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 antialiasing 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 30degree rotation. Figure
16 is a 2degree rotation and shrink by 1/2.
Figure 17 shows the
upperleft corner of the 2degree rotation magnified
by pixel replication to show the partialintensity
edge pixels which avoid edge aliasing. A trueperspective
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 trueperspective
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.


Figure
14. Thirtydegree rotation;
intermediate image.




Figure
15. Thirtydegree rotation;
final image.


Figure
16. Twodegree rotation
and shrink by 1/2.




Figure
17. Partialintensity edges of
the 2degree rotation.


Figure
18. Trueperspective mapping;
intermediate image.




Figure
19. Truerespective mapping;
final image.


Figure
20. Sinewave mapping
in both directions.

Conclusion
The
onedimensional resampling interpolation technique
is computationally simple, deterministically bounded,
and provides a highfidelity mapping from discrete
input pixels to discrete output pixels. It can
be combined in two passes over a twodimensional
image to effect twodimensional spatial transforms.
Realtime 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.279286. 
2. 
M.
Shantz, "Two Pass Warp Algorithm for
Hardware Implementation," Proc. SPIE.
Vol. 306, "Processing and Display of
Three Dimensional Data," 1982, pp.l60164. 
Footnotes
1.This is a rerendering
of: Karl Fant, "A Nonaliasing, RealTime
Spatial Transform Technique", IEEE Computer
Graphics and Applications, January 1986, pages
7180. 