704  shifts R g  left by 4 bits, to generate an n+5 bit signal R g *16. Shifter  706  shifts n+1 bits B g  left by 3 bits, to generate an n+4 bit signal B g *8. Adder  708  adds together the outputs of shifters  702 ,  704 ,  706 , number 32 (for rounding) and the data R g , and the adder subtracts B g . The output of adder  708  thus represents the n+6 bit value R g *19+B g *7+32. Shifter  710  shifts the output of adder  708  right by 6 bits (drops the six LSBs), thus generating the n bits signal P=R g * 19/64+B g * 7/64 rounded to the nearest integer. 
     FIG. 8  shows a circuit  800  that generates the signal P for α= 77/256 and β= 29/256. Shifter  802  shifts the n+1 bit signal R g  left by 2 bits, thus generating an n+3 bit signal R g *4. Shifter  804  shifts R g  left by 3 bits, thus generating an n+4 bit signal R g *8. Shifter  806  shifts R g  left by 6 bits, thus generating n+7 bit signal R g *64. 
   Shifter  808  shifts n+1 bits B g  left by 1 bit, thus generating an n+2 bits signal B g *2. Shifter  810  shifts B g  left by 5 bits, thus generating an n+6 bits signal B g *32. 
   Adder  812  receives the signals R g , B g  and the outputs of shifters  802 ,  804 ,  806 ,  808 , and  810 . Adder  812  inverts the signal B g  and the output of shifter  808 , and adds the inverted signals to R g  and the outputs of shifters  802 ,  804 ,  806 , and  810 , also adding a number 128 for rounding, thus generating an n+8 bit signal R g *77+B g *29+128. Shifter  814  shifts the output of adder  812  right by 8 bits (drops the eight LSBs), thus generating an n bit signal P=R g * 77/256+B g * 29/256 rounded to the nearest integer. 
     FIG. 9  shows a circuit  900  that generates the signal P for α= 306/1024 and β= 117/1024. Shifter  902  shifts the n+1 bit signal R g  left by 1 bit, thus generating an n+2 bit signal R g *2. Shifter  904  shifts R g  left by 4 bits, thus generating an n+5 bit signal R g *16. Shifter  906  shifts R g  left by 5 bits, thus generating an n+6 bit signal R g *32. Shifter  908  shifts R g  left by 8 bits, thus generating an n+9 bit signal R g *256. 
   Shifter  910  shifts n+1 bit signal B g  left by 2 bits, thus generating an n+3 bit signal B g *4. Shifter  912  shifts B g  left by 4 bits, thus generating an n+5 bit signal B g *16. Shifter  914  shifts B g  left by 7 bits, thus generating an n+8 bit signal B g *128. 
   Adder  916  receives the signal B g  and the outputs of shifters  902 ,  904 ,  906 ,  908 ,  910 ,  912 , and  914 . Adder  916  inverts the signal B g  and the outputs of shifters  910 ,  912 , and adds the inverted signals to the outputs of shifters  902 ,  904 ,  906 ,  908 ,  914  and to number 512 (for rounding), thus generating an n+10 bit signal R g *306+B g *117+512. Shifter  918  shifts the outputs of adder  916  right by 10 bits (drops the 10 LSBs), thus generating n bit signal P=R g * 306/1024+B g * 117/1024 rounded to the nearest integer. 
   The embodiments described above illustrate but do not limit the invention. The invention is not limited to the particular numbers of bits described above, or to integer computer arithmetic. Blocks  802 ,  804  and other shifters and adders described above can be replaced by circuits other than shifters and adders. These block can share circuitry. They can be implemented by software programmable computer processors. Numerous other modifications and variations are possible in accordance with the principles of the present invention. The invention is defined by the appended claims.