Abstract:
A method and apparatus for magnifying a portion of a digital image on a display screen in either of two ways. The first method includes a two pass scheme, where each of the passes represents an interpolation in x and y direction respectively, cubic interpolation in each direction is approximated using a one dimensional convolution filter followed by linear interpolation. The second method uses a two dimensional convolution filter first, followed by bilinear interpolation. All of the procedures that are used are accelerated using a hardware package which facilitates exceptionally fast execution.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     Not applicable. 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
     BACKGROUND OF THE INVENTION 
     The present invention relates to imaging methods and apparatus and more specifically to a method and apparatus which facilitates magnification or zooming of a portion of an image on a display. 
     Digital imaging is extremely important in many different applications. For example, digital imaging has proven invaluable in the medical imaging art where vast amounts of data are collected and used to generate images for observation on a video monitor or display. Although not so limited, in the interest of simplifying this explanation the present invention will be described in the context of medical imaging. A typical display includes a two-dimensional raster of pixels. For the purpose of this explanation, although raster pixels may be arranged in any of several different configurations, it will be assumed that pixels are arranged in distinct rows and columns. 
     To generate images on the display a processor collects all image data and generates intensity signals or pixel values for each display pixel. The pixel values are provided to a video driver which excites each pixel separately in accordance with an associated pixel value. From a distance the contrast between display pixel intensities is seen as an image. 
     Magnification of digital images in real time is needed in many different applications. For example, it may be advantageous to magnify a section of a medical image which includes a tumor. To this end, software has been developed which allows a user to select an image section for magnification and then magnifies the selected section. 
     One problem with image magnification has been selecting pixel intensities in a magnified image which reflect the initial image and result in a clean magnified image. For example, assume an intensity range between 0 and 100 where first and second adjacent pixel values correspond to intensities 55 and 92. Also assume that upon magnification, the area corresponding to the first and second pixels increases and covers 32 pixels (e.g. 16 pixels each). In this case, if the 16 magnified pixels corresponding to the first pixel and the 16 magnified pixels corresponding to the second pixel are provided with intensities 55 and 92, respectively, and all other magnified pixels are similarly magnified, the result is a highly granulated image which, in many cases, is not useful for the intended purpose of closer examination. 
     Instead of exciting pixels in the magnified image as indicated above (i.e. with either intensities 55 or 92), other solutions have been adopted by the industry with varying success and at varying costs. The most common methods for calculating new pixel intensities are nearest neighbor, bilinear and bicubic interpolation methods. 
     According to the nearest neighbor method, when points corresponding to first and second adjacent pixels on an initial image are separated by magnification so that the initial image points correspond to third and fourth pixels which are separated by a plurality of other pixels, the intensity of each of the other pixels (e.g. pixels between the third and fourth) are set equal to the intensity of the closest of the third or fourth pixel. This solution has the advantage of being computationally simple and therefore can be implemented easily using existing imaging hardware. Unfortunately, nearest neighbor methods only increase magnified image quality slightly and therefore are unacceptable for many applications. 
     Bilinear methods generally linearly fill in pixel intensities. For instance, in the example above where points corresponding to first and second adjacent pixels on an initial image are separated by magnification so that the initial image points correspond to third and fourth pixels which are separated by three other pixels and the first and second pixel intensities were 55 and 92, respectively, the other pixel intensities are linearly determined and are approximately 64, 73 and 83. High speed bilinear image magnification is now commonly available in accelerated graphics hardware. Unfortunately, while this solution generates a better image than the nearest neighbor methods, this solution requires much more processor time to perform necessary computations and still does not provide an extremely accurate magnification. 
     Bicubic interpolation methods generally take into account the intensities of more than just first and second adjacent pixels when determining the intensities of pixels in a magnified image which are between image points which correspond to the initial first and second pixels. In effect, these interpolation methods mathematically identify pixel intensities on one or more curves wherein the curves correspond to proximate initial pixel intensities of several pixels about an area. 
     These interpolation methods are extremely accurate and generate diagnostic quality magnified images. Unfortunately, these methods require massive amounts of processor time and therefore, in many cases, cannot be performed in real time because of processor limitation. For this reason high quality bicubic image magnification is not generally available. 
     One way to speed up calculations is to provide special hardware which is specifically designed to perform specific calculations. For example, many image processing systems include special hardware to perform either one or two dimensional high speed convolution filtering and linear or bilinear interpolation processes required in many imaging application. Unfortunately, hardware solutions have not yet been provided to facilitate bicubic interpolation. 
     Therefore, it would be advantageous to have a method and an apparatus which can be used with existing hardware to facilitate image magnification wherein resulting magnified images are of a quality which is essentially identical to the quality achievable using bicubic interpolation and wherein the method facilitates real time magnification. 
     BRIEF SUMMARY OF THE INVENTION 
     It has been recognized that specific hardware which already exists in many imaging work stations can be used to imitate bicubic interpolation thereby providing extremely accurate image magnification. Importantly, while the functions and calculations performed by the hardware are extremely computationally intensive, the special hardware can perform the functions and calculations extremely quickly thereby giving the appearance of real time zooming or magnification. 
     Generally, the hardware includes a convolver, an interpolator, and a processor for identifying convolver coefficients. After the coefficients are identified, the convolver convolves pixel intensities for each pixel in the original image region to be magnified thereby generating at least two intermediate values. The interpolator interpolates the intermediate values to generate output pixel intensities. 
     Specifically, according to a one dimensional method of zooming, the invention includes a method and apparatus to be used with a digital imaging system to facilitate magnification of a region of interest on an initial digital image wherein the region of interest includes a plurality of pixels of interest, adjacent halves of adjacent pixels of interest forming interpixel intervals of interest. Each pixel of interest is characterized by a pixel intensity. The system includes convolution filter and interpolation hardware. The method comprises the steps of, for each interpixel interval of interest, identifying adjacent pixels, for each adjacent pixel, identifying a convolution window and for each window, determining a convolution filter coefficient for each pixel in the window, convoluting the pixel intensities of the pixels in each window using the coefficients to generate an intermediate value, and interpolating the intermediate values to generate interpolated pixel intensities, the interpolated pixel intensities together forming an interpolated image. 
     Preferably, according to the one-dimensional method, the initial pixel array is arranged in y columns and x rows and the step of selecting includes the step of, for each interpixel interval of interest between pixels in adjacent columns, selecting first and second row convolution windows consisting of pixels which are in the same row as the interval of interest and wherein, the method further includes the steps of, for each interpixel interval between pixels in adjacent rows on the interpolated image, selecting first and second separate column convolution windows on the interpolated image, each column window including adjacent image pixels within the same column as, and proximate the interpixel interval of interest, for each window, determining a convolution filter coefficient for each pixel in the window and convoluting the pixel intensities in each window using the coefficients to generate an intermediate value, and interpolating the intermediate values to generate a final pixel intensity, the final pixel intensities together forming the final image. The invention also includes a two-dimensional method and apparatus for zooming which is similar to the one-dimensional method and apparatus. 
     In the description, reference is made to the accompanying drawings which form a part hereof, and in which there is shown a preferred embodiment of the invention. Such embodiment does not necessarily represent the full scope of the invention and reference is made therefore, to the claims herein for interpreting the scope of the invention. 
    
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
     FIG. 1 is a schematic view illustrating one dimensional convolution interpolation methods according to the present invention; 
     FIG. 2 is a is a schematic illustrating a convolution window in accordance with a second embodiment of the present invention; 
     FIG. 3 is a schematic view illustrating a relationship between intermediate intensity values and all pixel intensities according to the second embodiment to the invention; 
     FIG. 4 is a perspective view of a computer system used to facilitate the inventive methods; 
     FIG. 5 is a schematic diagram of hardware used in both the first and second embodiments to the present invention; 
     FIG. 6 is a schematic illustrating hardware according to a one dimensional convolution embodiment of the present invention; 
     FIG. 7 is a schematic diagram illustrating a region of interest and expanded region of interest according to the present invention; 
     FIG. 8 is similar to FIG. 6, albeit illustrating hardware required for a two-dimensional convolution system according to the present invention; 
     FIG. 9 is a flow chart illustrating operation of the hardware of FIG. 5; 
     FIG. 10 is a flow chart illustrating operation of the hardware of FIG. 6; and 
     FIG. 11 is a flow chart illustrating operation of the hardware of FIG.  8 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     A. Theory 
     Generally, it has been recognized that bicubic interpolation can be closely estimated or mimicked by performing a two-dimensional convolution of initial pixel intensities to generate intermediate values and then linearly interpolate between intermediate values to generate specific final image pixel intensities. While such convolution and interpolation procedures are computationally intensive, many imaging systems now include hardware which can perform required computations extremely quickly such that computations appear to be performed in real time. Specifically, the hardware includes convolution filters and linear or bilinear interpolators which can quickly provide the functions required to estimate pixel intensities which would be generated via software based bicubic interpolation. 
     Some systems only include hardware which can be used to perform either one or two-dimensional convolution and either linear or bilinear interpolation. The present invention includes a first method referred to herein as a one-dimensional convolution (ODC) method which can be used by hardware which is limited to one-dimensional convolution and linear interpolation to estimate output pixel intensities. In addition, the present invention also includes a second method referred to herein as a two-dimensional convolution (TDC) method which can be used by hardware which can perform two-dimensional convolution and bilinear interpolation to estimate output pixel intensities. Each of the one and two-dimensional convolution methods is described separately below. 
     In each of the methods below it will be assumed that display pixels are arranged in x horizontal rows and y vertical columns. In addition, it will be assumed that a command received from a user indicates that a specific image region should be magnified by a factor n where n is 5 (i.e. magnification is by a factor of 5). Thus, generally, the area of a first image which corresponds to a single pixel will be magnified such that the same area in the final image corresponds t o a b lock of pixels having 5×5 dimensions. The region to be magnified will be referred to herein as a region of interest and each pixel within the region of interest will be referred to as a pixel of interest. 
     1. One-Dimensional Convolution (ODC) Method 
     According to the ODC method, when a command is received to magnify region of interest by a factor of n, prior to convolution and interpolation an expanded region of interest which includes the region of interest and an additional number of rows above and below the region of interest is selected. The additional rows in the expanded must be processed in the x dimension so that data is generated for processing in the y dimension. This requirement will become more clear below. 
     After the expanded region has been selected, convolution and interpolation hardware is used to modify pixel data first in the x (i.e. row) dimension to generate an interpolated image. Thereafter, the interpolated image data is fed again to the convolution and interpolation hardware which modifies the pixel data in the y (i.e. column) dimension thereby generating a final or output image. Each pass through the hardware includes a two step process corresponding to each pixel processed during the pass. In the first step, for each pixel in the image being processed the convolution hardware selects a separate convolution window on the image being processed (i.e. on the initial image during the first pass and on the interpolated image during the second pass). During the first pass each window (i.e. a row window) is in the same row as a corresponding pixel of interest and during each second pass each window (i.e. a column window) is in the same column as a corresponding final pixel. In addition, each window includes pixels adjacent the pixel of interest. The number of pixels in each convolution window depends on a filter size m where m is typically a number of pixels (e.g. 5, 7, etc.). Hereinafter m will be assumed to be 5 unless indicated otherwise. 
     To better understand convolution windows and the inventive ODC method refer to FIG. 1 wherein six initial pixels P 1  through P 6  are illustrated. The pixels P 1  through P 6  are to be magnified by a factor of 5. For the purpose of this explanation, when referring to the ODC method, the term “interpixel interval” will be used to refer to adjacent halves of adjacent pixels. For example, in FIG. 1, an interpixel interval corresponding to pixels P 3  and P 4  is identified in phantom as P 34 . Similarly, there is an interpixel interval between each other two adjacent pixels. To magnify pixels P 1  through P 6 , according to the ODC method each interpixel interval is magnified by a factor of 5. To this end, two convolution windows including m pixels are selected which correspond to each interval. For example, where a filter size m is 5, for interval P 34 , the windows are first window  10   a  including pixels P 1  through P 5  and second window  10   b  including pixels P 2  through P 6 . The intensities of pixels P 1  through P 6  are identified as I 1  through I 6 . 
     When the 5 times magnification request is received, in effect, the area in the initial image corresponding to each interpixel interval is first expanded in the x dimension to cover five pixels. For example, in FIG. 1, the area corresponding to interpixel interval P 34  is expanded into interpolated pixels P 1 ′ through P 5 ′ having intensities O 1  through O 5  forming an output pixel block  12 . Similarly, the interval areas between other initial adjacent pixel pairs P 1  and P 2 , P 2  and P 3 , P 4  and P 5  and P 5  and P 6  are each expanded into five distinct interpolated pixel intensities. Thus, pixels P 1  through P 6  are first expanded in the x dimension to form 25 separate interpolated pixels (i.e. 5 output pixels corresponding to each interpixel interval), each output pixel characterized by a separate intensity. Thereafter, the same procedure is repeated in the y dimension to provide five separate output pixel intensities for the area between any two adjacent interpolated pixels. 
     Referring still to FIG. 1, to expand interval P 34  in the x dimension, convolution windows  10   a  and  10   b  are first selected about adjacent pixels P 3  and P 4 . After convolution windows  10   a  and  10   b  have been selected the convolution hardware convolutes each of the windows  10   a  and  10   b  separately, thereby generating two intermediate values b 1  and b 2 , respectively. The convolution equations for intermediate values b 1  and b 2  are expressed as:                  b   1     =       ∑     i   =   1     m            a   i          I   i                         and           (   1   )                 b   2     =       ∑     i   =   1     m            a   i          I     i   +   1                   (   2   )                                
     where a i  are coefficients of the convolution filter which depend on the magnification factor n, where, in this example, n is 5. 
       
     After each of intermediate values b 1  and b 2  have been identified, the intermediate values are provided to the linear interpolation hardware which interpolates intensities b 1  and b 2  across n output pixels. To this end, the interpolation hardware performs the following equations to determine intensities O k  of final pixels P 1 ′, P 2 ′, P 3 ′, P 4 ′ and P 5 ′ where k is 1 through n, in the present example 1 through 5. Where n is odd:                O   k     =           (     n   -   k   +   1     )     n          b   1       +         (     k   -   1     )     n          b   2                 (   3   )                                
     and, where n is even:                O   k     =           (     n   -   k   +   0.5     )     n          b   1       +         (     k   -   0.5     )     n          b   2                 (   4   )                                
     Substituting values b 1  and b 2  from Equations 1 and 2 into Equations 3 and 4 and simplifying yield, where n is odd:                O   k     =           (     n   -   k   +   1     )     n            ∑     i   =   1     m            a   i          I   i           +         (     k   -   1     )     n            ∑     i   =   1     m            a   i          I     i   +   1                       (   5   )                                
     and where n is even:                O   k     =           (     n   -   k   +   0.5     )     n            ∑     i   =   1     m            a   i          I   i           +         (     k   -   0.5     )     n            ∑     i   =   1     m            a   i          I     i   +   1                       (   6   )                                
     Equations 5 and 6 can be expressed as:                O   k     =       ∑       i   1     =   1       m   +   1              ∑       i   2     =   2     m            λ       i   1          i   2              a     i   2            I     i   1                     (   7   )                                
     where the λ i     1     i     2    values are a function of the coefficients with corresponding I values from either Equation 3 or Equation 4, depending on if n is odd or even. For example, if n is odd and i 2  is 1, λ i     1     i     2    is from Equation 3 or:                λ       i   1        1       =       n   -   k   +   1     n             (   8   )                                
     The only unknowns in Equation 7 are convolution filter coefficients a i     2   . Therefore, if coefficients a i   2  can be identified for a specific magnification factor n, Equations 1, 2 and 3 in the case of an odd factor n or, in the case of an even factor n, Equations 1, 2 and 4, can be performed to determine intensities O 1 -O 5  of pixels P 1 ′-P 5 ′. 
     To identify the convolution filter coefficients a i     2   , a set of equations similar to Equation 7 is first derived in accordance with conventional bicubic interpolation methods. Thereafter common coefficients of Equation 7 and the cubic Equations are set equal and solved for the convolution filter coefficients. To this end, as well known, where there are four equidistant points x 1 , x 2 , x 3  and x 4  on an image and a function f defines intensity values I 1 , I 2 , I 3  and I 4  of the four points, respectively, the intensity value of the function at some point x between points x 2  and x 3  can be determined as follows. First, assume dx is the normalized distance defined by:              dx   =       x   -     x   2           x   3     -     x   2                 (   9   )                                
     with distance dx so defined, the following cubic coefficients c 1  through c 4  can be calculated:                c   1     =         -     1   3          dx     +       1   2            (   dx   )     2       -       1   6            (   dx   )     3                 (   10   )                 c   2     =     1   -       1   2        dx     -       (   dx   )     2     +       1   2            (   dx   )     3                 (   11   )                 c   3     =     dx   +       1   2            (   dx   )     2       -       1   2            (   dx   )     3                 (   12   )                 c   4     =         -     1   6          dx     +       1   6            (   dx   )     3                 (   13   )                                
     The intensity value O k  of function f at point x, using cubic interpolation can therefore be expressed as:                O   k     =       f        (   x   )       =       ∑     i   =   1     4            c   1          I   1                   (   14   )                                
     Comparing Equations 7 and 14, and equating coefficients with corresponding input pixel intensities I i , a set of up to four linear equations for the convolution filter coefficients a i     2    can be expressed as:                  ∑       i   2     =   1     m            λ     ii   2            a     i   2           =     c   i             (   15   )                                
     All four cubic coefficients c; can be determined by solving Equations 10 through 13 and, therefore, the only unknown values in Equation 15 are the five coefficients a 1  through a 5 . 
     To generate additional equations for determining coefficients a 1  through a 5 , the procedure above is repeated for each of the output pixels. Thus, referring again to FIG. 1, because n is 5 and there are therefore five output pixels P 1 ′ through P 5 ′ in block  12 , four c i  values are generated for each separate pixel P 1 ′ through P 5 ′ to generate twenty c i  values and appropriate equations like Equation 15, a separate equation for each of the twenty c i  values. Thus, twenty (i.e. 4n) cubic equations are derived with only 5 (i.e. m) unknowns (i.e. a 1  through a 5 ). 
     There can be less than four equations if the pixels obtained by bilinear interpolation used by the convolution filter do not have any contribution from some input pixels I i . In this case, the number of equations is reduced by the number of such pixels. 
     The system of 4n equations can be expressed in matrix form as: 
       C=ΛA   (16) 
     where C is the vector of corresponding coefficients c i  from right sides of equations 15, Λ is the matrix of coefficients λ ij  of those linear equations, and A is a vector of m filter coefficients a i . For practical values of zoom factors n and filter sizes m the matrix of Equation 16 represents an over determined system of linear equations. In this case, the solution to matrix Equation 16 is determined by using the linear least squares method which takes the form: 
     
       
           A= (Λ T Λ) −1 Λ T   C   (17) 
       
     
     The linear least squares method and in particular Equation 17 are well known in mathematics and therefore will not be explained here in detail. For a general explanation of the linear least squares method reference should be had to any standard algebra text which teaches matrix algebra. 
     Upon determining the coefficients aj corresponding to the convolution filter that is to be applied, the whole procedure for zooming digital images is defined. 
       2 . Two-Dimensional Convolution (TDC) Method 
     According to the TDC method, when a command is received to magnify a region of interest by factor n, a two dimensional hardware convolution filter and a bilinear interpolator are used to convolve and interpolate initial pixel intensities simultaneously in both the x and y dimensions thereby generating a final output image. 
     Thus, unlike the ODC method which first processes values in one dimension and then processes values in a second dimension to magnify, the TDC method processes in both dimensions simultaneously. Other than the TDC method processes in both dimensions simultaneously. Other than this distinction and more complex mathematics which are required for simultaneous two-dimensional processing, the TDC method is very similar to the ODC method. 
     For the purpose of this explanation, when referring to the TDC method the term “interpixel interval” will be used to refer to adjacent quarters of four adjacent pixels. Referring to FIG. 2, the general pixel scheme for applying the TDC method to a region of interest is illustrated. The initial region of interest includes a pixel block of (m+1) (m+1) pixels. In FIG. 2, I 11  through I (m+1)(m+1)  denote source image pixel intensities corresponding to pixels of interest P 11  through P (m+1)(m+1) , bag through b 22  denote intermediate values obtained by applying a 2D convolution filter to intensities I 11  through I (m+1)(m+1) , numeral  20  outlines a convolution filter window including pixels P 11  through P mm  and O 11  through O nn  represent intensities of output pixels P 11 ′ through P nn ′ obtained by bilinear interpolation between intermediate pixels b 11 , b 12 , b 21 , and b 22 . Generally intensities I 11  through I mm  will be referred to as I ij . 
     Intermediate values b 11  through b 22  are obtained by applying the m×m convolution filter to the input pixels I ij  according to the following expressions:                b   11     =       ∑     i   =   1     m            ∑     j   =   1     m            a   ij          I   ij                   (   18   )                 b   12     =       ∑     i   =   1     m            ∑     j   =   1     m            a   ij          I     ij   +   1                     (   19   )                 b   21     =       ∑     i   =   1     m            ∑     j   =   1     m            a   ij          I     i   +     1      j                       (   20   )                 b   22     =       ∑     i   =   1     m            ∑     j   =   1     m            a   ij          I     i   +   ij   +   1                     (   21   )                                
     where a ij  are the coefficients of the two-dimensional convolution filter. 
     After intermediate values b 11  through b 22  are determined bilinear interpolation is applied to generate output intensities O 11  through O nn . FIG. 3 illustrates the relationship between intermediate values b 11 , b 12 , b 21 , and b 22  and the output pixels O 11  through O nn . Each of the output pixels O ij  occupies a different location relative to the intermediate values b 11  through b 22 . All the other output pixels generated by the linear interpolation occupy the same relative locations to some intermediate pixels. The values of the output pixels O 11  through O nn , with respect to the intermediate values are given by the following equations:                      O   ij     =                        (     n   -   i   +   1     )          (     n   -   j   +   1     )         n   2            b   11       +           (     n   -   i   +   1     )          (     j   -   1     )         n   2            b   12       +                                      (     i   -   1     )          (     n   -   j   +   1     )         n   2            b   21       +           (     i   -   1     )          (     j   -   1     )         n   2            b   22                       (   22   )                                
     for i, j=1, . . . , n (n odd), and                      O   ij     =                        (     n   -   i   +   0.5     )          (     n   -   j   +   0.5     )         n   2            b   11       +                                      (     n   -   i   +   0.5     )          (     j   -   0.5     )         n   2            b   12       +                                      (     i   -   0.5     )          (     n   -   j   +   0.5     )         n   2            b   21       +           (     i   -   0.5     )          (     j   -   0.5     )         n   2            b   22                       (   23   )                                
     for i, j=1, . . . , n (n even). 
     When the expressions for intermediate values b 11  through b 22  from Equations 18 through 21 are substituted in Equation 22, the following expression is obtained:                      O   ij     =                        (     n   -   i   +   1     )          (     n   -   j   +   1     )         n   2              ∑     k   =   1     m            ∑     l   =   1     m            a   kl          I   kl             +                                      (     n   -   i   +   1     )          (     j   -   1     )         n   2              ∑     k   =   1     m            ∑     l   =   1     m            a   kl          I     kl   +   1               +                                      (     i   -   1     )          (     n   -   j   +   1     )         n   2              ∑     k   =   1     m            ∑     l   =   1     m            a   kl          I     kl   +     1      l                 +                                    (     i   -   1     )          (     j   -   1     )         n   2              ∑     k   =   1     m            ∑     l   =   1     m            a   kl          I     k   +     1      l     +   1                             (   24   )                                
     which can be expresses as:                O   kl     =       ∑       i   1     =   1       m   +   1              ∑       i   2     =   1       m   +   1              ∑       i   3     =   1     m            ∑       i   4     =   1     m            λ       i   1          i   2          i   3          i   4              a       i   3          i   4              I       i   1          i   2                           (   25   )                                
     where λ i     1     i     2     i     3     i     4    are the constant coefficients from Equations 22. Similar expressions are obtainable using Equations 23. 
     As well known in the art, using known expressions for bicubic interpolation alternate expressions for pixel values O kl  can be obtained which can be collectively expressed as:                O   kl     =       ∑     i   =   1     4            ∑     j   =   1     4            c   ij          I   ij                   (   26   )                                
     where c ij  are the coefficients derived from the expressions for bicubic interpolation in the case of magnification by n. Comparing Equations 25 and 26 and equating coefficients with corresponding input pixel values I ij , a set of up to sixteen linear equations for the filter coefficients a i     3     i     4    are obtained which can be collectively expressed as:                  ∑       i   3     =   1     m            ∑       i   4     =   1     m            λ       iji   3          i   4              a       i   3          i   4               =     c   ij             (   27   )                                
     There can be less than sixteen equations if the pixels obtained by bilinear interpolation used by the convolution filter do not have any contribution from some input pixels I ij . In that case, the number of equations is reduced by the number of such pixels. Repeating this procedure for all of the output pixels O ij  (i, j=1, . . . , n) a system of (16n) linear equations for the convolution filter coefficients is obtained which can be expressed in matrix form as: 
     
       
           C=ΛA   (28) 
       
     
     where C is a vector of the corresponding coefficients c ij  from right side of Equations 26, Λ is the matrix of coefficients λ ijkl  of those linear equations, and A is the vector of m×m filter coefficients a 11  through a mm . For practical values of zoom factors n and filter sizes m by m this represents an over determined system of linear equations. In this case, the coefficients a ij  can be determined using the linear least squares method indicated by Equation 17 above: 
     Upon determining the coefficients a ij  of the convolution filter that is to be applied, the whole procedure for zooming digital images is defined. 
     B. Hardware and Operation 
     Referring now to FIG. 4, a conventional workstation  30  is illustrated which includes a computer  32 , a display  34  linked to the computer  32 , and two interface devices, a keyboard  36  and a mouse  38 . Keyboard  36  and mouse  38  allow a user to provide information to the computer including commands to control the computer  32 . Information including images, is displayed on display  34  for observation. For the purpose of this explanation it is assumed that an image is displayed via display  34  and that a workstation user uses either board  36  or mouse  38  to select a region of interest on the initially displayed image to be magnified by a factor of 5 (i.e. n=5). 
     In addition to other processing hardware, computer  32  must include hardware capable of either one-dimensional convolution filtering and linear interpolation or two-dimensional convolution filtering and bilinear interpolation. To this end, referring to FIG. 5, computer  32  includes an interpolation coefficient calculator  40 , a cubic coefficient calculator  42  and a linear least square determiner  44 . Referring also to FIG. 9, at block  160 , calculator  40  receives magnification factor n and uses factor n to determine interpolation coefficients. To this end, on one hand, where a hardware based convolver is only capable of one dimensional convolution and a hardware based interpolator is only capable of linear interpolation, calculator  40  uses the coefficient expressions from Equations 5 and 6 to determine interpolation coefficients at block  162 . For example, where n is 5 and m is 6 the four equations corresponding to Equation 15 above for the first output pixel (i.e. n=1) are: 
     
       
         λ 11   a   1 +λ 12   a   2 +λ 13   a   3 +λ 14   a   4 +λ 15   a   5   =c   1   (29) 
       
     
     
       
         λ 21   a   1 +λ 22   a   2 +λ 23   a   3 +λ 24   a   4 +λ 25   a   5   =c   2   (30) 
       
     
     
       
         λ 31   a   1 +λ 32   a   2+λ   33   a   3+λ   34   a   4 +λ 35   a   5   =c   3   (31) 
       
     
     
       
         λ 41   a   1 +λ 42   a   2 +λ 43   a   3 +λ 44   a   4 +λ 45   a   5   =c   1   (32) 
       
     
     Looking at Equation 5 for I, where i is 1 corresponding to Equation 29, there is only a single coefficient expression for a 1 , the expression being (n−k+1)/n and therefore coefficient λ 11  is known to be (n−k+1)/n and all other coefficients λ 12 , λ 13 , λ 14  and λ 15  are known to be zeros. Where i is 2 corresponding to Equation 30, referring to Equation 5, coefficient λ 21  is (k−1)/n, coefficient λ 22  is (n−k+1)/n and all other λ coefficients in Equation 30 are zero. Where i is 3 corresponding to Equation 31, coefficient λ 31  is zero, coefficient λ 32  is (k−1)/n, coefficient λ 33  is (n−k+1)/n and all other coefficients λ in Equation 31 are zeros. Where i is 4 corresponding to Equation 32 coefficients λ 41  and λ 42  are zeros, coefficient λ 43  is (k−1)/n, coefficient λ 44  is (n−k+1)/n and all other coefficients are zeros. The k value used in each of the coefficients corresponds to an output pixel and Equations like 29 through 32 are examined for each intended output pixel thereby generating 4n interpolation coefficients λ which are provided to determiner  44 . 
     On the other hand, where the hardware convolver is a two dimensional convolver and a bi-linear interpolator is provided, at block  162  calculator  40  uses Equations 22 or 23, depending on if factor n is odd or even, to determine interpolation coefficients λ ijkl  which are provided to determiner  44 . 
     Referring still to FIGS. 5 and 9, at block  164  cubic coefficient calculator  42  also receives factor n and determines bicubic coefficients. Where the system convolver facilitates only one dimensional convolution calculator  42  uses Equations 9 and 10 through 13 or other similar equations to determine coefficients c 1  through c 4  for each separate output pixel. Where the system can perform two dimensional convolution, calculator  42  generates sixteen coefficients c 1  through c 16  for each output pixel using conventional bicubic equations which are well known in the art. Thus, 16n coefficients are generated and provided to determiner  44 . 
     In addition to receiving interpolation coefficients λ and cubic coefficients c, determiner  44  also receives a value m indicating the filter dimensions. At blocks  166  and  168  determiner  44  forms the matrix equation indicated by Equation 17 and solves the equation to generate m or m×m coefficients a which are required in either Equations 1 and 2 or in Equations 18, 19, 20 and 21. Coefficients a i  or a ij  are provided as an output from determiner  44 . 
     The remaining hardware in the case of the ODC method is illustrated in FIG. 6 while the remaining hardware in the case of the TDC method is illustrated in FIG.  7 . Referring to FIG. 6, to facilitate the ODC method computer  32  (see FIG. 5) further includes a 1D convolver  56 , a linear interpolator  58  and a data buffer  60 . 
     Referring also to FIG. 10 at block  180 , convolver  56  receives each of the initial image pixel intensity I i  which corresponds to the expanded region of interest. Referring also to FIG. 7, a region of interest  80  (i.e. region of an image to be magnified) includes A rows and B columns. To convolve and interpolate convolution windows for each interval in the region of interest  80  in the x dimension (i.e. horizontal), the windows include m+1 pixel intensities corresponding to m+1 pixels which are adjacent, and in the same row as, the interval of interest. Thus, for intervals on the lateral edges of region  80  corresponding windows will extend laterally from the region of interest by as much as m/2 pixels. One such laterally extending window is identified by number  82 . Pixels which are laterally positioned with respect to the region of interest  80  are similar to intervals within region  80  in that they have not yet been convoluted and interpolated. Therefore, convolution windows which laterally extend during the first pass of an image through hardware include only non-convoluted pixels. 
     However, after intervals within a region of interest have been expanded in the x dimension, pixels in rows directly above and below the region of interest have not been expanded. Thus, upon the second pass of the image through the hardware to expand intervals in the y dimension, for intervals near the top and bottom of the region which has already been expanded in the x-dimension, suitable convolution windows cannot be selected. For example, for an interval in the top row of an x-dimension expanded region, when a convolution window including 5 pixels is chosen, while the pixels below the interval of interest and within the window have been expanded in the x-dimension like the interval of interest, the pixels above the interval of interest have not been similarly expanded in the x-dimension and therefore cannot be suitably used to convolute and interpolate the interval of interest. 
     To overcome this problem, referring to FIG. 7, the region of interest  80  is expanded to include m/2 rows of pixels above and m/2 rows of pixels below the initial region of interest resulting in expanded region  84 . Thereafter, after expansion in the x dimension, expansion in the y dimension can be performed on intervals in rows corresponding to the original region of interest using convolution windows for top and bottom located pixels which extend into region  84  above region  80  and similarly into the region below region  80 . 
     Referring still to FIGS. 6 and 10, in addition to receiving intensity signals I i  for each pixel within the expanded region  84 , at block  180  convolver  56  also receives convolution filter coefficients a i  from determiner  44  (see FIG.  5 ), filter size value m and factor n. Convolver  56  first selects pixel pairs adjacent each interval of interest and then selects a separate convolution window for each adjacent pixel on the expanded region of interest, and, at block  182 , forms Equations 1 and 2 generating intermediate values b 1  and b 2  for the windows. 
     At block  184  interpolator  58  receives values b 1  and b 2  and factor n and solves either Equation 3 or Equation 4 (depending on if factor n is odd or even) for each of k output pixels to determine pixel intensities O 1  through O k . Intensities O 1  through O k  are determined for each interval in the original region of interest  84  (see FIG. 7) and together intensities O 1 -O k  define an interpolated image (i.e. the interpolated region expanded in the x dimension). The interpolated image is temporarily stored in buffer  60 . Once the interpolated image has been completed, that image, including all pixel intensities, is provided to convolver  56  a second time. At block  186 , for every interval between adjacent pixels in adjacent rows of the interpolated image which corresponded to rows in the initial region of interest  80  (see FIG.  7 ), convolver  56  identifies the adjacent pixels and selects a convolution column window for each adjacent pixel and convolves the intensities in the windows according to Equations 1 and 2 generating intermediate values b 1  and b 2 . At block  188 , interpolator  58  receives values b 1  and b 2 , performs either Equation 3 or Equation 4 (depending on if n is odd or even) and generates final output intensities which together define the final and magnified image. Computer  32  (see FIG. 5) uses intensities O k  to drive display  34  to display the magnified region of interest. 
     Referring now to FIG. 9, the remaining hardware to facilitate the TDC method includes a 2D convolver  90  and a bi-linear interpolator  92 . Referring also to FIG. 11, at block  150  convolver  90  receives the image pixel intensities I i  comprising the region of interest, the convolution filter coefficients a i , factor n and filter value m. Convolver  90  selects four adjacent pixels for each interpixel interval and then selects a separate convolution window for each interpixel interval within the region of interest. At block  152  convolver  90  performs Equations 18 through 21 to determine intermediate values b 11 , b 12 , b 21  and b 22 . At block  152  interpolator  92  receives each of values b 11  through b 22  and factor n for each interval in the region of interest and interpolates each set of four values b 11  through b 22  to generate nxn pixel intensities O ij  by solving either Equation 22 or Equation 23, depending on if factor n is odd or even. Pixel intensities O ij  define the final output magnified image and can be used by computer  32  (see FIG. 5) to display the magnified image on display  34 . 
     It should be understood that the methods and apparatuses described above are only exemplary and do not limit the scope of the invention, and that various modifications could be made by those skilled in the art that would fall under the scope of the invention. 
     To apprise the public of the scope of this invention, we make the following claims.