Separable image warping methods and systems using spatial lookup tables

• US 5,204,944 A
• Filed: 07/28/1989
• Issued: 04/20/1993
• Est. Priority Date: 07/28/1989
• Status: Expired due to Fees

• Current Assignee: Trustees Of Columbia University In The City Of New York (Columbia University)
• Original Assignee: Trustees Of Columbia University In The City Of New York (Columbia University)
• Inventors: Boult, Terrance E., Wolberg, George
• Primary Examiner(s): Harkcom, Gary V.
• Assistant Examiner(s): Nguyen, Phu K.
• Application Number:
  US07/387,605
• Time in Patent Office: 1,362 Days
• Field of Search: 364/518, 364/521, 364/522, 364/949.4 M, 364/949.4 S, 395/133, 395/136, 395/162, 395/163, 395/164, 395/118, 395/119, 395/127, 340/728, 340/747, 340/750, 382/114
• US Class Current: 345/427
• CPC Class Codes: G06T 3/18 Image warping, e.g. rearranging pixels individually

Abstract

A two-dimensional image is subjected to a three-dimensional effect, such as folding, twisting or rotation, and transformed or warped into a modified output image. By successive one-dimensional resampling along two coordinates and parallel channel processing of transposed input image data with rejection of distorted pixel value data, bottleneck, shear and other distortions are addressed. Complete warping systems, resampling subsystems and related methods are disclosed.

Claims

1. An image system including at least one separable image warping system, for transforming input image data to output image data, comprising:

input image value means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

a first channel, for processing input image value data to derive image values representing preliminary values of output image pixels, comprising;

shear resampler means coupled to said x value and y value means and responsive to conditions indicative of shear for processing first and second coordinate data for developing at first and second outputs, respectively, scaled coordinate data representing first and second coordinate values of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image, and also coupled to said input image value means for similarly processing image data for developing at a third output scaled input image data representing the same magnification of the input image;

x intensity resampler means coupled to said first and third shear resampler outputs for developing intermediate pixel value data representing said scaled input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate;

coordinate resampler means coupled to said first and second shear resampler outputs for developing at a first output modified second coordinate data representing said scaled second coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate, and for developing at a second output shear data representing a measure of compression variations, along the direction of said first coordinate, in the output image as compared to the input image;

y intensity resampler means coupled to said x intensity resampler means and said first shear resampler output for developing preliminary output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate;

bottleneck resampler means coupled to said first and second coordinate resampler outputs for developing bottleneck image value data representing said shear data after resampling to give effect to image compression variations along the direction of said second coordinate;

transposing means coupled to said input image value, x value and y value means for developing transposed input image, transposed first coordinate and transposed second coordinate data representative of said input and output images respectively after rotation of coordinates of said images to a second alignment;

a second channel, for processing said transposed input image value data in parallel with said first channel to derive image values representing preliminary values of output image pixels of said rotated input image, comprising;

shear resampler means, x intensity resampler means, coordinate resampler means, y intensity resampler means and bottleneck resampler means for respectively providing the same functions as said corresponding elements of said first channel;

comparator means coupled to the bottleneck resampler means of said first and second channels for comparing the respective bottleneck image value data for providing a control signal representative of which of said preliminary output pixel values, as provided at the outputs of said first and second channels, should be selected on a predetermined basis for inclusion in the final output image data for each pixel of the output image; and

selector means coupled to the y intensity resamplers of said first and second channels and responsive to said control signal for providing final output image data wherein the value for each pixel represents the value for that pixel from either the first or second channel as selected in response to said control signal.

2. The image system in accordance with claim 1, further comprising:

means for providing first input color image component values;

means for providing second input color image component values;

means for providing third input color image component values;

first said at least one separable image warping system including, second and third warping systems, each coupled to means for providing one of said first, second and third input color image component values; and

means for supplying common data to each of said warping systems representing the value of first and second coordinates to which each pixel of a plurality of pixels in said input image is to be transferred in said output image.

11. The image system in accordance with claim 1, further comprising:

means for providing first input color image component values;

means for providing second input color image component values;

means for providing third input color image component values;

first said at least one separable image warping system including second and third warping systems, each coupled to means for providing one of said first, second and third input color image component values; and

means for supplying common data to each of said warping systems representing the value of first and second coordinates to which each pixel of a plurality of pixels in said input image is to be transferred in said output image.

3. A transformation channel for use in a separable image warping system for transforming input image data to output image data, comprising:

input image value means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

shear resampler means coupled to said x value means and responsive of conditions indicative of shear for processing first coordinate data for developing at a first output scaled coordinate data representing first coordinate values of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image, and also coupled to said y value means for processing second coordinate data for developing at a second output scaled second coordinate data having a plurality of pixel values for each pixel value of the basic output image;

x intensity resampler means coupled to said image value means and to said first shear resampler output for developing intermediate pixel value data representing said input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate;

coordinate resampler means coupled to said first and second shear resampler outputs for developing at a first output modified second coordinate data representing said scaled second coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate, and for developing at a second output shear data representing a measure of compression variations, along the direction of said first coordinate, in the output image as compared to the input image;

y intensity resampler means coupled to said x intensity resampler means and said first coordinate resampler output for developing preliminary output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate; and

bottleneck resampler means coupled to said first and second coordinate resampler outputs for developing bottleneck image value data representing shear data after resampling to give effect to image compression variations along the discretion of said second coordinate.

4. A separable image warping system for transforming input image data to output image data, comprising:

input image means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

a first channel, for processing input image value data to derive image values representing preliminary values of output image pixels, comprising;

intensity resampler means coupled to said image value means and to said x value and y value means for providing a two-pass transformation for developing preliminary output pixel values representing said input image pixel values after successive resampling to give effect to image compression variations along the direction of said first and second coordinates; and

bottleneck resampler means coupled to said x value and y value means for resampling coordinate data for developing bottleneck image value data indicative of individual pixels in said preliminary output pixel data that have been subjected to positive compression variations exceeding a predetermined measure;

transposing means coupled to said input image value, x value and y value means for developing transposed input image, transposed first coordinate and transposed second coordinate data representative of said input and output images respectively after rotation of coordinates of said images to a second orientation;

a second channel, for processing said transposed input image value data in parallel with said first channel to derive image values representing preliminary values of output image pixels of said rotated input image, comprising;

intensity resampler means and bottleneck resampler means for providing the same functions as said corresponding elements of said first channel;

comparator means coupled to the bottleneck resampler means of said first and second channels for comparing the respective bottleneck image value data for providing a control signal representative of which of said preliminary output pixel values, as developed in said first and second channels, should be selected on a predetermined basis for inclusion in the final output image data, for each pixel of the output image; and

selector means coupled to the intensity resampler means of said first and second channels and responsive to said control signal for providing final output image data wherein the value for each pixel represents the value for that pixel from either the first or second channel as selected in response to said control signal.

5. A transformation channel for use in a separable image warping system for transforming input image data to output image data, comprising:

input image means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

a first channel, for processing input image value data to derive image values representing preliminary values of output image pixels, comprising;

intensity resampler means coupled to said image value means and to said x value and y value means for providing two-pass transformation for developing preliminary output pixel values representing said input image pixel values after successive resampling to give effect to image compression variations along the direction of said first and second coordinates; and

bottleneck resampler means coupled to said first x value and y value means for resampling coordinate data for developing bottleneck image value data indicative of individual pixels in said preliminary output pixel data that have been subjected to positive compression variations exceeding a predetermined measure.

6. A two-pass transformation system for use in a separable image warping system, for transforming input image data to output image data, comprising:

input image means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

x intensity resampler means coupled to said image value means and to said x value means for developing intermediate pixel value data representing said input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate;

coordinate resampler means coupled to said x value and y value means for developing modified second coordinate data representing second coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate; and

y intensity resampler means coupled to said x intensity resampler means and said coordinate resampler means for developing output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate.

7. A shear data processor for a separable image warping system, for transforming input image data to output image data, comprising:

input image value means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

shear resampler means coupled to said x value and y value means and responsive to conditions indicative of shear, for processing first and second coordinate data to develop, at respective first and second outputs, scaled coordinate data representing first and second coordinate values of a magnification of a basic output image, having a plurality of pixel values in place of each pixel value of the basic output image;

and, also coupled to said input image value means, means for processing image data in correspondence with the processing of the coordinate data, to develop, at a third output, output image data representing the same magnification of the basic output image.

8. A bottleneck data processor for a separable image warping system, for transforming input image data to output image data, comprising:

x value means for supplying data representing the value of a first coordinate to which each pixel of an input image is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

coordinate resampler means coupled to said x value and y value means for developing at a first output modified second coordinate data representing said second coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate, and for developing at a second output shear data representing a measure of compression variations, along the direction of said first coordinate, in the output image as compared to the input image; and

bottleneck resampler means coupled to said first and second coordinate resampler outputs for developing bottleneck image value data representing said shear data after resampling to give effect to image compression variations along the direction of said second coordinate.

9. A separable image warping system, for transforming input image data to output image data, comprising:

input image means for supplying luminance data each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying XLUT data;

y value means for supplying YLUT data;

a first channel, for processing luminance data to derive image values representing preliminary values of output image pixels, comprising;

shear resampler means coupled to said x value and y value means for processing XLUT data for developing at a first output scaled XLUT data representing XLUT of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image, and for processing XLUT data for developing at a second output scaled second coordinate data having a plurality of pixel values for each pixel value of the basic output image;

x intensity resampler means coupled to said input image means and to said first shear resampler output for developing intermediate luminance data representing said input luminance data after resampling to give effect to image compression variations along the x direction;

coordinate resampler means coupled to said first and second shear resampler outputs for developing at a first output modified YLUT data representing said scaled YLUT after resampling to give effect to image compression variations along the x direction, and for developing at a second output shear data representing a measure of compression variations along the direction of said first coordinate, in the output image as compared to the input image;

y intensity resampler means coupled to said x intensity resampler means, said first shear resampler output and a first z channel for developing preliminary luminance data representing said intermediate luminance data after resampling to give effect to image compression variations along the y direction;

bottleneck resampler means coupled to said first and second coordinater resample outputs for developing bottleneck image value data representing said shear data after resampling to give effect to image compression variations along the direction of said y coordinate;

transposing means coupled to said input image, x value and y value means for developing transposed input image and transposed XLUT and YLUT data representative of said input and output images respectively after transposing their coordinates to a second orientation;

a second channel, for processing said transposed luminance data in parallel with said first channel to derive data representing preliminary luminance values of output image pixels of said transposed input image, comprising;

shear resampler means, x intensity resampler means, coordinate resampler means, y intensity resampler means and bottleneck resampler means each for providing the same functions as corresponding elements of said first channel;

comparator means coupled to the bottleneck resampler means of said first and second channels for comparing the respective bottleneck image value data for providing a control signal representative of which of said preliminary luminance values, as provided at the outputs of said first and second channels, should be selected on a predetermined basis for inclusion in the final output image data for each pixel of the output image; and

selector means coupled to the y intensity resamplers of said first and second channels and responsive to said control signal for providing final output luminance data wherein the luminance value for each pixel represents the luminance for that pixel from either the first or second channel as selected in response to said control signal.

10. An image system including at least one separable image warping system, for transforming input image data to output image data, comprising:

input image value means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

z value means for supplying data representing values of a third coordinate relevant to the transformation of input image data to output image data;

a first channel, for processing input image value data to derive image values representing preliminary values of output image pixels, comprising;

shear resampler means coupled to said x value means and y value means for processing first coordinate data for developing at a first output scaled first coordinate data representing first coordinate values of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image, and for processing second coordinate data for developing at a second output scaled second coordinate data having a plurality of pixel values for each pixel value of the basic output image;

x intensity resampler means coupled to said input image value means and to said first shear resampler output for developing intermediate pixel value data representing said input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate;

coordinate resampler means coupled to said first and second shear resampler outputs for developing at a first output modified second coordinate data representing said scaled second coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate, and for developing at a second output shear data representing a measure of compression variations along the direction of said first coordinate, in the output image as compared to the input image;

y intensity resampler means coupled to said x intensity resampler means, said first coordinate resampler output and a first z channel for developing preliminary output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate;

bottleneck resampler means coupled to said first and second coordinate resample outputs for developing bottleneck image value data representing said shear data after resampling to give effect to image compression variations along the direction of said second coordinate;

a first z channel, for processing third coordinate values, comprising;

terminal means for supplying scaled first coordinate data as developed at said first shear resampler output of said first channel;

z shear resampler means coupled to said x value, y value and z value means for developing sealed third coordinate data representing third coordinate values of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image;

z coordinate resampler means coupled to said terminal means and to said z shear resampler means for developing modified third coordinate data representing said scaled third coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate; and

means for coupling said modified third coordinate data to the y intensity resampler of said first channel for enabling said resampler to process foldover data for deleting foldover data representative of image portions not visible in the final output image;

transposing means coupled to said input image value, x value, y value and z value means for developing transposed input image, transposed first coordinate, transposed second coordinate and transposed third coordinate data representative of said input and output images respectively after rotation of their coordinates to a second orientation;

a second channel, for processing said transposed input image value data in parallel with said first channel to derive image values representing preliminary values of output image pixels of said rotated input image, comprising;

shear resampler means, x intensity resampler means, coordinate resampler means, y intensity resampler means and bottleneck resampler means for providing the same functions as said corresponding elements of said first channel;

a second z channel, in parallel with said first z channel, for processing transposed third coordinate values, comprising;

terminal means, z shear resampler means and z coordinate resampler means for providing the same functions as the corresponding elements of said first z channel; and

means for coupling the resulting modified transposed third coordinate data to the y intensity resampler of said second channel for providing the same function as the corresponding element of said first z channel;

comparator means coupled to the bottleneck resampler means of said first and second channels for comparing the respective bottleneck image value data for providing a control signal representative of which of said preliminary output pixel values, as provided at the outputs of said first and second channels, should be selected on a predetermined basis for inclusion in the final output image data for each pixel of the output image; and

selector means coupled to the y intensity resamplers of said first and second channels and responsive to said control signal for providing final output image data wherein the value for each pixel represents the value for that pixel from either the first or second channel as selected in response to said control signal.

12. A separable image warping system, for transforming input image data to output image data, comprising:

input image means for supplying luminance data each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying XLUT data;

y value means for supplying YLUT data;

z value means for supplying ZLUT data;

a first channel, for processing luminance data to derive image values representing preliminary values of output image pixels, comprising;

shear resampler means coupled to said x value and y value means for processing XLUT data for developing at a first output scaled XLUT data representing XLUT of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image, and for similarly processing XLUT data for developing at a second output scaled second coordinate data having a plurality of pixel values for each pixel value of the basic output image;

x intensity resampler means coupled to said input image means and to said first shear resampler output for developing intermediate luminance data representing said input luminance data after resampling to give effect to image compression variations along the x direction;

coordinate resampler means coupled to said first and second shear resampler outputs for developing at a first output modified YLUT data representing said scaled YLUT after resampling to give effect to image compression variations along the x direction, and for developing at a second output shear data representing a measure of compression variations along the direction of said first coordinate, in the output image as compared to the input image;

y intensity resampler means coupled to said x intensity resampler means, said first shear resampler output and a first z channel for developing preliminary luminance data representing said intermediate luminance data after resampling to give effect to image compression variations along the y direction;

bottleneck resampler means coupled to said first and second coordinate resampler outputs for developing bottleneck image value data representing said excised y coordinate values after resampling to give effect to image compression variations along the direction of said y coordinate;

a first z channel, for processing ZLUT data, comprising;

terminal means for supplying scaled XLUT data as developed at said first shear resampler output of said first channel;

z shear resampler means coupled to said x value, y value and z value means for developing scaled XLUT data representing XLUT of a magnification of said output image and having a plurality of pixel values in place of each pixel value of the basic output image;

z coordinate resampler means coupled to said terminal means and to said z shear resampler means for developing modified ZLUT data representing said scaled ZLUT after resampling to give effect to image compression variations along the z direction; and

means for coupling said modified ZLUT data to the y intensity resampler of said first channel for enabling said resampler to process foldover data for deleting foldover data representative of image portions not visible in the final output image;

transposing means coupled to said input image, x value, y value and z value means for developing transposed nput image and transposed XLUT, YLUT and ZLUT data representative of said input and output images respectively after transposing coordinates of said images to a second orientation;

a second channel, for processing said transposed luminance data in parallel with said first channel to similarly derive data representing preliminary luminance values of output image pixels of said transposed input image, comprising;

shear resampler means, x intensity resampler means, coordinate resampler means, y intensity resampler means and bottleneck resampler means each for providing the same functions as corresponding elements of said first channel;

a second z channel, in parallel with said first z channel, for processing transposed ZLUT data, comprising;

terminal means, z shear resampler means and z coordinate resampler means for providing the same functions as the corresponding elements of said first z channel; and

means for coupling the resulting modified ZLUT data to the y intensity resampler of said second channel for providing the same function as the corresponding element of said first z channel.

comparator means coupled to the bottleneck resampler means of said first and second channels for comparing the respective bottleneck image value data for providing a control signal representative of which of said preliminary luminance values, as provided at the outputs of said first and second channels, should be selected on a predetermined basis for inclusion in the final output image data for each pixel of the output image; and

selector means coupled to the y intensity resamplers of said first and second channels and responsive to said control signal for providing final output luminance data wherein the luminance value for each pixel represents the luminance for that pixel from either the first or second channel as selected in response to said control signal.

13. A separable image warping system, for transforming input image data to output image data, comprising:

input image value means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

z value means for supplying data representing values of a third coordinate relevant to the transformation of input image data to output image data;

x intensity resampler means coupled to said input image value means and to said x value means for developing intermediate pixel value data representing said input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate;

y intensity resampler means coupled to said x intensity resampler means, said y value means and a z intensity resampler for developing preliminary output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate;

z coordinate resampler means coupled to said x value and z value means for developing modified third coordinate data representing said scaled third coordinate values after resampling to give effect to image compression variations along the direction of said first coordinate; and

means for coupling said modified third coordinate data to the y intensity resampler for enabling said resampler to process foldover data for deleting foldover data representative of image portions not visible in the final output image.

14. A method for separably warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input image data, in response to output image first coordinate values to adjust for image compression variations along the direction of said first coordinate of an input image, for developing intermediate pixel data;

(b) resampling output image second coordinate values as in step (a), for developing transformed coordinate data representing second coordinate data resampled in response to said first coordinate values;

(c) resampling said intermediate pixel data, in response to transformed coordinate data developed in step (b) to adjust for image compression variations along the direction of a second coordinate, for developing first image data representing preliminary output pixel values;

(d) repeating steps (a), (b) and (c) substituting transposed image data, first coordinate values and second coordinate values for said image data, first coordinate values and second coordinate values, respectively, as referred to in said steps, for developing second image data representing preliminary output pixel values;

(e) determining, for each output image pixel, which pixel value from either step (c) or step (d) to select for use in the output image data; and

(f) assembling output image data by selection of values for individual output pixels from steps (c) and (d), according to said step (e) determination.

18. The method in accordance with claim 14, 15, 16 or 17, wherein the resampling recited is carried out independently with respect to output image first and second coordinate values for each pixel of the output image.

19. The method in accordance with claim 14, 15 or 17, additionally comprising the steps of:

initially developing scaled coordinate data representing first and second coordinates of a magnification of the desired output image, developing scaled input image data representing a corresponding magnification of the input image, and substituting such scaled data for the corresponding unscaled data as called for in the steps of this claim.

15. A method for separably warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input pixel values in two-passes to successively adjust for image compression variations along the directions of two coordinates of a first orientation of an input image for developing a first output representing preliminary pixel values of an output image;

(b) repeating step (a) substituting transposed input pixel values resampled to adjust for compression variations along the directions for two coordinates of a second orientation of said input image for developing a second such output;

(c) resampling data representing the values of coordinate positions to which pixels of said input image are to be transformed in said output image, for developing a control signal indicative of which pixel value from either step (a) or step (b) to select for use in the output image data; and

(d) assembling output image data by selection of values for individual output pixels from steps (a) and (b) in response to said control signal.

16. A method for separably warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input image data, in response to output image first coordinate values to adjust for image compression variations along the direction of said first coordinate of an input image, for developing intermediate pixel data;

(b) resampling output image second coordinate values as in step (a), for developing transformed coordinate data representing second coordinate data resampled in response to said first coordinate values;

(c) resampling output image third coordinate values as in step (a), for developing transformed coordinate data representing third coordinate data resampled in response to said first coordinate values;

(d) resampling said intermediate pixel data, in response to transformed coordinate data developed in step (b), to adjust for image compression variations along the direction of a second coordinate and removing pixel values for foldover image portions not visible in the final output image, in response to the transformed third coordinate data developed in step (c), for developing first image data representing preliminary output pixel values;

(e) repeating steps (a), (b), (c) and (d) substituting transposed image data, first coordinate values, second coordinate values, and third coordinate values for said image data, first coordinate values, second coordinate values and third coordinate values, respectively, referred to in said steps (a), (b), (c) and (d), for developing second image data representing preliminary output pixel values;

(f) determining, for each output image pixel, which pixel value from either step (d) or step (e) to select for use in the output image data; and

(g) assembling output image data by selection of values for individual output pixels from steps (d) and (e), according to said step (f) determination.

20. The method in accordance with claim 16 additionally comprising the step of initially developing scaled coordinate data representing first, second and third coordinates of a magnification of the desired output image, developing scaled input image data representing a corresponding magnification of this input image, and substituting such scaled data for the corresponding unscaled data as called for in the steps of the claim.

17. A method for separably warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input image data, in response to output image first coordinate values to adjust for imae compression variations along the direction of said first coordinate of an input image, for developing intermediate pixel data;

(b) resampling output image second coordinate values as in step (a), for developing transformed coordinate data representing second coordinate data resampled in response to said first coordinate values; and

(c) resampling said intermediate pixel data, in response to transformed coordinate data developed in step (b) to adjust for image compression variations along the direction of a second coordinate, for developing first image data representing output pixel values.

21. A bottleneck compensation method for separably warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input pixel values in two-passes to successively adjust for image compression variations along the directions of two coordinates of a first orientation of an input image for developing a first output representing preliminary pixel values of an output image, which pixel values may have been distorted in said successive resampling;

(b) repeating step (a) substituting transposed input pixel values resampled to adjust for compression variations along the directions for two coordinates of a second orientation of said input image for developing a second such output, including pixel values which may have been distorted in said successive resampling; and

(c) assembling output image data by selection of values for individual output pixels from steps (a) and (b), wherein the value for each pixel represents the value for that pixel selected in preference to pixel values subject to relatively higher distortion.

22. A method for warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling output image second coordinate values, in response to output image first coordinate values to adjust for image compression variations along the direction of said first coordinate of an input image, for developing transformed coordinate data representing second coordinate data resampled in response to said first coordinate values; and

(b) resampling said image pixel data, in response to transformed coordinate data developed in step (a) to adjust for image compression variations along the direction of a second coordinate, for developing image data representing output pixel values.

23. A method for warping input image data, representing an image value for each of a plurality of pixels of an input image, to provide output image data, comprising the steps of:

(a) resampling input image data, in response to output image first coordinate values to adjust for image compression variations along the direction of said first coordinate of an input image, for developing intermediate pixel data;

(b) resampling output image second coordinate values as in step (a), for developing transformed coordinate data representing second coordinate data resampled in response to said first coordinate values;

(c) resampling output image third coordinate values as in step (a), for developing transformed coordinate data representing third coordinate data resampled in response to said first coordinate values; and

(d) resampling said intermediate pixel data, in response to transformed coordinate data developed in step (b), to adjust for image compression variations along the direction of a second coordinate and removing pixel values for foldover image portions not visible in the final output image, in response to the transformed third coordinate data developed in step (c), for developing image data representing preliminary output pixel values.

24. A resampler system for use in a separable image warping system, for transforming input image data to output image data, comprising:

input image means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

x intensity resampler means coupled to said image value means and to said x value means for developing intermediate pixel value data representing said input image pixel values after resampling to give effect to image compression variations along the direction of said first coordinate, and for developing a plurality of intermediate values for individual pixels subject to foldover, representing visible and hidden pixel values; and

y intensity resampler means coupled to said x intensity resampler means for developing output pixel value data representing said intermediate pixel values after resampling to give effect to image compression variations along the direction of said second coordinate.

25. A resampler system for use in a separable image warping system, for transforming input image data to output image data, comprising:

input image means for supplying data representing an image value for each pixel of a plurality of pixels in a two-dimensional image;

x value means for supplying data representing the value of a first coordinate to which each said pixel is to be transferred in an output image;

y value means for supplying data representing the value of a second coordinate to which each said pixel is to be transferred in said output image;

coordinate resampler means coupled to said x value and y value means for developing modified second coordinate data representing second coordinate values after resampling to give effect to image compression variations along the direction is said first coordinate; and

y intensity resampler means coupled to said coordinate resampler means and said image value means for developing output pixel value data representing input image pixel values after resampling to give effect to image compression variations along the direction of said second coordinate.

26. A bottleneck compensation system for an image warping system for transforming input image data to output image data, comprising:

first channel means for providing preliminary output pixel values representing input pixel values after successive resampling along the direction of first and second coordinates to give effect to image compression variations, which pixel values may have been distorted in said successive resampling;

second channel means for providing preliminary output pixel values representing said input image pixel values after successive resampling effectively along the direction of third and fourth coordinates to give effect to image compression variations, which pixel values may have been distorted in said successive resampling; and

selector means coupled to said first and second channel means for providing final output image data wherein the value for each pixel represents the value for that pixel selected from either the first or second channel means to provide final output image pixel values selected in preference to pixel values subject to relatively higher distortion.

27. A bottleneck compensation system as in claim 26, in which the effective resampling along third and fourth coordinates in the second channel means is accomplished by the resampling along said first and second coordinates of input image data that has been transposed by rotation of the coordinates of the input image to a second orientation.

28. A method for resampling input image data, representing an image value for each of a plurality of pixels of an input image, in response to output image coordinate values to adjust for image compression variations along the direction of the same coordinate of the input image, comprising the steps of:

(a) determining whether the next input pixel portion of input data fits within the boundaries of an output pixel, along the direction of said coordinate;

(b) determining, for an input pixel portion that so fits as determined in step (a), the midpoint value of said pixel portion, along said direction;

(c) deriving the area of said input pixel portion by multiplying the midpoint value from step (b) by a figure representative of the length of said pixel portion, along said direction;

(d) determining, for an input pixel portion that does not so fit as determined in step (a), the part of the input pixel portion that fits within said output pixel, along said direction;

(e) determining the midpoint value of the input pixel part determined in step (d), along said direction;

(f) deriving the area of said input pixel part by multiplying the midpoint value from step (e) by a figure representative of the length of said pixel part, along said direction;

(g) accumulating the areas determined in steps (c) and (f) to obtain the resamples value for a complete output pixel; and

(h) repeating the foregoing steps to resample additional input image data to provide output image pixel value data.

29. A method for resampling input image data successively to adjust for image compression variations along two coordinates, comprising:

applying the steps of claim 28 first with respect to the direction of a first coordinate and repeating said steps with respect to the direction of a second coordinate, whereby said input image data is successively resampled in two coordinate directions.

30. A method for resampling input image data as in claim 29, in which said two coordinates are orthogonal.

31. A method for resampling input image data as in claim 28 or 29, in which the image value of the input image which is resampled is the intensity value of input pixels.

32. A method for resampling input image data as in claim 28 or 29, in which the image value of the input image which is resampled is a color value of the input pixels.

33. A method for resampling second coordinate data, representing values of a second coordinate to which each pixel of an input image is to be transferred in an output image, in response to first coordinate values to adjust for image compression variations along the direction of said first coordinate, comprising the steps of:

(a) determining whether the next input pixel portion of input data fits within the boundaries of an output pixel, along the direction of said coordinate;

(b) determining, for an input pixel portion that so fits as determined in step (a), the midpoint value of said pixel portion, along said direction;

(c) determining, for an input pixel protion that does not so fit as determined in step (a), the part of the input pixel portion that fits within said output pixel along said direction;

(d) determining the midpoint value of the input pixel part determined in step (c), along said direction; and

(e) accumulating the midpoint value information from steps (b) and (d) as resampled second coordinate value data for the respective pixels, reflecting effects of image compression variations in the first coordinate direction.

34. A resampler for developing pixel value data representing input image pixel values after resampling to give effect to image compression variations along the direction of a first coordinate, comprising:

first means for determining whether the next input pixel portion in input data fits within the boundaries of an output pixel, along the direction of said first coordinate, and for input pixel portions that do not so fit, for determining the part of the input pixel portion that does fit within said output pixel, along said direction;

first multiplication means coupled to said first means and responsive to input pixel portions that so fit, for determining the midpoint value of said pixel portion, along said direction, and for deriving the area of said input pixel portion by multiplying said midpoint value by a figure representative of the length of said pixel portion, along said direction;

second multiplication means, coupled to said first means, for determining the midpoint value of said pixel part, along said direction, and for deriving the area of said input pixel part by multiplying said midpoint value by a figure representative of the length of said pixel part, along said direction; and

accumulation means, coupled to said first and second multiplication means, for accumulating the derived areas to obtain resampled values for complete output pixels;

whereby pixel values input to the resampler are resampled to provide at the output pixel values reflecting image compression values in said first coordinate direction.

36. A resampler for giving effect to image compression variations along the direction of a second coordinate, comprising a resampler as in claim 34 or 35 wherein the respective means are arranged for developing pixel value data representing input image pixel values after resampling to give effect to image compression variations along the direction of a second coordinate;

whereby said resampler may be arranged in series with a resampler in accordance with claim 34 or 35 to comprise an image warping system for successively resampling an input image in two coordinates.

35. An image system including at least one resampler for developing pixel value data representing image pixel values after resampling to give effect to image compression variations along the direction of a first coordinate, comprising:

first means for determining whether the next input pixel portion in input data fits within the boundaries of an output pixel, along the direction of said first coordinate;

second means coupled to said first means and responsive to input pixel portions that so fit, for determining the midpoint value of said pixel portion, along said direction;

first multiplication means, coupled to said second means, for deriving the area of said input pixel portion by multiplying said pixel portion midpoint value by a figure representative of the length of said pixel portion, along said direction;

third means, coupled to said first means and responsive to input pixel portions that do not so fit, for determining the part of the input pixel portion that does fit within said output pixel, along said direction;

fourth means, coupled to said third means, for determining the midpoint value of said pixel part, along said direction;

second multiplication means, coupled to said fourth means, for deriving the area of said input pixel part by multiplying said pixel part midpoint value by a figure representative of the length of said pixel part, along said direction; and

accumulation means, coupled to said first and second multiplication means, for accumulating the derived areas to obtain resampled values for complete output pixels;

whereby pixel values input to the resampler are resampled to provide at the output pixel values reflecting image compression values in said first coordinate direction.

37. The image system in accordance with claim 35, in which said at least one resampler comprising:

a first resampler and

a second resampler, coupled to said first resampler, wherein the respective means are arranged for developing pixel value data representing input image pixel values from said first resampler after second resampling to give effect to image compression variations along the direction of a second coordinate.

38. A coordinate resampler for developing coordinate value data representing second coordinate data, for the transfer of input pixel data to output pixel data, after resampling to give effect to image compression variations along the direction of a first coordinate, comprising:

first means for determining whether the next input pixel portion of input data fits within the boundaries of an output pixel, along the direction of said first coordinate;

second means coupled to said first means and responsive to input pixel portions that so fit, for determining the midpoint value of said pixel portion, along said direction;

third means, coupled to said first means and responsive to input pixel portions that do not so fit, for determining the part of the input pixel portion that does fit within said output pixel, along said direction;

fourth means coupled to said third means, for determining the midpoint value of said pixel part, along said direction; and

accumulation means coupled to said second and fourth means and responsive to said midpoint values, for providing resampled second coordinate value data for the respective pixels, reflecting effects of image compression variations in said first coordinate direction.