Method for image processing

A method for image processing to facilitate expression with more reality of sense of distance by blurring is provided. In the method according to the invention, image data having far and near distance information of the image are processed by a digital low-pass filter having a cut-off frequency corresponding to the distance information, so that a blurred image based on the distance information is formed.

FIELD OF THE INVENTION
 The invention relates to a method for image processing in a computer and a
 computer adapted thereto.
 BACKGROUND OF THE INVENTION
 One of the methods for expressing a sense of volume or distance is
 blurring. For example, when a photograph is taken, an object just in focus
 is taken clear and degree of blurring is greater as the object is more
 separated from the focus. It is known that this blurring provides a
 photograph with a sense of distance.
 One of the methods for expressing blurring by means of computer is
 dispersed beam tracing method. Another method to express blurring more
 simply is a method disclosed in the patent specifications of Toku-Kai-Hei
 6-36025 and Toku-Kai-Hei 2-190898 in which the diameter of a blurring
 circle around each pixel is calculated based on the information of
 distance for the pixel and, thereby, spreading area of blurring extended
 to neighboring pixels is determined to show the sense of distance.
 Calculation of blurring process is carried out with dots of 3 by 3 in
 Toku-Kai-Hei 2-190898 and with dots of 5 by 5 in Toku-Kai-Hei 6-36025.
 Of the methods referred to above, the dispersed beam tracing method
 requires a lot of time for calculation, such as for calculating on a
 multitude of beams to build up a blurred image related to one pixel. The
 method of calculating the effect of neighboring pixels on the pixel of
 interest can be carried out in shorter period than dispersed beam tracing
 method but it is associated with other problems.
 For example, in the method disclosed in Toku-Kai-Hei 6-36025, the effect of
 neighboring pixels is taken into account with the use of a kernel of 5 by
 5 to accomplish sense of distance, but pixels that can be considered with
 respect to the latter are those only two pixels remote from the pixel of
 interest. For calculation of data required to express a pixel, data on a
 kernel of 25 pixels including neighboring pixels have to be added after
 processing, resulting in heavy burden of calculation. 25 weight
 coefficients involved in the kernel have to be calculated for each pixel,
 thereby the burden of calculation increases. Weight coefficients for dots
 other than that in the center of the kernel are calculated by the
 following equation:
EQU fw.sub.i,j
 =.vertline.(1-fw.sub.33)/(5.times.5-1).vertline..times.df.sub.i,j
 The equation above can be transformed as follows:
EQU fw.sub.i,j =(df.sub.33.times.df.sub.i,j)/(5.times.5-1)
 This equation indicates multiplication of distance by distance. Therefore,
 repeating of filter processing seems to be needed, with the times of
 filtration set to control the degree of blurring. This repeated processing
 also increases burden of calculation.
 In the method of Toku-Kai-Hei 2-190898, effect of surrounding pixels is
 taken into account based on a mask register of 3.times.3, but it is the
 effect of only pixels directly adjacent to the pixel of interest that can
 be dealt with. Moreover, addition of data on mask registers of 9 pixels
 including surrounding pixels subsequent to the processing is required to
 accomplish the calculation of data for expressing a pixel of interest. In
 this method, it is rather easy to select a suitable mask register based on
 the distance information but it is difficult to design the contents of
 mask register (weight coefficients) in accordance with the distance.
 For a very far object, the effect of pixels remote as much as 10 pixels
 from the pixel of interest may contribute in view of characteristics of
 human vision. To encounter such an effect, the mask register has to be
 extended to 21.times.21, thereby addition of data on a mask register of
 441 pixels consisting of surrounding pixels subsequent to the processing
 is required to calculate the data for expressing blurring of a pixel of
 interest, resulting in an unreasonable burden of calculation. A remarkable
 difficulty is expected to arise also in designing the contents of masks
 register in accordance with distance. Thus, real expression of sense of
 distance is limited in practical application of the method.
 SUMMARY OF THE INVENTION
 Accordingly, it is an object of the invention to provide a method for
 processing image data capable of expressing sense of distance with more
 reality by blurring of the image and improved in the efficiency of
 processing, whereby the burden of calculation is decreased.
 It is another object of the invention to provide a method for processing
 image data capable of expressing sense of distance by blurring of the
 image having compatibility with a conventional rendering system.
 It is still another object of the invention to provide a digital computer
 adapted to the method for processing image data capable of expressing
 sense of distance with more reality by blurring of the image.
 According to the invention, a method for processing image data having far
 and near distance information comprises the step of:
 generating images including a blurred image in accordance with the far and
 near distance information by applying image processing by a digital low
 pass filter to the image data;
 wherein the digital low pass filter has a cut-off frequency corresponding
 to the far and near distance information.
 According to the invention, a method for processing image data having far
 and near distance information of images preferably comprises the step of:
 generating a blurred image based on an image selected from the images by
 applying image processing by a digital low pass filter to the image data
 of the selected image to remove higher frequency components therefrom;
 wherein the digital low-pass filter has a cut-off frequency lower than the
 highest frequency of frequency components in the image data of the
 selected image, represented by the formula;
EQU f.sub.d =kf.sub.m;
 f.sub.d representing the cut-off frequency, f.sub.m representing the
 highest frequency of frequency components in the image data to be
 processed contain and k representing a positive variable less than 1.
 According to the invention, a digital computer in which image data having
 far and near distance information are processed so as to generate images
 including a blurred image comprises: a digital low-pass filter for
 generating the blurred image based on the far and near distance
 information, having a cut-off frequency represented by the formula:
EQU f.sub.d =kf.sub.m :
 f.sub.d representing the cut-off frequency, f.sub.m representing the
 highest frequency of frequency components in the image data to be
 processed and k representing a positive variable less than 1; and
 a medium for recording a program to execute the process to generate the
 images including the blurred image. information means the information of
 distance between the object and the observer, including a camera such as
 still camera, TV camera etc.
 The effect of blurring is determined by controlling the highest frequency
 of frequency components contained in the distance information of the
 image. Concerning the control, the present invention permits assignment of
 the distance to a blurred image freely and quantitatively by approximating
 the characteristics of human vision by a digital filter having frequency
 characteristics in accordance with the distance information of the blurred
 image.
 The cut-off frequency f.sub.d of a digital low-pass filter used in the
 invented method is determined in relation to the highest frequency f.sub.m
 of frequency components which the original image contain, by the following
 equation:
EQU f.sub.d =kf.sub.m :
 where k is a positive variable not larger than 1. The variable k is the
 reciprocal of the ratio of the number of dots in the processed image of
 minimum discernible dimension to the number of dots in the original image
 of minimum discernible dimension.

DESCRIPTION OF THF PREFERRED EMBODIMENTS
 Before explaining preferred embodiments of the invention, the general flow
 of an image data processing consisting of A/D conversion, computer
 processing and D/A conversion will be explained with reference to FIG. 1.
 Image data used in a computer are digital data. All of the information in
 nature is continuous, i.e. analog, data. Thus, analog data are converted
 to digital data by way of an A/D converter (analog/digital converter) so
 as to be processed by a computer. In order to display digital data in a
 display apparatus such as video monitor, the digital data are converted to
 electric analog data, again, by way of a D/A converter (digital/analog
 converter). FIG. 1 shows the flow to output an image information
 consisting of digitizing (A/D conversion) of an analog image, processing
 of the digital information by a computer and converting the digital
 information to an analog information again (D/A conversion) to output an
 analog image.
 An image is expressed by a group of dots in order to be indicated on
 display of a video monitor or a computer display. FIGS. 2A and 2B show the
 idea of relation between dots and the analog signal corresponding thereto.
 FIG. 2A indicates a row of dots being light, dark, light, dark and so on,
 respectively. FIG. 2B indicates a row of dots being light, light, light,
 dark, dark, dark, and so on, respectively. The analog signal of image in
 FIG. 2A consists of waves having a shorter cycle, while that in FIG. 2B
 consists of waves having a longer cycle.
 Before explaining the preferred embodiments, further, the basic principle
 of the present invention will be explained. When a person tries to
 recognize a small object, any object smaller than a limit is recognized
 merely as a lump. The dimension of the smallest object recognizable can be
 determined by the smallest angle between the lines connecting the
 periphery of the object and an eye. A near object and a far object of the
 same dimension are different in relation to the smallest angle of vision.
 The farther an object is seen, the broader is the minimum area of the
 object which is recognized merely as a small point.
 Sampling is an operation of converting an input signal to discrete signals
 in the time axis by generating pulses at a certain cycle. Quantization is
 conversion of an input signal to dispersed signals in the amplitude axis.
 Sampling signal y(t) is represented by the formula:
EQU y(t)=.EPSILON.x(t).delta.(t.multidot.nT)
 where T is a fixed cycle, .EPSILON. represents the summation with respect
 to time from -infinity to +infinity, and x(t) is analog data of the input
 signal. In general, x(t) is an analog signal of voltage versus time.
 .delta. is a delta function, that is, a function being:
EQU .delta.(t.multidot.nT)=1 (for t=nT)
EQU =0 (for t.noteq.nT)
 Where the signal x(t) is a signal of band width W (radian/second), cycle T
 is represented by:
EQU T=.pi./W
 A function f is defined based on T as:
EQU f=1/(2T)=W/(2.pi.)
 This f is the highest frequency of the frequency components contained in
 the image information.
 In the cases of FIG. 2A and FIG. 2B, the frequency of analog signal in FIG.
 2A is three times as high as that in FIG. 2B. This suggests that the
 number of dots in the image having the smallest dimension of object
 recognizable by human eye changes in accordance with the highest frequency
 of frequency components as the latter changes.
 Image data used by a computer can be grasped as a group of image data
 obtained in the sampling frequency. Concerning these data, the highest
 frequency f.sub.m of components contained in the original data is equal to
 a half of the 'sampling frequency fs, that is, f.sub.s /2 according to the
 sampling rule.
 The dimension of an object recognizable by human eye is equal to the size
 of one dot in the expressed image. When the highest frequency of
 components contained in the image data changes from f.sub.m to f.sub.m /2,
 the dimension of object recognizable by human eye changes to the size of
 two dots in the expressed image. By processing the original image data so
 as to express an image of an object as if the object is located farther,
 the effect of farther distance is obtained. Such processing can be
 accomplished by altering the image by blurring to an image discernible in
 the unit of 1.5 dots or 2 dots, in place of one dot. But the distance
 information of an image discernible in the unit of 2 dots expresses a
 fairly far scene. Usually, the distance effect is produced to such an
 extent that 10 dots in the basic image are recognized as 8 or 9 dots in
 the blurred image. In such a case, assuming that an original image of
 minimum discernible dimension consists of 4 dots, the minimum discernible
 dimension of the blurred image after processing is that of 5 dots.
 Where the cut-off frequency f.sub.d of a low-pass filter is expressed by
 the equation:
EQU f.sub.d =kf.sub.m (0&lt;k.ltoreq.1),
 the image data subject to low pass filter processing with k-1 have the
 highest frequency of frequency components similar to those of the original
 image data, that is, an image without low-pass filter processing,
 indicating a state in which a processed image is recognizable by every one
 dot. If k=0.5, image data for an object so distant as being recognizable
 by every 2 dots are expressed subsequent to the processing. If k=0.8,
 image data for an object so distant as to be recognizable by every 10 dots
 are altered by blurring processing to those for an image recognizable by
 every 10/0.8 dots. In other words, dots of number equal to the reciprocal
 of k is recognized as one dot. Thus, data for an image blurred so as to
 arouse a sense of distance associated with a specified distance can be
 produced by changing the value of k according to the specified distance.
 As explained above, the effect of out-of-focus can be accomplished by
 controlling the highest frequency contained in the image data which is
 initially the sampling frequency. Accordingly, the invention makes it
 possible to assign any distance to an image quantitatively by simulating
 the human vision in the frequency control by means of a digital filter
 having frequency characteristics corresponding to the assigned distance.
 Image data used in a computer may be taken at various
 Image data used in a computer may be taken at various frequencies. In most
 cases, however, no information related to the sampling frequency can be
 obtained, arising a problem. According to the invention, given image data
 can be converted from those of an image recognizable by every one dot to
 those recognizable by l/k dot. The problem above cannot be a serious
 obstacle by using the mathematics below.
 An example of digital low pass filter is Butterworth filter. This filter
 may be of order either of even number or of odd number. Whichever the
 order is, it makes no difference, thus, an embodiment using a Butterworth
 filter of even order will be explained in the following. The transfer
 function of a digital Butterworth filter having an order n=2 m, a cut-off
 frequency .omega..sub.d =2.pi.f.sub.d, and a sampling frequency f.sub.s
 =1/t.sub.s, is represented by EQUATION 1.
 ##EQU1##
 Provided: f.sub.d =kf.sub.m, .omega..sub.d =2.pi.f.sub.d, .omega..sub.m
 =2.pi.f.sub.m, .omega..sub.d =k.omega..sub.m =k.omega..sub.s /2, the term
 t.sub.s.omega.cp=t.sub.s.omega..sub.d is represented as follows:
EQU t.sub.s.omega.cp=t.sub.s.omega..sub.d =t.sub.s
 k.omega.s/2=(1/f.sub.s)(2.pi.kf.sub.s /2)=k.pi.
 As the sampling frequency t.sub.s is not present in the last expression of
 Equation 1, it is found that the transfer function has no relation to the
 sampling frequency, being represented only by the ratio k of the cut-off
 frequency f.sub.d to a half of the sampling frequency f.sub.m.
 FIG. 3 shows the data processing performed by a digital low pass filter of
 order n=2 m. Data are processed by the filter in series of m steps.
 Low-pass filters include Tchebycheff filter, Anti- Tchebycheff filter and
 elliptic function filter, in addition to Butterworth filter. Transfer
 function for any of them is expressed by the parameter k above.
 An example of transfer function to which the invention is applied in
 practice will be shown. In this example, a processing, called pre-warp, is
 performed upon frequency conversion from analog to digital. Pre-warp
 processing is represented by the following equation.
EQU .omega..sub.a =(2/t.sub.s)tan(.omega..sub.d t.sub.s /2)
 This equation means that a digital filter having a cut-off frequency
 .omega..sub.d and sampling frequency t.sub.s is designed based on an
 analog filter having a cut off frequency .omega..sub.a. But for this
 processing, errors in the transfer characteristics from those expected
 occur as the frequency approaches from a lower angular frequency to cut
 off frequency .omega..sub.d. Referring to Equation 1, the term
 t.sub.s.omega.cp=t.sub.s.omega..sub.d above is to be:
EQU t.sub.a (2/t.sub.s)tan(.omega..sub.d t.sub.s /2)=2 tan(k.pi./2)
 Examples of the coefficients in transfer function H.sub.2m (d,z) by
 EQUATION 1 are shown in TABLE 1 below.
 TABLE 1
 TRANSFER FUNCTION
 ##EQU2##
 EXAMPLES OF COEFFICIENTS
 FOR SECOND-ORDER FILTER AND k = 0.9
 C.sub.0p = 0.8008, A.sub.1p = 1.561, A.sub.2p = 0.6414
 FOR SECOND-ORDER FILTER AND k = 0.8
 C.sub.0p = 0.6389, A.sub.1p = 1.143, A.sub.2p = 0.4128
 FOR SECOND-ORDER FILTER AND k = 0.7
 C.sub.0p = 0.5050, A.sub.1p = 0.7478, A.sub.2p = 0.2722
 FOR SECOND-ORDER FILTER AND k = 0.6
 C.sub.0p = 0.3913, A.sub.1p = 0.3695, A.sub.2p = 0.1958
 FOR SECOND-ORDER FILTER AND k = 0.5
 C.sub.0p = 0.2929, A.sub.1p = 0.13*10.sup.-5 .apprxeq. 0,
 A.sub.2p = 0.1716
 The transfer function of a digital low pass filter is a function of
 distance information z, as shown in EQUATION 1. thereby, blurring in
 relation to distance of the processed image can be expressed by
 correlating the value of z with coefficient k.
 A preferred embodiment of the present invention will be explained in more
 detail with reference to an example. In the following example, C Language
 is used for presentation. In the example, processing according to the
 invention is performed in the procedures shown in FIGS. 4 to 7. FIG. 4
 shows the image data processing to generate a blurred image according to
 the invention. The image data processing includes the steps of preparatory
 processing (S41), idling process (S42), processing of real data for one
 line of dots (S43) and a step of judgment "Has processing for all lines
 finished?" (S344).
 FIG. 5 shows in detail the step of preparatory processing S41 in FIG. 4 for
 preparing the z-value threshold table. The preparatory step comprises the
 steps of providing image data for v+1 lines of dots (S51), providing
 z-value data for v+1 lines (S52), preparation of z-value threshold table
 (S53), setting of idling counter (S54), holding a work memory (S55) and
 holding z-value buffers (S56).
 The preparatory processing shown in FIG. 5 will be explained in more
 detail. Steps 51 to 56 are included in preparatory processing S41. Image
 data are assumed to include w+l dots, lengthwise, and v+1 dots, across.
 Image data for v+1 lines of dots are provided in step S51.
 The data for a single line are:
EQU d.sub.0, d.sub.1, d.sub.2, d.sub.3, . . . d.sub.w.
 z-value data for v+1 lines are provided in step S52. Data of z representing
 the distance of each pixel are provided for v+1 lines of dots, the data
 for a single line being:
EQU z.sub.0, z.sub.1, z.sub.2, . . . z.sub.w.
 A threshold table for z_tbl[n+1][5] is prepared in step 53. The matrix
 z_tbl for a Butterworth filter or a Tchebycheff filter is represented in
 the form shown in TABLE 2 below.
 TABLE 2
 VALUES OF CUT-OFF
 z VARIABLES A0 A1 C0
 z0 k0 a00 a10 c00
 z1 k1 a01 a11 c01
 z2 k2 a02 a12 c02
 . . . . .
 . . . . .
 . . . . .
 zn kn a0k a1k c0k
 Dual arrangement z[i][j] in C Language corresponds to Z.sub.ij in
 mathematical matrix expression, thus i and j correspond to the line and
 the row, respectively. For example, with i being 2 and j being 0 to 4, the
 relation between Table 2 and z_tbl is:
EQU z_tbl[2][0]=z.sub.2, z_tbl[2][1]=k.sub.2, z_tbl[2][2]=a.sub.02
 z_tbl[2][3]=a.sub.12, z_tbl[2][4]=c.sub.02
 TABLE 2 is prepared by calculating all of a.sub.0 k, a.sub.1 k and c.sub.0
 k in the system program based on the definition of cut-off variables k
 corresponding to n+1 values of z which fulfill:
EQU z.sub.0 &gt;;z.sub.1 &gt;;z.sub.2 &gt;;&gt;;Z.sub.n.
 In case where an anti-Tchebycheff filter or an elliptic function filter is
 used, another filter coefficient b.sub.1 is necessary. For example, the
 cut-off coefficient k.sub.n for z.sub.n, the the last value of z, should
 be 0.99 which causes no blurring.
 In step 54 of setting of idling counter, values of 0, 1, 2, 3, . . . , x
 can be set.
 In step 55 of holding work memories, p sets of memories m.sub.1 and
 m.sub.2, respectively, are required according to the order of filter 2*p.
 If the order is 1, only one memory m.sub.1 is enough. In step 56 of
 holding z-value buffers, 0 to x buffers zb for values of z are required
 according to the value in the idling counter.
 After steps 51 to 56 have been finished, processing by a filter of second
 order is performed, an example of which will be shown below for one line
 of dots.
 FIG. 6 shows the detail of idling process shown in FIG. 4. The idling
 process comprises the steps of initializing of work memory (S61),
 initializing of z-value buffers (S62), value setting of idling counter
 (S63), setting of filter coefficient in accordance with z-value of the
 first pixel in the line (S64) and filtration of data for the first pixel
 in the line (S65).
 Initializing including the following is required:
 (1) m, and m.sub.2 are initialized to zero.
 (2) zb[0], zb[1], . . . , zb[x-1] are initialized to Z.sub.n.
 (3) count in the idling counter is substituted for the register variable
 "offset".
 (4) initializing others.
 Data processing in the idling process S42 is performed as shown in the
 following:

for(I=0; I&lt;;offset;I++).vertline.
 set_coef(z[0]); // to set a filter coefficient according to
 z-value of the first pixel.
 filter(d[0]); // to input data for the first pixel to the filter.
 zb[offset -1 -I] = z[0]; // to do nothing if offset=0.
 .vertline.
 "for(i=0;i&lt;;k;i++).vertline.. . . .vertline." represents to make a loop
 while "i&lt;k" is satisfied during counting up of i one by one starting from
 0, that is, processing .vertline.. . . .vertline. is repeated k times.
 "set_coef(z[0])" and "filter(d[0])" are functions based on user definition,
 the contents being shown later.
 FIG. 7 shows the processing of real data for one line of dots shown in FIG.
 4 as step 43. The processing comprises the steps of reading z values of
 dots to be expressed (S71), a judgment "Is it larger than z value having
 appeared offset value times earlier?" (S72), renewal of z value to new one
 (S73), selection of filter coefficient according to z value (S74),
 performing filtration (S75) and another judgment "Is the last dot in the
 line reached?" (S76).
 The real data processing for dots in one line in step S43 is performed as
 shown in the following.

for(I=0; I&lt;;offset;I++).vertline.
 set_coef(z[0],0,d[0]); // Setting the filter coefficient
 in accordance with the z value
 of the first pixel.
 filter(d[0]); // Inputting the first image data
 to the filter
 zb[offset -1 -I] = z[0]; // Nothing is done if
 offset = 0
 (Modification of real data processing related to one line)
 for(I=0; I&lt;;w+1;I++).vertline.
 if(offset).vertline.
 zf=max(z)[I],z.sub.b [offset-1]); // selecting a z value
 for more distant scene.
 for(j=offset-1;j&gt;;0;j--) z.sub.b [j]=z.sub.b [j-1];
 z.sub.b (0)= z[I];
 .vertline. else zf= z[I];
 set_coef(zf, 1, d[I]); // Setting the filter coefficient
 in accordance with the distance
 of the pixel.
 // If it is desired to disenable distance effect,
 it may be set_coef(Z.sub.n, 1, d[I])
 out = filter(d[I]);
 The modification above has made it possible to solve the problem related to
 changing of the filter coefficients. It is desirable practically to make
 the time for processing equal, in no relation to whether the coefficients
 are changed or not.
 (Final modification of the processing of set_coef(zf))
 The pointer of coefficient is passed to system variable coef_pnt. At the
 same time, the two memories in the filter may be changed to suitable
 values if the coefficient is changed. It is assumed that the present
 filter coefficients are contained in a.sub.0, a.sub.1 and c.sub.0.

set_coef(zf, jitu, d) .vertline.
 for(I=0; I&lt;;n+1;I11).vertline.//m.sub.1 and m.sub.2 are adjusted
 for any case
 if(zf&gt;;=z_tbl[I][0]) .vertline.
 coef_pnt = &;z_tbl[I][2];
 if(jitu == 0) .vertline. // the case of idling process
 a.sub.0 = *coef_pnt; a.sub.1 = *(coef_pnt+1);
 c.sub.0 = *(coef_pnt+2);
 break;
 .vertline.
 a.sub.0d = *coef_pnt; a.sub.1d = *(coef_pnt+1);
 c.sub.0d = *(coef_pnt+2);
 m.sub.1 = ((c.sub.0 -c.sub.0d)*d+c.sub.0 *(2-a.sub.0)*m.sub.1)
 / c.sub.0d / (2-a.sub.0d);
 m.sub.2 = c.sub.0 *(1-a.sub.1)*m.sub.2 /(1-a.sub.1d)/c.sub.0d ;
 a.sub.0 = a.sub.0d ; a.sub.1 = a.sub.1d ;
 c.sub.0 = c.sub.0d ;
 break;
 .vertline.
 .vertline.
 .vertline.
 (Final modification of idling process in the one-line processing)
 (1) Initializing: Initialize ml and m.sub.2 to zero; return to initial
 setting.
 (2) Initialize z.sub.b [0], . . . , z.sub.b [x-1] to Z.sub.n. and so forth.
 (Final modification of real data processing in the one-line processing)
 The modification in the foregoing is applicable.
 According to the invention, amount of calculation for one pixel to be
 outputted is enough to accomplish blurring by means of a filter. For
 example, in calculation to express the effect of 10 pixels, there is no
 influence on the calculation load, only the change in cut-off variables
 being involved. In an application to accomplish the blurring effect only
 in one direction, either transverse or longitudinal, the image memory for
 output may be omitted.
 According to the invention, only a round of filtration is enough to obtain
 blurring effect in any degree, and the process is comprehensive because
 the human vision is simulated in setting the degree of blurring effect.
 Focusing at a certain distance can be accomplished by setting cut-off
 variable k for a particular value of z to about 0.99, upon producing the
 threshold table of z values.
 In producing the threshold table of z values, it is admitted to assign
 k=0.99 for a certain value of z and to assign other values of k enough to
 obtain blurring effect for other z values related to more remote and
 closer distances, respectively. Thus, the phenomenon of blurring upon
 looking at a very close object can be easily expressed.
 According to the invention, the range to which the blurring effect extends
 can be controlled not only by cut-off variables but also by means of an
 idling counter to some extent, whereby system with greater liberty can be
 provided. Further, the process of the present invention is adapted to
 supplement a conventional rendering system, being added behind the latter.
 Although the invention has been described with respect to the specific
 embodiments for complete and clear disclosure, the appended claims are not
 to be thus limited but are to be construed as embodying all modification
 and alternative constructions that may occur to one skilled in the art
 which fairly fall within the basic teaching herein set forth.