Abstract:
Image processing methods and apparatus by which images in the spatial domain can be represented in the frequency domain through the use of discrete cosine transforms, conveniently operated on to achieve scaling and filtering effects while in the frequency domain, and then retransformed to the spatial domain or stored, displayed, reproduced or transmitted to distant locations for subsequent reuse. The scaling techniques which the invention utilizes can be for image enlargement by interpolation or image reduction by decimation. In the case of decimation, a filtering operation preferably is first performed in the frequency domain to avoid artifacts in the decimation process. The filtering operation is mathematically equivalent to a linear convolution in the spatial domain as a consequence of the properties of the DCT transformation. In both interpolation and decimation procedures, use is made of a hybrid inverse discrete cosine transform in which the series of cosine terms are evaluated at values arrived at by scaling ratio considerations, rather than the usual sampling index increments. As a consequence, image data points other than original image data in the spatial domain can be created or replaced by an approximation technique which involves transforming the image data points to frequency space by a series of terms that can be considered to be continuous over the range of the sampling index corresponding to the original image data.

Description:
RELATED APPLICATIONS 
     This application is related to U.S. Pat. No. 5,168,375. Furthermore, this application is a continuation of application Ser. No. 08/159,795 filed by the same inventors on Nov. 30, 1993, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates generally to improved methods and apparatus for image processing. More particularly, the invention relates to novel methods and apparatus for using discrete cosine transformations to: enlarge an image by interpolation; reduce an image by decimation; and/or filter an image in the frequency domain by a method equivalent to a mathematical convolution in the spatial domain. 
     2. Description of the Prior Art 
     Images can be thought of as two-dimensional representations of some visual reality that is distributed in space and/or time. Ordinarily, they are what the human visual system perceives as variations in external stimuli such as brightness, color, and sometimes depth cues. Over the years many techniques have been developed to capture and reproduce images, such as representing images as continuous, discrete, or digital signals which can be manipulated, processed or displayed through the use of computers or other special purpose electronic hardware in electronic form, images can be enhanced to create special visual effects, restored, coded for transmission to distant locations, reconstructed, displayed, or converted to some other tangible form. 
     Processing an electronic image may include such operations as filtering, sharpening, smoothing, compressing, de, compressing, enlarging, reconstructing, and printing the image, or any portion thereof, in various image processing systems such as an electronic camera, camcorder, printer, computer or any other imaging device. 
     Image processing can occur in either the spatial domain or the frequency domain. An image is said to reside in the spatial domain when the values of the parameters used to describe it, such as brightness, have a direct correspondence with spatial location. In the frequency domain, the image in the spatial domain may be represented by a series of frequency components in the form of trigonometric functions which, when summed for each image data point (i.e., pixel) yield the value of the parameter used to characterize the image of that point in the spatial domain, and such a representation may be extended to cover all points of an image. 
     In the spatial domain, original image data may be conveniently represented as image data points in a first spatial matrix designated, s(j,i), for a two-dimensional case where the lower case, s, designates the spatial domain, i is the index of rows and j is the index of columns. In the frequency domain, matrices can also be used to mathematically describe an image as a set of the transform coefficients (also referred to as frequency coefficients) which represent frequency data in a transform matrix conventionally designated, S(v,u), where the upper case, S, designates the frequency domain and, u is the index of rows and v is the index of columns. 
     Spatial image data points may be transformed to frequency space using transformations such as Fourier transforms or discrete cosine transforms (DCTs). When the transformation involved is a discrete cosine transformation, the frequency domain is referred to as the DCT domain and the frequency coefficients are referred to as DCT coefficients. Conventionally, transforming data from the spatial domain to the frequency domain is referred to as a forward transformation, whereas transforming data from the frequency domain to the spatial domain is referred to as an inverse transformation. Hence, a forward discrete cosine transformation is defined as a transform that maps an image from the original image data points s(j,i) in the spatial domain to DCT coefficients S(v,u) in the DCT domain according to the basis function of the forward DCT, whereas an inverse discrete cosine transformation (or IDCT) is defined as a transform that maps the DCT coefficients S(v,u) from the DCT domain to reconstructed image data points s&#39;(j,i) in the spatial domain according to the basis function of the IDCT. 
     The use of DCT and IDCT transforms for compressing or decompressing images to reduce memory storage requirements and/or increase transfer and computational speeds is well-known and, in fact, the practice has been adopted as standard in industry by such groups as The Joint Photographic Experts Group (JPEG), which was created as part of a joint effort of the Consultative Committee on International Telegraphy and Telephony (CCITT) and The International Standards Organization (ISO). Today, most image processing programs support both loading and saving files that conform to JPEG standards for files or formats, and there is even custom hardware on the market for compressing and decompressing in JPEG format. 
     Even so, the application of DCT transforms to image processing operations other than compression is not well-known and offers an opportunity to conveniently utilize available custom hardware or general purpose computers to achieve other processing effects such as scaling, filtering, or the conditioning of image data for efficient transmission while suppressing or reducing artifacts. 
     Consequently, it is a primary object of this invention to provide methods and apparatus which utilize discrete cosine transforms for image processing operations other than compression. 
     It is another object of the present invention to provide methods and apparatus which utilize discrete cosine transforms for enhancing compression while reducing artifacts. 
     Other objects of the invention will, in part, appear hereinafter and, in part, be obvious when the following detailed description is read in conjunction with the drawings. 
     SUMMARY OF THE INVENTION 
     Image processing methods and apparatus include ways in which images in the spatial domain can be represented in the frequency domain through the use of discrete cosine transforms, conveniently operated on to achieve scaling and filtering effects while in the frequency domain, and then retransformed to the spatial domain or stored, displayed, reproduced or transmitted to distant locations for subsequent reuse. 
     The scaling techniques which the invention utilizes can be for image enlargement by interpolation or image reduction by decimation. In the case of decimation, a filtering operation preferably is first performed in the frequency domain to avoid artifacts in the decimation process. The filtering operation is mathematically equivalent to a linear convolution in the spatial domain as a consequence of the properties of the DCT transformation. 
     In both interpolation and decimation procedures, use is made of a hybrid inverse discrete cosine transform in which the hybrid inverse cosine basis function is determined from the series of cosine terms, evaluated at values arrived at by scaling, i.e. resampling, ratio considerations rather than the usual sampling index increments. As a consequence, image data points other than those comprising original image data in the spatial domain can be generated by an approximation technique which involves representing the image in frequency space by a series of frequency coefficients that can be considered to be continuous over the range of the sampling index of the original image data. 
     In use, the hybrid technique involves first transforming spatial image data to frequency space through the use of a standard DCT. Here, a DCT basis matrix, and its transpose for the two-dimensional case, are matrix product multiplied by the image data in matrix form to generate a matrix of DCT coefficients. A scaling ratio is chosen and a hybrid IDCT basis matrix is generated along with its transpose. The results of these two operations are mathematically combined to generate a reconstructed image data matrix through an inverse transform step. The reconstructed image data matrix represents the new image which may be either enlarged or reduced in size. 
     The scaling operations can be carried out so that the magnification is the same along orthogonal azimuths or different along both since the operations can be mathematically decoupled by reason of the orthogonal property of the DCT basis functions. 
     To perform the filtering operation for sharpening or smoothing, use is made of a discrete odd cosine transform of a symmetric filtering kernel in the spatial domain. When decimating, the filtering operation is preferably performed to avoid artifacts and adjacent image data matrices are preferably overlapped by at least one row and column. 
     All operations are preferably performed in JPEG format and special purpose hardware adapted to the JPEG standard may also be beneficially employed to process image data in 8×8 or 16×16 sized blocks. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The aforementioned aspects and other features of the invention are described in detail in conjunction with the accompanying drawings in which the same reference numerals are used throughout for denoting corresponding elements and wherein: 
     FIG. 1 is a plot of discrete cosine normalized basis functions for the one dimensional case of a DCT as given by equation (1); 
     FIG. 2A is a graph of reconstructed image data points s&#39;(i) vs. i for an eight element one dimensional matrix with a 1:1 scaling ratio; 
     FIG. 2B is a graph (related to image enlargement) of reconstructed image data points s&#39;(x) vs. x with a 10:8 scaling ratio over the same range of image data as in FIG. 2A; 
     FIG. 2C is a graph (related to image reduction) of reconstructed image data points s&#39;(x) vs. x with a 2:5 scaling ratio over the same range of image data as in FIG. 2A; 
     FIG. 3 is a block diagram of a first embodiment of a method for scaling an image in two-dimensional form where the scaling ratio may be changed, but is the same in both dimensions; 
     FIG. 4 is a block diagram of a second embodiment of a method for scaling an image in two dimensional form where the scaling ratio may be changed by different amounts in both dimensions; 
     FIG. 5 is a schematic block diagram of an apparatus for scaling an image according to the method of FIG. 3; 
     FIG. 6 is a schematic block diagram of steps for filtering an image in the frequency domain by a method mathematically equivalent to a convolution in the spatial domain; 
     FIG. 7 illustrates overlapping between adjacent groups of pixels in an image; 
     FIG. 8 is a schematic block diagram of an apparatus for filtering an image according to the method of FIG. 6; and 
     FIG. 9 is a schematic block diagram of a method for reducing an image by decimation. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     This invention relates generally to improved methods and apparatus for image processing. More particularly, the invention relates to novel methods and apparatus for using discrete cosine transformations to: enlarge an image by interpolation; reduce an image by decimation; and/or filter an image in the frequency domain by a method equivalent to a mathematical convolution in the spatial domain. 
     The following mathematical discussion, in part, sets forth certain fundamentals relating to forward and inverse discrete cosine transforms (DCTs). 
     A forward DCT is defined as a mathematical process for transforming image data points from the spatial domain to the frequency or, more particularly, DCT domain. Image data points s(i) in one dimensional form may be transformed from the spatial domain to DCT coefficients S(u) for the frequency domain according to equation (1). ##EQU1## for 0≦u≦(N-1), where: S(u) represents the DCT coefficients; 
     s(i) represents the image data point; 
     N represents the number of image data points; ##EQU2## for u=0; and C u  =1 for u≠0. 
     The DCT coefficients S(u) are derived from equation (1) where the cosine normalized basis terms are shown in FIG. 1 for N=16, where 0≦u≦15 and 0≦i≦15. The value for S(0) is determined for u=0 by summing each of the image data points s(i) for 0≦i≦(N-1) times the cosine terms as represented in FIG. 1. The value for S(1) is determined as the summation of image data points s(i) times the cosine terms for u=1. This procedure, which indexes first on u and then on i, is continued to derive all sixteen DCT coefficients S(0) through S(15). 
     The various terms of equation (1) can alternatively be expressed in matrix notation, where each cosine term represents an element of a one dimensional matrix defined as a forward DCT basis matrix FB, each image data point represents an element of a first spatial matrix s(i) of image data points, and each DCT coefficient represents an element of a DCT matrix S(u) of DCT coefficients. 
     An inverse discrete cosine transformation is defined as a mathematical process for transforming DCT coefficients from the DCT domain to reconstructed image data points in the spatial domain. DCT coefficients S(u) in one dimensional form are transformed from the DCT domain to reconstructed image data points s&#39;(i) in the spatial domain according to equation (2). ##EQU3## for 0≦i≦(N-1), where: S(u) represents the DCT coefficients; 
     s&#39;(i) represents the reconstructed image data points; 
     N represents the number of DCT coefficients; ##EQU4## for u=0; and C u  =1 for u≠0. 
     If the DCT coefficients S(u) of equation (1) are derived from a set of image data points s(i), and the reconstructed image data points s&#39;(i) of equation (2) are derived from the corresponding DCT coefficients S(u), then s(i)≡s&#39;(i) and the process is referred to as lossless since the reconstructed image data points s&#39;(i) are identical to the original image data points s(i), within limits. The reconstructed image data points s&#39;(i) are derived from equation (2) where the cosine terms are shown in FIG. 1 for N=16, where 0≦i ≦15 and 0≦u≦15. The value for s&#39;(0) is determined for i=0 by summing each of the DCT coefficients S(u) times the cosine terms as represented in FIG. 1. The value for s&#39;(1) is determined as the summation of DCT coefficients S(u) times the cosine terms for i=1. This procedure is continued, indexed as before, to derive all sixteen reconstructed image data points s&#39;(0) through s&#39;(15). 
     Note that the conventional inverse DCT of equation (2) for transforming data in one dimensional form includes the same cosine argument resulting in the same basis function used in the conventional forward DCT of equation (1) so that the reconstructed image data points s&#39;(i) coincide with the original image data points s(i). However, there is no recognition in the prior art for one to use the IDCT for determining reconstructed image data points s&#39;(i) that fall between the original image data points s(i) in the spatial domain. 
     The above examples for a one dimensional DCT and IDCT can be extended, as known by those skilled in the art, to multi-dimensional formats. For instance, Section A.3.3 of ISO/IEC 10918-1 of the draft international standards for digital compression using discrete cosine transforms defines the forward DCT in two dimensional form as: ##EQU5## for 0≦u≦7 and 0≦v≦7, while defining the IDCT in two dimensional form as: ##EQU6## for 0≦i≦7 and 0≦j≦7, where S(v,u) represents DCT coefficients; 
     s(j,i) represents original image data points; 
     s&#39;(j,i) represents the reconstructed image data points; ##EQU7## for u=0; C u  =1 for u≠0; ##EQU8## for v=0; and C v  =1 for v≠0; 
     According to equation (3), a first spatial matrix s(j,i) (representing a two dimensional 8×8 group of original image data points in the spatial domain) can be forward DCT transformed to an 8×8 DCT matrix S(v,u) in the frequency domain having 64 DCT coefficients that can be related to the 64 image data points through mapping. An 8×8 forward DCT basis matrix, i.e. forward transform matrix, is derived from the cosine expression of equation (3) by indexing over the full range of values for j, i, v and u. 
     Once the image data points s(j,i) are transformed into DCT coefficients S(v,u) in the DCT domain, the number of DCT coefficients can be reduced by compression which is defined, generally, as the process of reducing either the bandwidth or the number of bits necessary to represent an image and, more specifically, as the process of decreasing the number of DCT coefficients S(v,u) in the DCT domain by removing a selected set of the DCT coefficients. The selected set are determined to be non-essential to image reproduction according to some predetermined criteria. Typically, the set of DCT coefficients selected for removal includes zero or near zero values or those terms which represent high frequency content that the human visual system cannot perceive thus no necessary information is lost. 
     Another aspect of image processing in the DCT domain is sharpening anal smoothing an image by a filtering procedure mathematically equivalent to a convolution in the spatial domain. A convolution of two discrete signals in the spatial domain occurs by multiplying the two discrete signals point-by-point then summing the products over appropriate limits. Sharpening is defined as the process of enhancing blurry images, particularly by emphasizing high frequency components representing edges in an image. Smoothing, on the other hand, is defined as the process for either softening the edges of the image or alternatively decreasing high frequency components. 
     Filtering an image in the DCT domain using a process similar to a mathematical convolution in the spatial domain is disclosed in an article entitled &#34;Discrete Cosine Transform Filtering&#34; by Chitprasert &amp; Rao, Signal Processing 19(1990) pgs. 233-245, where a kernel is processed through a 2N point discrete Fourier transform (DFT) yielding 2N complex numbers. A kernel is defined as the signal values of a filter to perform a specific operation such as sharpening or smoothing in the spatial or frequency domain. The imaginary parts of the 2N complex numbers are determined, then discarded, while the real and even parts of the 2N complex numbers are retained for further calculations. Determination of the imaginary parts of the complex numbers requires computational time and effort even though the imaginary parts are not required for filtering the image. However, this filtering process is computationally intensive and, as will be seen, its requirement for use of complex numbers will be overcome by the invention. 
     Resampling 
     According to the invention, images may be scaled utilizing hybrid procedures based on, but advantageously different from, the foregoing mathematical description. 
     An image can be scaled (i.e. resampled) in accordance with a scaling ratio, i.e. resampling ratio defined for each dimension as the number of output pixels desired after a transformation divided by the number of input pixels available before the transformation. For example, for one dimensional image data where the number of input pixels N is eight pixels per cm and the number of output pixels N&#39; is ten pixels per cm, the scaling ratio R=N&#39;/N is 10:8. To enlarge an image then, a scaling ratio greater than one is typically selected (e.g., 10:8). To reduce an image, a scaling ratio less than one is selected (e.g., 6:8) and when a scaling ratio of one is selected (e.g, 8:8), the output image is typically the same size as the input image. 
     Examples of scaling one dimensional image data are shown in FIGS. 2A, 2B and 2C. FIG. 2A is a plot of the values from a second spatial matrix, s&#39;(i), of image data points in one-dimension (designated by open circles), which were derived from equation (2) for the inverse transformation of a DCT matrix S(u) of eight DCT coefficients for 0≦i≦7. The IDCT of equation (2) for one dimensional data, or equation (4) for two dimensional data, does not reconstruct image data points which fall between the discrete index values for the original image data points. However, there are instances in image processing when calculation of such intermediate values is desirable, e.g., image enlargement by interpolation and image reduction by decimation. 
     A reconstructed matrix of image data points s&#39;(x) for one dimensional data can be derived from a hybrid IDCT which is given by equation (5). In other words, the hybrid IDCT of equation (5) provides for a continuous variable, x, indexed by Δx for the range of 0≦i≦(N-1), so that the reconstructed image data points s&#39;(x) may be determined at non-integer values of x (e.g., x 1  =0.5, x 2  -6.3, x 3  =4.99712, etc.). Instead of indexing i with an interval of Δi equal to the interger value of 1 (the sampling rate for the original image data), x can be indexed at some other sampling rate, Δx, which can be smaller or larger than one in floating point form, as the case may be. 
     Due to the scaling, some of the reconstructed image data points will correspond to spatial locations different than the original image data points of the first spatial matrix s(i). In this manner, reconstructed image data points s&#39;(x) can be determined which will fall between the discrete index values of the original image data points s(i). ##EQU9## for 0≦x≦(N-1), where: s&#39;(x) represents reconstructed image data points; 
     S(u) represents the DCT coefficients; 
     N represents the number of DCT coefficients; 
     x is a continuous variable over the range 0≦x≦(N-1); ##EQU10## for u=0; and C u  =1 for u≠0. 
     An infinite number of reconstructed image data points, s&#39;(x), can be theoretically derived to construct, for example, the curves of FIGS. 2B and 2C, while keeping within the appropriate range of the function s&#39;(x) of equation (5). The index values, x, of the resampled image data points (as shown in FIGS. 2B and 2C) are determined by first selecting a scaling ratio, R. The scaling ratio is selected for each dimension of image data, e.g., one dimensional image data is scaled by a single scaling ratio, two dimensional image data is scaled in the first dimension by a first dimension scaling ratio and in the second dimension by a second dimension scaling ratio, etc., where the various resampling ratios may be the same or different. 
     The end points of contiguous groups of image data points (i.e., end points of each row and column of the image data points in matrix notation) are sometimes omitted during scaling but can be retained or recaptured if desired. 
     FIG. 2B shows an example of scaling up by resampling (or upsampling) in one dimension, where both end points have arbitrarily been retained. The spacing of the x index values in FIG. 2B is determined for a scaling ratio of 10:8, meaning ten reconstructed image data points s&#39;(x) are sought for each eight image data points s(i) of the original image. In other words, the source axis, i.e. the i axis of FIG. 2A, is divided into seven segments (for eight image data points) and the target axis, i.e. the x axis of FIG. 2B, is divided into nine segments (for ten image data points). Hence, s&#39;(x) for 0≦x≦N is solved according to the relationship 1/9=Δx/7, where Δx=7/9. In other words the ten image data points s&#39;(x) of FIG. 2B, derived according to the 10:8 scaling ratio, are resampled for s(x) at 0, 7/9, 14/9, 21/9, 28/9, 35/9, 42/9, 49/9, 56/9, and 7 with respect to the original i-axis. In general, the new sampling interval for interpolation including both end points is given by: ##EQU11## 
     FIG. 2C shows an example of scaling down by resampling (or downsampling) in one dimension where a 2:5 scaling ratio is chosen and the zero index point (i.e. end point) is ignored. The scaling ratio of 2:5 indicates that only 0.4 of the number of originally sampled points are to be resampled. The spacing between resampled image data points is determined as Δx 2.5. Hence, the resampled points designated by the index values for x fall within the given range of 0≦i≦7 and x equals 2.5 and 5.0, omitting the zero end point in the given range as shown in FIG. 2C. The pitch between adjacent pixels is the inverse of the sampling ratio. 
     Enlarging An Image By Interpolation 
     One special case of scaling by resampling is enlarging an image by interpolation, defined as the process of generating (by approximation) a greater number N&#39; of reconstructed image data points s&#39;(x) than the original number N of image data points s(i). 
     FIG. 3 is a block diagram of a preferred embodiment of a method of scaling by resampling an image of two dimensional data where the first dimension scaling ratio and the second dimension scaling ratio are equal and do not equal one. The forward DCT section is denoted by reference number 10 whereas the hybrid IDCT section is denoted by the reference number 12. A first spatial matrix s(j,i), as in the forward DCT equation (3), is selected in block 20 as an 8×8 group of pixels in the spatial domain example (shown below). Of course, it will be understood that the 8×8 matrix is usually but a subset of a larger set of original image data which has been remapped into smaller submatrices that serve as the input to the inventive methods and/or apparatus. In JPEG format, the adopted standard submatrix size may be 8×8 or 16×16. However, the pixel group, in the most general sense, can be of any convenient size that can be selected as a program parameter or directly by an operator. 
     
         __________________________________________________________________________FIRST SPATIAL MATRIX, s(j,i)__________________________________________________________________________100.00120.00     140.00          160.00               100.00                    100.00                         100.00                              100.00100.00100.00     120.00          140.00               160.00                    100.00                         100.00                              100.00120.00100.00     100.00          120.00               140.00                    160.00                         100.00                              100.00140.00120.00     100.00          100.00               120.00                    140.00                         160.00                              100.00160.00140.00     120.00          100.00               100.00                    120.00                         140.00                              160.00100.00120.00     140.00          160.00               100.00                    100.00                         100.00                              100.00100.00100.00     120.00          140.00               160.00                    100.00                         100.00                              100.00120.00100.00     100.00          120.00               140.00                    160.00                         100.00                              100.00__________________________________________________________________________ 
    
     Each element of the first spatial matrix s(j,i) represents an image data point corresponding to a pixel of an image. The pixel data is thus stored in the spatial domain in matrix form. 
     The following 8×8 forward DCT basis matrix FB is produced in block 22 by evaluating the cosine terms of equation (3) for the two dimensional forward DCT by indexing first on u and v and then on i and j. 
     
         ______________________________________FORWARD DCT BASIS MATRIX, FB______________________________________0.3540.354   0.354   0.354 0.354 0.354 0.354 0.3540.4900.416   0.278   0.098 -0.098                            -0.278                                  -0.416                                        -0.4900.4620.919   -0.191  -0.462                      -0.462                            -0.191                                  0.191 0.4620.416-0.098  -0.490  -0.278                      0.278 0.490 0.098 -0.4160.354-0.354  -0.354  0.354 0.354 -0.354                                  -0.354                                        0.3540.278-0.490  0.098   0.416 -0.416                            -0.098                                  0.490 -0.2780.191-0.462  0.462   -0.191                      -0.191                            0.462 -0.462                                        0.1910.098-0.278  0.416   -0.490                      0.490 -0.416                                  0.278 -0.098______________________________________ 
    
     A first intermediate matrix I 1  as shown below is an 8×8 matrix generated in multiplier 25 by matrix product multiplication of the 8×8 forward DCT basis matrix FB times the 8×8 first spatial matrix s(j,i) (i.e., I 1  =FB*s(j,i)). 
     
         __________________________________________________________________________FIRST INTERMEDIATE MATRIX, I.sub.1__________________________________________________________________________332.340318.198     332.340          367.696               360.624                    346.482                         318.198                              304.056-6.2032.301     6.553          8.504               -6.553                    -10.806                         1.951                              -5.853-40.782-22.304     9.239          36.955               24.546                    -11.480                         -46.194                              -27.716-12.56723.678     41.801          36.245               -41.801                    -59.923                         -5.556                              16.66735.35521.213     -7.071          -28.284               -35.355                    21.213                         35.355                              21.213-11.920-4.710     -1.105          7.210               1.105                    -2.500                         8.315                              -24.944-6.0691.585     3.827          15.307               -33.128                    27.716                         -19.134                              -11.48016.1723.444     -2.920          -12.728               2.920                    9.283                         -9.808                              29.424__________________________________________________________________________ 
    
     In block 24, the following 8×8 first transpose matrix, FB T , is generated by transposing the above forward DCT basis matrix, FB. 
     
         ______________________________________FIRST TRANSPOSE MATRIX, FB.sup.T______________________________________0.3540.490   0.462   0.416 0.354 0.278 0.191 0.0980.3540.416   0.191   -0.098                      -0.354                            -0.490                                  -0.462                                        -0.2780.3540.278   -0.191  -0.490                      -0.354                            0.098 0.462 0.4160.3540.098   -0.462  -0.278                      0.354 0.416 -0.191                                        -0.4900.354-0.098  -0.462  0.278 0.354 -0.416                                  -0.191                                        0.4900.354-0.278  -0.191  0.490 -0.354                            -0.098                                  0.462 -0.4160.354-0.416  0.191   0.098 -0.354                            0.490 -0.462                                        0.2780.354-0.490  0.462   -0.416                      0.354 -0.278                                  0.191 -0.098______________________________________ 
    
     The 8×8 first intermediate matrix I 1  is next matrix product multiplied in multiplier 26 times the 8×8 first transpose matrix, FB T , to generate the following 8×8 DCT matrix S(v,u) of DCT coefficients as defined in equation (3) (i.e. S(v,u)=I 1  *FBT). 
     
         __________________________________________________________________________DCT MATRIX, S(v,u)__________________________________________________________________________947.50010.632     -50.581          16.730               17.500                    9.417                         2.010                              -6.588-3.5736.265     -4.483          -12.875               -3.573                    7.684                         -6.609                              -0.299-27.48410.490     -72.730          -21.370               22.537                    -8.165                         5.732                              -5.382-0.51533.688     11.396          -86.570               -0.515                    19.912                         -14.895                              -6.95522.500-6.111     63.647          19.165               -27.500                    11.044                         3.402                              -9.918-10.0941.955     -19.490          4.306               -10.094                    12.679                         -11.975                              2.475-7.5579.356     -9.268          -1.511               -17.453                    9.149                         22.730                              -38.91212.653-5.906     23.158          3.529               12.653                    17.875                         16.480                              -2.374__________________________________________________________________________ 
    
     Once the DCT matrix S(v,u) is determined as shown above, equation (4) is typically used to generate a second spatial matrix s&#39;(j,i) of reconstructed image data points derived from the inverse discrete cosine transformation of DCT matrix S(v,u). 
     The inventive method of FIG. 3 allows resampling to generate an infinite number of reconstructed image data points s&#39;(y,x) in two dimensional form over the given ranges of the hybrid IDCT of equation (6). The values of real number indices x and y are determined according to the scaling ratios previously discussed for the one dimensional IDCT example of FIG. 2B. ##EQU12## for 0≦x≦(N-1) and 0≦y≦(M-1), where s&#39;(y,x) is a two dimensional matrix of reconstructed image data points; 
     S(v,u) represents a two dimensional matrix of DCT coefficients; 
     N represents the number of DCT coefficients in the first dimension; and 
     M represents the number of DCT coefficients in the second dimension. 
     Since the first dimension scaling ratio equals the second dimension scaling ratio for the preferred embodiment of FIG. 3, then N&#39;=M&#39; for equation (6) where N&#39; and M&#39; are the number of output pixels for the first and second dimensions, respectively, as previously described for the one-dimensional case. 
     The conventional size of a two dimensional IDCT basis matrix is 8×8 in conformance with the above JPEG standard of equation (4). To enlarge each pixel group size by 25%, the first dimension scaling ratio and the second dimension scaling ratio are each selected in block 23 as 10:8. In other words, N&#39;=M&#39;=10, which means that the first dimension hybrid IDCT basis matrix IB 1HYB  generated in block 30 will be a 10×8 matrix as shown below. 
     
         ______________________________________FIRST DIMENSION HYDRID IDCT BASIS MATRIX, IB.sub.1HYB______________________________________0.3540.490   0.462   0.416 0.354 0.278 0.191 0.0980.3540.438   0.269   0.033 -0.211                            -0.403                                  -0.469                                        -0.4660.3540.346   -0.022  -0.376                      -0.498                            -0.313                                  0.065 0.4030.3540.221   -0.304  -0.490                      -0.129                            0.376 0.462 0.0330.3540.076   -0.477  -0.221                      0.410 0.346 -0.304                                        -0.4380.354-0.076  -0.477  0.221 0.410 -0.346                                  -0.304                                        0.4380.354-0.221  -0.304  0.490 -0.129                            -0.376                                  0.462 -0.0330.354-0.346  -0.022  0.376 -0.498                            0.313 0.065 -0.4030.354-0.438  0.269   -0.033                      -0.211                            0.403 -0.496                                        0.4660.354-0.490  0.462   -0.416                      0.354 -0.278                                  0.191 -0.098______________________________________ 
    
     In multiplier 32, a 10×8 second intermediate, matrix I 2  (shown below) is generated by matrix product multiplication of the 10×8 first dimension hybrid IDCT basis matrix IB 1HYB  times the 8×8 DCT matrix S(v,u), i.e. I 2  =IB 1HYB  *S(v,u). 
     
         __________________________________________________________________________SECOND INTERMEDIATE MATRIX, I.sub.2__________________________________________________________________________325.26925.279     -31.543          -38.234               0.000                    19.038                         -2.242                              -18.350323.1919.042     -50.960          -14.983               23.295                    1.504                         -15.971                              17.360331.110-6.306     -39.125          24.903               25.473                    -17.401                         14.082                              0.969333.036-8.914     -21.492          50.581               -9.094                    4.728                         10.906                              -13.185350.419-10.303     25.909          42.521               -19.705                    17.362                         -16.121                              11.431368.808-2.886     65.467          6.308               -1.316                    0.561                         1.025                              4.607340.87320.272     4.966          -32.098               -1.258                    12.495                         7.145                              -21.580316.67220.676     -58.083          -31.431               11.035                    14.608                         -13.333                              -0.591330.009-2.581     -41.594          8.729               30.114                    -12.970                         -3.501                              17.860332.340-8.810     -29.958          43.294               7.071                    -8.612                         20.063                              -13.185__________________________________________________________________________ 
    
     The first dimension hybrid IDCT basis matrix IB 1HYB  is transposed in block 28 to generate the following 8×10 second transpose matrix, IB 1HYB   T . 
     
         __________________________________________________________________________SECOND TRANSPOSE MATRIX IB.sub.1HYB.sup.T__________________________________________________________________________0.354   0.354  0.354      0.354          0.354              0.354                  0.354                      0.354                          0.354                              0.3540.490   0.438  0.346      0.221          0.076              -0.076                  -0.221                      -0.346                          -0.438                              -0.4900.462   0.269  -0.022      -0.304          -0.477              -0.477                  -0.304                      -0.022                          0.269                              0.4620.416   0.033  -0.376      -0.490          -0.221              0.221                  0.490                      0.376                          -0.033                              -0.4160.354   -0.211  -0.498      -0.129          0.410              0.410                  -0.129                      -0.498                          -0.211                              0.3540.278   -0.403  -0.313      0.376          0.346              -0.346                  -0.376                      0.313                          0.403                              -0.2780.191   -0.496  0.065      0.462          -0.304              -0.304                  0.462                      0.065                          -0.496                              0.1910.098   -0.466  0.403      0.033          -0.438              0.438                  -0.033                      -0.403                          0.466                              -0.098__________________________________________________________________________ 
    
     Finally in multiplier 33, the 10×10 second spatial matrix s&#39;(y,x) of equation (6), which represents the reconstructed image data points of the 8×8 first spatial matrix s(j,i), is determined as follows by matrix product multiplication of the 10×8 second intermediate matrix I 2  times the 8×10 second transposed matrix IB 1HYB   T  (i.e., s&#39;(y,x)=I 2  *IB 1HYB   T ). 
     
         __________________________________________________________________________SECOND SPATIAL MATRIX, s&#39;(y,x)__________________________________________________________________________100.00    118.34   125.30       154.46           155.73               105.72                   92.67                       105.79                           96.93                               100.0096.22    98.35   118.02       129.86           149.88               156.06                   108.90                       87.44                           108.79                               95.59113.21    98.81   100.44       112.07           129.44               154.30                   152.30                       111.85                           89.58                               108.17123.37    110.47   94.56       105.07           116.50               125.40                   155.93                       152.35                           106.50                               90.01144.37    127.55   111.72       94.88           99.18               117.57                   127.34                       152.46                           158.46                               107.25162.18    144.32   128.00       107.74           94.88               101.76                   114.48                       131.37                           151.18                               158.55121.70    131.43   127.96       146.68           137.88               93.04                   98.25                       115.04                           105.72                               125.7787.55    103.05   121.01       147.51           162.07               134.40                   96.59                       92.71                           98.21                               85.41107.94    96.93   109.44       114.68           135.47               164.35                   132.97                       95.27                           104.81                               106.94120.00    105.18   94.00       108.12           121.13               136.02                   161.82                       137.88                           90.84                               100.00__________________________________________________________________________ 
    
     The second spatial matrix s&#39;(y,x) provides reconstructed image data points which are used for reproduction of the image in block 36. Note that the original image of the above two dimensional interpolation example was represented by an 8×8 pixel group which was increased to a 10×10 pixel group by resampling using interpolation, according to the first dimension hybrid IDCT basis matrix IB 1HYB  derived from equation (6) as described above. This method facilitates image enlargement with excellent fidelity without compromising resolution. 
     The above preferred embodiment of the method of FIG. 3 is appropriate for two dimensional data when: the first dimension scaling ratio equals one and the second dimension scaling ratio does not equal one; the first dimension scaling ratio does not equal one and the second dimension scaling ratio equals one; and the first dimension scaling ratio does not equal one, the second dimension scaling ratio does not equal one, and the first dimension scaling ratio equals the second dimension scaling ratio. When different scaling ratios are selected by independent resampling of the image data points for each dimension, the method of FIG. 3 is modified as reflected in FIG. 4. 
     FIG. 4 is a block diagram, including a forward DCT section 10 and a hybrid IDCT section 12, of a second embodiment of a method for scaling an image in two dimensional form where the first dimension scaling ratio does not equal one, the second dimension scaling ratio does not equal one, and the first dimension scaling ratio does not equal the second dimension scaling ratio. For example, for an 8×8 first spatial matrix s(j,i), the first dimension scaling ratio is selected in block 23 as 10:8 (N&#39;=10) whereas the second dimension scaling ratio is selected as 12:8 (M&#39;=12). In this case, the 10×8 first dimension hybrid IDCT basis matrix IB 1HYB  is generated in block 30, and the 8×12 second dimension hybrid IDCT basis matrix IB 2HYB  is generated in block 38. Multiplier 44 produces the 10×8 second intermediate matrix I 2  by matrix product multiplication of the 10×8 first dimension hybrid IDCT basis matrix IB 1HYB  times the 8×8 DCT matrix S(v,u)(i.e. I 2  =IB 1HYB  *S(v,u)). In multiplier 46, the second spatial matrix s&#39;(j,i) is generated by matrix product multiplication of the 10×8 second intermediate matrix I 2  times the 8×12 second dimension hybrid IDCT basis matrix IB 2HYB  (i.e. s&#39;(y,x)=I 2  *IB 2HYB ). From the second spatial matrix s&#39;(j,i), the image can be printed, processed, or otherwise reconstructed in block 36. 
     An apparatus corresponding to the method of FIG. 3 is shown in FIG. 5. A forward discrete cosine transform processor 50 transforms the first spatial matrix s(j,i) into DCT matrix S(v,u) in matrix multiplier 54. After processing the DCT matrix S(v,u) in the frequency domain, for instance for compression in a processor not shown, the second spatial matrix s&#39;(y,x) of reconstructed image data points is generated in matrix multiplier 66 of the hybrid IDCT processor 62. 
     A digital representation of an image (not shown) is represented by the first spatial matrix s(j,i) and input to the forward DCT processor 50, which includes a cosine processor 52, a transposer 51 and a matrix multiplier 54. The cosine processor 52 generates the forward DCT basis matrix FB in the DCT domain by processing the cosine terms of equation (3). The transposer 51 produces the first transpose matrix FB T  by transposing the forward DCT basis matrix FB. The matrix multiplier 54 initially produces the first intermediate matrix I 1  by matrix product multiplication of the forward DCT basis matrix FB times the first spatial matrix s(j,i). The matrix multiplier 54 then generates the DCT matrix S(v,u) of DCT coefficients in the frequency domain by matrix product multiplication of the first intermediate matrix I 1  times the first transpose matrix FB T . 
     If desired, the DCT matrix S(v,u) can be decreased in size (i.e. decimated) in a microprocessor or other appropriate circuitry (not shown). Interpolation is typically provided in the spatial domain in the form of linear interpolation. Interpolation or increasing the number of reconstructed image data points s&#39;(y,x) can be accomplished in the hybrid IDCT processor 62, which includes a resampling cosine processor 64, a transposer 76 and a matrix multiplier 66. The resampling cosine processor 64 produces a first dimension hybrid IDCT basis matrix IB 1HYB  in response to the first dimension scaling ratio received at terminal 68 along with the second dimension scaling ratio. Note for this example that the first dimension scaling ratio does not equal one, the second dimension scaling ratio does not equal one, and the first dimension scaling ratio equals the second dimension scaling ratio. The elements of the first dimension hybrid IDCT basis matrix IB 1HYB  are derived from the first dimension cosine terms of equation (6). The transposer 76 produces the second transposed matrix IB 1HYB   T  by transposing the first dimension hybrid IDCT basis matrix IB 1HYB . The second intermediate matrix I 2  is generated by matrix product multiplication of the first dimension hybrid IDCT basis matrix IB 1HYB  matrix times the DCT matrix S(v,u) in matrix multiplier 66. Finally, the matrix multiplier 66 generates the second spatial matrix s&#39;(y,x) by matrix product multiplication of the second intermediate matrix I 2  times the second transpose matrix IB 1HYB   T . The image can now be printed or otherwise reconstructed using the reconstructed image data of s&#39;(y,x). 
     The above methods of FIG. 3 and 4 for enlarging an image by interpolation can also be used for reducing an image by decimation using downsampling, where the scaling ratios are selected to be less than one, and the image data is well behaved for this application. However for best results, the method of reducing an image by decimation using downsampling is generally preceded by low pass filtering to remove high frequency components which might cause aliasing. 
     Filtering 
     Typically, filtering an image in the DCT domain in a manner similar to a mathematical convolution in the spatial domain involves operations of the DCI matrix S(v,u) of DCT coefficients with a matrix of kernel values represented as a DCT kernel matrix of DCT kernel coefficients. 
     Convolution of an image, as understood by those of ordinary skill in the art, is a process of filtering in the spatial domain by multiplying two discrete signals point-by-point then summing the products over appropriate limits. Convolution generally results in filtering by sharpening or smoothing an image. Sharpening enhances blurry images, particularly by enhancing high frequency components representing edges in an image, whereas smoothing softens the edges of the image. The kernel is defined as the signal values of a filter used for performing a specific filtering operation such as sharpening or smoothing in the spatial or frequency domain. The kernel may be selected in the spatial or frequency domain according to a designer&#39;s predetermined criteria for sharpening, smoothing, or the like. 
     Known methods for filtering an image in the frequency domain, in a manner similar to a mathematical convolution in the spatial domain, require computation of complex numbers when processing the kernel as previously described. The following filtering method performed in the DCT domain obviates the need for the calculation of imaginary numbers when processing the kernel, thus minimizing computational complexity, time and effort. 
     Image data is filtered in the DCT domain in a manner mathematically equivalent to a convolution in the spatial domain for one dimensional data by: (1) generating a DCT matrix S(u) of DCT coefficients by taking a discrete even cosine transform (DECT) of a first spatial matrix s(i) of image data points; (2) generating a DOCT matrix H(u) of DOCT coefficients by taking a discrete odd cosine transform (DOCT) of a kernel matrix h(i); (3) generating a mask multiplied matrix by mask multiplying S(u) with H(u); and (4) generating a second spatial matrix s&#39;(i) of reconstructed image data points by taking an IDCT of the mask multiplied matrix. It will be understood by those skilled in the art that a discrete even cosine transformation (DECT) is conventionally referred to as a discrete cosine transformation (DCT). Also, in a preferred embodiment the kernel h(i) is odd and symmetric. 
     Since images contain varying amounts of high frequency content, aliasing often results during transformation operations. When the amount of high frequency content is problematic, then the above filtering procedure would further include the steps of: selecting an overlap of adjacent first spatial matrices s(i) prior to generating DCT matrices S(u); and generating filtered saved regions s&#39; S  (i) of the second spatial matrices s&#39;(i)by discarding certain selected elements of the second spatial matrices s&#39;(i). 
     Mathematically, the one dimensional expression for filtering in a manner equivalent to a mathematical convolution is defined by equation (7). 
     
         s(i)⊕h(i)⃡S(u)▪H(u)              (7) 
    
     where 
     s(i) represents the first spatial matrix of image data points; 
     h(i) represents the kernel matrix in the spatial domain; 
     s(i)⊕h(i) represents the filtering or convolution of s(i) with h(i); 
     H(u) represents the DOCT matrix derived from h(i); 
     S(u) represents the DCT matrix derived frown s(i); 
     i and u are integer indices; 
     ⊕ is the convolution operator; and 
     ▪ designates mask multiplication. 
     The above DCT matrix S(u) for one dimensional data is derived from equation (1), whereas the one dimensional DOCT matrix H(u) is derived as follows from equation (8) where h(i) is an odd symmetric kernel of size k. ##EQU13## for 0≦u≦(N-1); where N represents the size of the DCT matrix used; 
     h(i)=0 for |i|&gt;(k-1)/2; 
     d i  =1/2 for i=0; 
     d i  =1 for i=1, 2 . . . (N-1); 
     i, u and N are integers; and 
     k is the kernel size. 
     Of course, the above mathematical filtering analysis could be readily extended to image data presented in any number of dimensions. 
     A preferred embodiment of a method of filtering two dimensional image data in the DCT domain, in a manner equivalent to a mathematical convolution in the spatial domain, is depicted in FIG. 6 which includes a forward DCT section 10 having steps identical to those previously described for the interpolation method of FIG. 3. As an example, the, 8×8 first spatial matrix s(j,i) of 64 image data points listed below is selected in block 20 as shown below. 
     
         __________________________________________________________________________FIRST SPATIAL MATRIX, s(j,i)__________________________________________________________________________100.00120.00     140.00          160.00               100.00                    100.00                         100.00                              100.00100.00100.00     120.00          140.00               160.00                    100.00                         100.00                              100.00120.00100.00     100.00          120.00               140.00                    160.00                         100.00                              100.00140.00120.00     100.00          100.00               120.00                    140.00                         160.00                              100.00160.00140.00     120.00          100.00               100.00                    120.00                         140.00                              160.00100.00120.00     140.00          160.00               100.00                    100.00                         100.00                              100.00100.00100.00     120.00          140.00               160.00                    100.00                         100.00                              100.00120.00100.00     100.00          120.00               140.00                    160.00                         100.00                              100.00__________________________________________________________________________ 
    
     The following 8×8 forward DECT basis matrix FB E  is derived from the cosine terms of equation (3) in block 22. 
     
         ______________________________________FORWARD DECT BASIS MATRIX, FB.sub.E______________________________________0.3540.354   0.354   0.354 0.354 0.354 0.354 0.3540.4900.416   0.278   0.098 -0.098                            -0.278                                  -0.416                                        -0.4900.4620.919   -0.191  -0.462                      -0.462                            -0.191                                  0.191 0.4620.416-0.098  -0.490  -0.278                      0.278 0.490 0.098 -0.4160.354-0.354  -0.354  0.354 0.354 -0.354                                  -0.354                                        0.3540.278-0.490  0.098   0.416 -0.416                            -0.098                                  0.490 -0.2780.191-0.462  0.462   -0.191                      -0.191                            0.462 -0.462                                        0.1910.098-0.278  0.416   -0.490                      0.490 -0.416                                  0.278 -0.098______________________________________ 
    
     The following 8×8 first intermediate matrix I 1  is generated in multiplier 25 by matrix product multiplication of the 8×8 forward DECT matrix FB E  times the 8×8 first spatial matrix s(j,i). 
     
         __________________________________________________________________________FIRST INTERMEDIATE MATRIX, I.sub.1__________________________________________________________________________332.340318.198     332.340          367.696               360.624                    346.482                         318.198                              304.056-6.2032.301     6.553          8.504               -6.553                    -10.806                         1.951                              -5.853-40.782-22.304     9.239          36.955               24.546                    -11.480                         -46.194                              -27.716-12.56723.678     41.801          36.245               -41.801                    -59.923                         -5.556                              16.66735.35521.213     -7.071          -28.284               -35.355                    21.213                         35.355                              21.213-11.920-4.710     -1.105          7.210               1.105                    -2.500                         8.315                              -24.944-6.0691.585     3.827          15.307               -33.128                    27.716                         -19.134                              -11.48016.1723.444     -2.920          -12.728               2.920                    9.283                         -9.808                              29.424__________________________________________________________________________ 
    
     In block 24, the following 8×8 first transpose matrix FB E   T  is generated by transposing the above 8×8 forward DCT basis matrix FB E . 
     
         ______________________________________FIRST TRANSPOSE MATRIX, FB.sub.E.sup.T______________________________________0.3540.490   0.462   0.416 0.354 0.278 0.191 0.0980.3540.416   0.191   -0.098                      -0.354                            -0.490                                  -0.462                                        -0.2780.3540.278   -0.191  -0.490                      -0.354                            0.098 0.462 0.4160.3540.098   -0.462  -0.278                      0.354 0.416 -0.191                                        -0.4900.354-0.098  -0.462  0.278 0.354 -0.416                                  -0.191                                        0.4900.354-0.278  -0.191  0.490 -0.354                            -0.098                                  0.462 -0.4160.354-0.416  0.191   0.098 -0.354                            0.490 -0.462                                        0.2780.354-0.490  0.462   -0.416                      0.354 -0.278                                  0.191 -0.098______________________________________ 
    
     The first intermediate matrix I 1  is matrix product multiplied in multiplier 26 times the first transpose matrix FB E   T  to generate the,, following DCT matrix S(v,u) of DCT coefficients as defined in equation (3) (i.e. S(v,u)=I 1  *FB E   T ) 
     
         __________________________________________________________________________DCT MATRIX, S(v,u)__________________________________________________________________________947.50010.632     -50.581          16.730               17.500                    9.417                         2.010                              -6.588-3.5736.265     -4.483          -12.875               -3.573                    7.684                         -6.609                              -0.299-27.48410.490     -72.730          -21.370               22.537                    -8.165                         5.732                              -5.382-0.51533.688     11.396          -86.570               -0.515                    19.912                         -14.895                              -6.95522.500-6.111     63.647          19.165               -27.500                    11.044                         3.402                              -9.918-10.0941.955     -19.490          4.306               -10.094                    12.679                         -11.975                              2.475-7.5579.356     -9.268          -1.511               -17.453                    9.149                         22.730                              -38.91212.653-5.906     23.158          3.529               12.653                    17.875                         16.480                              -2.374__________________________________________________________________________ 
    
     The above described steps complete the determination of the two dimensional forward DECT matrix S(v,u) necessary in equation (7). Next, the two dimensional forward DOCT matrix H(v,u) must be determined. The following 5×5 odd, symmetrical kernel h(j,i) in the spatial domain was arbitrarily selected in block 124 to illustrate the preferred embodiment of the filtering process. 
     
         ______________________________________KERNEL MATRIX, h(j,i)______________________________________0.006    0.025   0.037       0.025 0.0060.025    -0.330  -0.709      -0.330                              0.0250.037    -0.709  4.782       -0.709                              0.0370.025    -0.330  -0.709      -0.330                              0.0250.006    0.025   0.037       0.025 0.006______________________________________ 
    
     In order to facilitate multiplication of the 5×5 kernel matrix h(j,i) with other 8×8 matrixes (selected for this example and in conformance with the international JPEG standard matrix size), the 5×5 kernel matrix h(j,i) is padded in block 122 to generate an 8×8 padded kernel matrix h P  (j,i) in the spatial domain. The padding is accomplished by inserting the lower right quadrant of the odd, symmetric kernel into the upper left quadrant of the padded kernel matrix h P  (j,i), then setting the remaining elements of h P  (j,i) to zero as shown below. 
     
         ______________________________________PADDED KERNEL MATRIX, h.sub.p (j,i)______________________________________4.782 -0.709   0.037  0.000 0.000                            0.000 0.000 0.000-0.709 -0.330   0.025  0.000 0.000                            0.000 0.000 0.0000.037 0.025    0.006  0.000 0.000                            0.000 0.000 0.0000.000 0.000    0.000  0.000 0.000                            0.000 0.000 0.0000.000 0.000    0.000  0.000 0.000                            0.000 0.000 0.0000.000 0.000    0.000  0.000 0.000                            0.000 0.000 0.0000.000 0.000    0.000  0.000 0.000                            0.000 0.000 0.0000.000 0.000    0.000  0.000 0.000                            0.000 0.000 0.000______________________________________ 
    
     The two dimensional DOCT matrix H(v,u) is derived as follows from equation (9). ##EQU14## for 0≦u≦(N-1) and 0≦v≦(N-1), where h P  (j,i) is the two dimensional padded kernel matrix; 
     N is the number of elements of h P  (j,i) in the first dimension; 
     M is the number of elements of h P  (j,i) in the second dimension; 
     d i  =1/2 for i=0; 
     d i  =1 for i=1, 2, . . . (N-1); 
     d j  =1/2 for j=0; 
     d j  =1 for j=1,2, . . . (M-1); 
     i, j, u, v, N, M are integers; and ##EQU15## An 8×8 forward DOCT basis matrix FB o  (listed below) is derived in block 18 from the cosine terms of equation (9). 
     
         ______________________________________FORWARD DOCT BASIS MATRIX, FB.sub.O______________________________________1.0002.000   2.000   2.000 2.000 2.000 2.000 2.0001.0001.848   1.414   0.765 -0.000                            -0.765                                  -1.414                                        -1.8481.0001.414   -0.000  -1.414                      -2.000                            -1.414                                  0.000 1.4141.0000.765   -1.414  -1.848                      0.000 1.848 1.414 -0.7651.000-0.000  -2.000  0.000 2.000 -0.000                                  -2.000                                        0.0001.000-0.765  -1.414  1.848 -0.000                            -1.848                                  1.414 0.7651.000-1.414  0.000   1.414 -2.000                            1.414 -0.000                                        -1.4141.000-1.848  1.414   -0.765                      0.000 0.765 -1.414                                        1.848______________________________________ 
    
     In multiplier 128, the following 8×8 second intermediate matrix 12 is generated by matrix product multiplication of the 8×8 forward DOCT basis matrix FB o  times the 8×8 padded kernel matrix h p  (j,i). 
     
         ______________________________________SECOND INTERMEDIATE MATRIX, I.sub.2______________________________________3.439-1.319   0.099  0.000 0.000 0.000 0.000 0.0003.525-1.283   0.092  0.000 0.000 0.000 0.000 0.0003.779-1.175   0.072  0.000 0.000 0.000 0.000 0.0004.187-0.996   0.047  0.000 0.000 0.000 0.000 0.0004.708-0.759   0.025  0.000 0.000 0.000 0.000 0.0005.272-0.492   0.009  0.000 0.000 0.000 0.000 0.0005.785-0.243   0.002  0.000 0.000 0.000 0.000 0.0006.145-0.065   0.000  0.000 0.000 0.000 0.000 0.000______________________________________ 
    
     The 8×8 forward DOCT basis matrix FB o  is transposed in block 130 to produce the following 8×8 second transposed matrix, FB 0   T . 
     
         ______________________________________SECOND TRANSPOSED MATRIX, FB.sub.O.sup.T______________________________________1.0001.000   1.000   1.000 1.000 1.000 1.000 1.0002.0001.848   1.414   0.765 -0.000                            -0.765                                  -1.414                                        -1.8482.0001.414   -0.000  -1.414                      -2.000                            -1.414                                  0.000 1.4142.0000.765   -1.414  -1.848                      0.000 1.848 1.414 -0.7652.000-0.000  -2.000  0.000 2.000 -0.000                                  -2.000                                        0.0002.000-0.765  -1.414  1.848 -0.000                            -1.848                                  1.414 0.7652.000-1.414  0.000   1.414 -2.000                            1.414 -0.000                                        -1.4142.000-1.848  1.414   -0.765                      0.000 0.765 -1.414                                        1.848______________________________________ 
    
     Multiplier 136 generates the following 8×8 forward DOCT matrix H(v,u) by matrix product multiplication of the 8×8 second intermediate matrix I 2  times the second transpose matrix FB o . 
     
         ______________________________________FORWARD DOCT MATRIX, H(v,u)______________________________________1.0001.143   1.574   2.289 3.240 4.307 5.303 6.0161.1431.284   1.710   2.413 3.341 4.377 5.339 6.0251.5741.710   2.117   2.778 3.635 4.576 5.441 6.0532.2892.413   2.778   3.357 4.092 4.882 5.596 6.0953.2403.341   3.635   4.092 4.658 5.253 5.780 6.1444.3074.377   4.576   4.882 5.253 5.635 5.967 6.1945.3035.339   5.441   5.596 5.780 5.967 6.128 6.2366.0166.025   6.053   6.095 6.144 6.194 6.236 6.265______________________________________ 
    
     At this point, both functions S(v,u) and H(v,u) located on the right side of equation (7) have been determined. Thus by mask multiplying, the 8×8 forward DCT matrix S(v,u) times the 8×8 forward DOCT matrix H(v,u) in the mask multiplier 144, the following 8×8 third intermediate matrix, I 3 , is generated. 
     
         __________________________________________________________________________THIRD INTERMEDIATE MATRIX, I.sub.3__________________________________________________________________________947.50012.147     -79.602           38.292                 56.698                       40.562                             10.658                                  -39.631-4.0828.044     -8.283           -31.065                 -11.935                       33.631                             -35.284                                  -1.801-43.25317.941     -153.998           -59.356                 81.912                       -37.366                             31.191                                  -32.580-1.17881.280     31.652           -290.610                 -2.105                       97.215                             -83.353                                  -42.39272.897-20.417     231.332           78.419                 -128.091                       58.017                             19.667                                  -60.943-43.4778.557     -89.196           21.021                 -53.022                       71.447                             -71.460                                  15.329-40.08049.954     -50.429           -8.453                 -100.885                       54.596                             139.289                                  -252.66676.113-35.583     140.178           21.509                 77.744                       -110.722                             102.776                                  -14.871__________________________________________________________________________ 
    
     To reconstruct the filtered image, the DCT coefficients of the third intermediate matrix I 3  are first transformed to the spatial domain. The 8×8 second spatial matrix s&#39;(j,i) of reconstructed image data points shown below is generated in block 140 by performing an inverse discrete cosine transformation as in equation(4) on the 8×8 third intermediate matrix, I 3 . The filtered image can then be printed or otherwise reproduced from the reconstructed image data points of the second spatial matrix s&#39;(v,u). 
     
         __________________________________________________________________________SECOND SPATIAL MATRIX, s&#39;(j,i)__________________________________________________________________________100.00    118.34   125.30       154.46           155.73               105.72                   92.67                       105.79                           96.93                               100.0096.22    98.35   118.02       129.86           149.88               156.06                   108.90                       87.44                           108.79                               95.59113.21    98.81   100.44       112.07           129.44               154.30                   152.30                       111.85                           89.58                               108.17123.37    110.47   94.56       105.07           116.50               125.40                   155.93                       152.35                           106.50                               90.01144.37    127.55   111.72       94.88           99.18               117.57                   127.34                       152.46                           158.46                               107.25162.18    144.32   128.00       107.74           94.88               101.76                   114.48                       131.37                           151.18                               158.55121.70    131.43   127.96       146.68           137.88               93.04                   98.25                       115.04                           105.72                               125.7787.55    103.05   121.01       147.51           162.07               134.40                   96.59                       92.71                           98.21                               85.41107.94    96.93   109.44       114.68           135.47               164.35                   132.97                       95.27                           104.81                               106.94120.00    105.18   94.00       108.12           121.13               136.02                   161.82                       137.88                           90.84                               100.00__________________________________________________________________________ 
    
     As earlier noted, the above method is adequate when the image is relatively free from aliasing due to high frequency components. However, many images require the addition of an overlapping procedure in the above filtering method which is directed at eliminating aliasing due to high frequency components. When the high frequency components do not effectively disturb the desired image quality, then the overlapping procedure can be excluded and the above second spatial matrix s&#39;(j,i) is adequate to print or otherwise reproduce the original image. In the event that aliasing is a problem, then an overlapping procedure such as the one described below in conjunction with FIG. 7 is instituted together with the above method wherein certain reconstructed image data points of the second spatial matrix s&#39;(j,i) are discarded, resulting in a filtered saved region designated as s&#39; S  (j,i). 
     For a kernel size of 5×5, i.e. k=5, a k-1 overlap of four pixels between pixel groups is used to provide a 4×4 saved region of filtered pixels for an 8×8 group of pixels to be processed. Thus, for filtering with overlap in a manner similar to a mathematical convolution, each 8×8 pixel group of the image preferably overlaps each adjacent pixel group by four pixels. 
     FIG. 7 illustrates a preferred embodiment of the overlapping procedure where an image 150 is represented by pixels arranged in twelve rows and twelve columns. A first 8×8 pixel group 152 of 64 image data points spans rows 1-8 and columns 1-8 as designated by the dark, solid line. A second 8×8 pixel group 154 of 64 image data points spans rows 1-8 and columns 5-12 as designated by a dotted line. Each pixel group represents a two dimensional first spatial matrix s(j,i) for filtering according to the above method of FIG. 8 including overlapping. The overlap region is shown in FIG. 7 as the area including rows 1-8 and columns 5-8. Although the example of FIG. 7 shows overlapping of matrices which are horizontally adjacent in the image 150, the same method applies to matrices which are vertically adjacent in the image 150. 
     After the first spatial matrix s(j,i) (defined as first pixel group 152) is processed and filtered in the DCT domain, the second spatial matrix s&#39;(j,i) of reconstructed image data points is generated by an IDCT of third intermediate matrix I 3  in block 140 of FIG. 6. Then, in block 142, a perimeter or border area of the second spatial matrix s&#39;(j,i) is determined as (k-1)/2 pixels wide and the reconstructed image data points within the perimeter area are discarded. The remaining pixels constitute the saved matrix s&#39; S  (j,i) of filtered reconstructed pixels. In FIG. 7, all reconstructed image data points for block 152 located in rows and columns 1, 2, 7 and 8 are discarded, resulting in the saved matrix s S  &#39;(j,i) of reconstructed image data points located in rows 3-6, columns 3-6. For second pixel group 154, the second spatial matrix s&#39;(j,i) of reconstructed image data points includes rows 1-8, columns 5-12. The reconstructed filtered pixels discarded in block 142 for second pixel group 154 include all reconstructed pixels in rows 1, 2, 7, 8 and columns 5, 6, 11 and 12. The saved matrix s&#39; S  (j,i) of second pixel group 154 includes reconstructed image data points located at rows 3-6, columns 7-10. 
     In the case where filtering the image in the frequency domain by a method similar to a mathematical convolution in the spatial domain, is combined with scaling the image, the overlap is determined by taking into consideration the kernel size k. 
     A filtering system which operates according to the method of FIG. 6 is depicted in the schematic block diagram of FIG. 8. The system includes a forward DECT processor 50, a forward DOCT processor 158, a zero pad processor 159, a mask multiplier 156, and an IDCT processor 160. The IDCT processor 160 could readily be replaced with a hybrid IDCT processor as previously described in conjunction with FIGS. 3 and 4. 
     DECT processor 50 includes a cosine processor 52, a transposer 51 and a matrix multiplier 54. The first spatial matrix s(j,i) is selected as an 8×8 matrix of image data points and the forward DECT basis matrix FB E  is generated in block 52 as an 8×8 matrix from the cosine terms of equation (3). The first transposed matrix FB E   T  is produced as an 8×8 matrix by transposing the forward DECT basis matrix FB E  in transposer 51. The first intermediate matrix I 1  is generated as an 8×8 matrix in matrix multiplier 54 by matrix product multiplication of the forward DECT basis matrix FB E  times the first spatial matrix s(j,i). The forward DECT matrix S(v,u) is then generated in the matrix multiplier 54 as an 8×8 matrix by matrix product multiplication of the 8×8 first intermediate matrix I 1  times the 8×8 first transposed matrix FB E   T . 
     DOCT processor 158 includes a cosine processor 118, a transposer 115, and a matrix multiplier 157. Since the first spatial matrix s(j,i) was selected as an 8×8 matrix of image data points in the forward DECT processor 50, forward DOCT basis matrix FB o  is also generated as an 8×8 matrix in cosine processor 118 from the cosine terms of equation (9). The second transposed matrix FB o   T  is produced in transposer 115 as an 8×8 matrix by transposing the forward DOCT basis matrix FB o . The kernel matrix h(j,i) is input to a zero pad processor 159 which generates the 8×8 padded kernel matrix h p  (j,i). Matrix multiplier 157 generates the second intermediate matrix I 2  as an 8×8 matrix by matrix product multiplication of the forward DOCT basis matrix FB o  times the padded kernel matrix h P  (j,i). The forward DOCT matrix H(v,u) is then generated as an 8×8 matrix in matrix multiplier 157 by matrix product multiplication of the second intermediate matrix I 2  times the second transposed matrix FB o   T . 
     The mask multiplier 156 produces the third intermediate matrix I 3  as an 8×8 matrix by mask multiplication of the forward DECT matrix S(v,u) with the forward DOCT matrix H(v,u). 
     The IDCT processor 160 includes a cosine processor 162, a transposer 164 and a matrix multiplier 166. The cosine processor 162 generates an 8×8 IDCT basis matrix IB from the cosine terms of equation (4). A third transposed matrix IB T  is produced as an 8×8 matrix in transposer 164 by transposing the IDCT basis matrix IB. A fourth intermediate matrix I 4  is generated in matrix multiplier 166 as an 8×8 matrix by matrix product multiplication of the IDCT basis matrix IB times the third intermediate matrix I 3 . Finally, the second spatial matrix s&#39;(j,i) of reconstructed image data points is generated in the matrix multiplier 166 by matrix product multiplication of the fourth intermediate matrix I 4  times the third transposed matrix IB T . The second spatial matrix s&#39;(j,i) can be used to print or otherwise reproduce the filtered image. However, if aliasing due to noise is a problem, then overlapping hardware (not shown) could be added to prevent aliasing by the previously discussed methods of overlapping adjacent pixel groups. 
     Reducing An Image By Decimation 
     At times it may be desirable to reduce the size of an image for printing or display applications. This can be accomplished by a method referred to as decimation, defined as reducing the image size by eliminating data points. Key features of the above resampling and filtering methods are incorporated into the method of reducing an image by decimation. In fact, reducing an image by decimation is a special case of scaling where the image is scaled down by downsampling using a scaling ratio less than one. 
     Typically, low pass filtering of an image is performed prior to reducing the image by decimation. The low pass filtering reduces high frequency artifacts. However, pre-decimation filtering is not necessary in the case when the spatial aliasing is inconsequential to accurate image reproduction. 
     The schematic block diagram of FIG. 9 shows the steps of a method for reducing an image by decimation as described in the following preferred embodiment for two dimensional image data. Repetition of some of the detailed explanations of previously described mathematics and steps will be omitted for the sake of brevity. 
     FIG. 9 shows a forward DECT section 10, a filtering section 220, and a decimation section 224. The forward DECT section 10 corresponds to the forward DCT section 10 of FIG. 3, the filtering section 220 includes the filtering steps of FIG. 6, and the decimation section is related to the hybrid IDCT section 12 of FIG. 3. 
     The two dimensional DECT matrix S(v,u), generated as described for the example of enlarging an image by interpolation in FIG. 3, is mask multiplied in the mask multiplier 144 with the forward DOCT matrix H(v,u) to generate a third intermediate matrix I 3  (i.e. a mask multiplied matrix) as previously described for FIG. 6. Decimation section 224 down samples the two dimensional hybrid IDCT basis matrix in either one or both dimensions so that fewer reconstructed image data points s&#39;(y,x) will be generated than the original number of image data points s(j,i). The reduced image (not shown) is then printed, processed or otherwise reconstructed from the reconstructed image data points s&#39;(y,x). 
     For the preferred embodiment of a method for reducing an image by decimation as shown in FIG. 9, the forward DECT section 10 includes the step of selecting the first spatial matrix s(j,i) as an 8×8 matrix in block 20. The 8×8 forward DECT basis matrix FB E  is determined in block 22 from the cosine terms of equation (3). In multiplier 25, the 8×8 first intermediate matrix I 1  is generated by matrix product multiplication of the forward DCT basis matrix FB E  times the first spatial matrix s(j,i). The first transpose matrix FBE E   T  is generated as an 8×8 matrix in block 24 by transposing the forward DECT basis matrix FB E . Multiplier 26 produces the 8×8 forward DECT matrix S(v,u) by matrix product multiplication of I 1  times the first transposed matrix FBE E   T . For this example, the values of the first spatial matrix s(j,i), the forward DECT basis matrix FB E , the first intermediate matrix I 1 , the first transpose matrix FB E   T , and the forward DECT matrix S(v,u) are the same as those listed above for the example of FIG. 3. 
     The filtering section 220 includes the step of selecting in block 124 a filter kernel or convolution kernel represented as h(j,i) in the spatial domain. The kernel is selected according to predetermined criteria such as a desire to sharpen or smooth the image. For our example, the following odd, symmetric 3×3 kernel h(j,i) was arbitrarily selected for low pass filtering. 
     
         ______________________________________KERNEL, h(j,i)______________________________________0.08            0.08   0.080.08            0.33   0.080.08            0.08   0.08______________________________________ 
    
     The 8×8 padded kernel matrix h(j,i) shown below is generated in block 122 to conform with the size of the 8×8 first spatial matrix chosen in block 20. 
     
         ______________________________________PADDED KERNEL MATRIX, h.sub.P (j,i)______________________________________0.33 0.08    0.00    0.00  0.00  0.00  0.00  0.000.08 0.08    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.000.00 0.00    0.00    0.00  0.00  0.00  0.00  0.00______________________________________ 
    
     The following 8×8 forward DOCT basis matrix FB o  of padded kernel matrix h P  (j,i) is derived from the cosine terms of equation (9) as previously described in the filtering example of FIG. 6. 
     
         ______________________________________FORWARD DOCT BASIS MATRIX, FB.sub.O______________________________________1.0002.000   2.000   2.000 2.000 2.000 2.000 2.0001.0001.848   1.414   0.765 -0.000                            -0.765                                  -1.414                                        -1.8481.0001.414   -0.000  -1.414                      -2.000                            -1.414                                  0.000 1.4141.0000.765   -1.414  -1.848                      0.000 1.848 1.414 -0.7651.000-0.000  -2.000  0.000 2.000 -0.000                                  -2.000                                        0.0001.000-0.765  -1.414  1.848 -0.000                            -1.848                                  1.414 0.7651.000-1.414  0.000   1.414 -2.000                            1.414 -0.000                                        -1.4141.000-1.848  1.414   -0.765                      0.000 0.765 -1.414                                        1.848______________________________________ 
    
     In multiplier 128, the 8×8 forward DOCT basis matrix FB o  is matrix product multiplied times the 8×8 padded kernel matrix h p  (j,i) to produce the 8×8 second intermediate matrix I 2  shown below. 
     
         ______________________________________SECOND INTERMEDIATE MATRIX, I.sub.2______________________________________0.5000.250    0.000  0.000 0.000 0.000 0.000 0.0000.4870.237    0.000  0.000 0.000 0.000 0.000 0.0000.4510.201    0.000  0.000 0.000 0.000 0.000 0.0000.3970.147    0.000  0.000 0.000 0.000 0.000 0.0000.3330.083    0.000  0.000 0.000 0.000 0.000 0.0000.2700.020    0.000  0.000 0.000 0.000 0.000 0.0000.215-0.035   0.000  0.000 0.000 0.000 0.000 0.0000.179-0.071   0.000  0.000 0.000 0.000 0.000 0.000______________________________________ 
    
     The following 8×8 second transposed matrix FB o   T  is generated by transposing the 8×8 forward DOCT basis matrix FB o  in block 130. 
     
         ______________________________________SECOND TRANSPOSED MATRIX, FB.sub.O.sup.T______________________________________1.0001.000   1.000   1.000 1.000 1.000 1.000 1.0002.0001.848   1.414   0.765 -0.000                            -0.765                                  -1.414                                        -1.8482.0001.414   -0.000  -1.414                      -2.000                            -1.414                                  0.000 1.4142.0000.765   -1.414  -1.848                      0.000 1.848 1.414 -0.7652.000-0.000  -2.000  0.000 2.000 -0.000                                  -2.000                                        0.0002.000-0.765  -1.414  1.848 -0.000                            -1.848                                  1.414 0.7652.000-1.414  0.000   1.414 -2.000                            1.414 -0.000                                        -1.4142.000-1.848  1.414   -0.765                      0.000 0.765 -1.414                                        1.848______________________________________ 
    
     The forward DOCT matrix H(v,u) shown below is generated as an 8×8 matrix in multiplier 136 by matrix product multiplication of the 8×8 second intermediate matrix I 2  times the 8×8 second transposed matrix, FB o   T . 
     
         ______________________________________FORWARD DOCT MATRIX, H(v,u)______________________________________1.0000.962   0.854   0.691 0.500 0.309 0.146 0.0380.9620.926   0.823   0.669 0.487 0.306 0.152 0.0490.8540.823   0.736   0.605 0.451 0.297 0.167 0.0790.6910.669   0.605   0.510 0.397 0.285 0.189 0.1250.5000.487   0.451   0.397 0.333 0.270 0.215 0.1790.3090.306   0.297   0.285 0.270 0.255 0.242 0.2330.1460.152   0.167   0.189 0.215 0.242 0.264 0.2790.0380.049   0.079   0.125 0.179 0.233 0.279 0.310______________________________________ 
    
     The following 8×8 third intermediate matrix, 13, is generated in mask multiplier 144 by mask multiplication of the 8×8 forward DECT matrix, S(v,u), with the 8×8 forward DOCT matrix, H(v,u). 
     
         __________________________________________________________________________THIRD INTERMEDIATE MATRIX, I.sub.3__________________________________________________________________________947.50010.206     -42.994          11.543               8.750                    2.919                         0.301                              -0.264-3.4305.826     -3.971          -8.626               -1.751                    2.382                         -0.991                              -0.015-23.3628.602     -53.820          -13.036               10.141                    -2.449                         0.974                              -0.431-0.35522.571     6.951          -44.151               -0.206                    5.575                         -2.830                              -0.90411.250-2.995     28.641          7.666               -9.075                    2.982                         0.749                              -1.785-3.1290.606     -5.847          1.206               -2.725                    3.170                         -2.874                              0.569-1.1341.403     -1.576          -0.287               -3.840                    2.196                         5.910                              -10.8950.506-0.295     1.853          0.459               2.277                    -4.111                         4.614                              -0.736__________________________________________________________________________ 
    
     At this point, a standard inverse DCT (as in equation (4)) of the above third intermediate matrix I 3  would result in an 8×8 second spatial matrix s&#39;(j,i) of reconstructed image data points. If overlapping of adjacent groups of pixels is deemed necessary to prevent aliasing (as previously discussed in accordance with the filtering examples of FIGS. 6 and 7), then the overlap for filtering with the above 3×3 kernel matrix h(j,i) would be k-1 or two pixels, amounting to a one pixel wide perimeter area (k-1)/2 of the second spatial matrix s&#39;(j,i) to be discarded. The discarded elements of s&#39;(j,i) would include all reconstructed image data points in rows 1, 8 and columns 1, 8. The filtered saved reconstructed matrix s&#39; S  (j,i) would be a 6×6 matrix including 36 reconstructed image data points located in rows 2-7, columns 2-7. In other words, a one pixel wide perimeter of the low pass filtered version of the second spatial matrix s&#39;(j,i) would be discarded to prevent aliasing. 
     Rather than immediately performing a standard IDCT on the third intermediate matrix I 3  to generate a full size reconstructed filtered image as discussed above, a reduced image can be obtained when scaling down or downsampling by selecting the first dimension scaling ratio in block 23A to be less than one (i.e. 6:8 where N&#39;=6), and selecting the second dimension scaling ratio in block 23B to be less than one (i.e. 6:8 where M&#39;=6), wherein the first dimension scaling ratio equals the second dimension scaling ratio for the preferred embodiment. Conventionally, selection of the scaling ratios has been limited to values which reduce (or enlarge) an image without undue mathematical computation. For instance, a scaling milo of 1:2 is typical, since an 8×8 first spatial matrix s(j,i) of image data points is mathematically related to a 4×4 second spatial matrix of sixteen reconstructed image; data points by the integer multiple of two. In contrast, a scaling ratio of 2:5 would be atypical. However, according to a feature of the invention, any scaling ratio can be selected for each dimension (as previously described in the methods of FIGS. 3 and 4 for enlarging an image by interpolation). Values of reconstructed image data points of the second spatial matrix s&#39;(y,x) which fall between the indexed locations of the image data points of the first spatial matrix s(j,i) are determined by solving s&#39;(y,x) for values of x and y falling within the appropriate ranges for u and v in equation (6). For instance, when selecting a 2:5 resampling ratio for each dimension, the values of s&#39;(y,x) for the first dimension interpolated to a curve (in the range of 0 to 7) are evaluated at x=2.5 and x=5 for the first group of eight pixels, as shown and earlier discussed in relation to FIG. 2C. 
     When choosing both the first and second dimension scaling ratios as 2:5, the following 2×8 hybrid IDCT basis matrix IB 1HYB  is generated in block 30 from the cosine terms of equation (6). 
     
         ______________________________________HYBRID IDCT BASIS MATRIX, IB.sub.1HYB______________________________________0.3540.191   -0.354  -0.462                      0.000 0.462 0.354 -0.1910.354-0.278  -0.191  0.490 -0.354                            -0.098                                  0.462 -0.416______________________________________ 
    
     A 2×8 fourth intermediate matrix 14 as shown below is generated in multiplier 32 of decimation section 224 by matrix product multiplication of the 2×8 hybrid IDCT basis IB 1HYB  matrix times the 8×8 third intermediate matrix I 3 . 
     
         ______________________________________FOURTH INTERMEDIATE MATRIX, I.sub.4______________________________________340.816 -7.912  -3.756  27.802                       -3.784                             2.806                                  0.759 -2.975335.834 13.183  -11.445 -15.831                       2.292 4.933                                  -0.365                                        -4.601______________________________________ 
    
     An 8×2 third transposed matrix IB 1HYB   T  (not shown) is generated in block 28 by transposing the hybrid IDCT basis matrix, IB 1HYB . Finally in multiplier 33, the following 2×2 second spatial matrix s&#39;(j,i) of reconstructed filtered image data points is generated by matrix product multiplication of the 2×8 fourth intermediate matrix I 4  times the 8×2 third transposed matrix, IB 1HYB   T . 
     
         ______________________________________SECOND SPATIAL MATRIX, s&#39;(j,i)______________________________________   109.601      139.698   135.647      109.952______________________________________ 
    
     In block 36, the image can be printed or otherwise reproduced using the reconstructed image data points of second spatial matrix, s&#39;(j,i). 
     The ordering of steps in the above examples is not absolute or unyielding in each case. One of ordinary skill in the art could easily ascertain when the ordering of steps is critical. Also, any number of dimensions of image data points could be processed according to the teachings of the invention, i.e. the above examples using one and two dimensional image data points are exemplary rather than limiting. Additionally, the numerous features of the invention can be combined or separately applied as desired. For instance, an image represented by two dimensional image data could be scaled for reduction in one dimension and enlargement in the other dimension. Finally, the above embodiments of novel methods and devices are preferred examples of the many variations and modifications which would be apparent to one of ordinary skill in the art in keeping with the invention as claimed.