Summary
The
four target corner points are determined by applying
the traditional coordinate transform equations.
The remainder of the transform is accomplished
by providing output location and size factor values
to the interpolation algorithm for each column
of the first pass and each row of the second pass.
Because
the interpolation from the input image is complete,
contiguous, and continuous, there are no resampling
artifacts in the interior of the output image.
Because each row and each column is mapped with
subpixel position phasing, there is no edge aliasing
of the output image. The output image is generated
directly in the form that anti-aliasing attempts
to achieve under conventional approaches to spatial
image transform. Furthermore, the transformation
is achieved with relatively simple computations.
Figure
13, Figure 14 and Figure
15 illustrate a 30-degree rotation. Figure
16 is a 2-degree rotation and shrink by 1/2.
Figure 17 shows the
upper-left corner of the 2-degree rotation magnified
by pixel replication to show the partial-intensity
edge pixels which avoid edge aliasing. A true-perspective
mapping and various arbitrary mappings have also
been developed within the technique. Figure
18 and Figure 19 show
the intermediate an final images of a true-perspective
mapping. Figure 20 shows
a mapping of the image driven by a sine wave.
The
following images have been through much abuse
and are included here only to provide continuity
with the original article. Please do not take
these images as indicative of the quality of the
algorithm as quality may vary with monitor and
printer.
The images in Figures 13 through 20 are reproduced
with the permission of Jean Grapp.
Figure
13. The original image.
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Figure
14. Thirty-degree rotation;
intermediate image.
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Figure
15. Thirty-degree rotation;
final image.
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Figure
16. Two-degree rotation
and shrink by 1/2.
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Figure
17. Partial-intensity edges of
the 2-degree rotation.
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Figure
18. True-perspective mapping;
intermediate image.
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Figure
19. True-respective mapping;
final image.
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Figure
20. Sine-wave mapping
in both directions.
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Conclusion
The
one-dimensional resampling interpolation technique
is computationally simple, deterministically bounded,
and provides a high-fidelity mapping from discrete
input pixels to discrete output pixels. It can
be combined in two passes over a two-dimensional
image to effect two-dimensional spatial transforms.
Real-time systems using the technique can be implemented
more economically and perform higher fidelity
transforms than systems using the standard technique
of mapping points first and then interpolating
intensities.
References
1. |
E.
Catmul and A.R. Smith, "3D Transformations
of Images in Scanline Order, Computer Graphics
(Proc. SIGGRAPH 80), Vol. 14, No. 3, July
1980, pp.279-286. |
2. |
M.
Shantz, "Two Pass Warp Algorithm for
Hardware Implementation," Proc. SPIE.
Vol. 306, "Processing and Display of
Three Dimensional Data," 1982, pp.l60-164. |
Footnotes
1.This is a rerendering
of: Karl Fant, "A Nonaliasing, Real-Time
Spatial Transform Technique", IEEE Computer
Graphics and Applications, January 1986, pages
71-80. |