“Planar 2-pass texture mapping and warping” by Smith

The 2-pass tranformation replaces a 2-D (2-dimensional) transformation with a sequence of orthogonal, simpler 1-D transformation. It may be used for the closely related processes of texture mapping and warping in computer graphics and image processing. First, texture maps onto planar quadric and superquadric surfaces and, second, planar bicubic and biquadratic warps of images are shown to be 2-pass transformable. A distinction between serial and parallel warps is introduced to solve a confusion in terms between computer graphics and image processing. It is shown that an n-th order serial polynomial warp is equivalent to an (n2+n)-th order parallel polynomial warp. It is also shown that the serial equivalent to a parallel polynomial warp is generally not a polynomial warp, being more complicated than a polynomial. The unusual problem of bottlenecking and the usual one of antialiasing are discussed in the 2-pass context.

1. Barr, Alan H. Superquadrics and Angle-Preserving Transformations, IEEE Computer Graphics and Applications, January, 1981, pp. 11-23.
2. Bennett, Philip P., and Steven A. Gabriel, System for Spatially Transforming Images, United States Patent 4,472,732, September 18, 1984. (Assigned to Ampex Corporation. This is one of the patents for the very successful ADO (Ampex Digital Optics) product. See also U. S. Patents 4,463,372 and 4,468,688.)
3. Catmull, Edwin, and Alvy Ray Smith. 3-D Transformations of lmages in Scanline Order, Computer Graphics, Vol. 14, No. 3, pp. 279-285, July, 1980. (SIGGRAPH 80 Conference Proceedings).
4. Fant, Karl M. A Nonaliasing, Real-Time Spatial Transform Technique, IEEE Computer Graphics and Applications, January, 1986, pp. 71-80. (See also the Letters to the Editor section of IEEE Computer Graphics and Applications, March, 1986, pp. 66-67, and July, 1986, pp. 3 and 8.)
5. Fraser, Donald, Robert A. Schowengerdt, and Ian Briggs. Rectification of Multichannel Images in Mass Storage Using Image Transposition, Computer Vision, Graphics, and Image Processing, Volume 29, Number 1, January, 1985, pp. 23-36. (See also Corrigendum, Volume 31, Number 3, September, 1985, p. 395.)
6. Paeth, Alan. A Fast Algorithm for General Raster Rotation, Graphics Interface ’86 Proceedings, May, 1986, pp. 77-81.
7. Smith, Alvy Ray. Projected Superquadrics are 2-Pass Transformable, Technical Memo 54, Computer Graphics Department, Computer Division, Lueasfilm Ltd., August, 1982 (now Technical Memo 54, Pixar).
8. Smith, Alvy Ray. A 2-Pass Solution to the Planar Biquadratic Patch, Technical Memo 128, Computer Graphics Department, Computer Division, Lucasfilm Ltd., May, 1985 (now Technical Memo 128, Pixar).
9. Smith, Alvy Ray. A 2-Pass Solution to the Planar Bicubic Patch, Technical Memo 132, Computer Graphics Department, Computer Division, Lucasfilm Ltd., June, 1985 (now Technical Memo 132, Pixar).
10. Smith, Alvy Ray. Serial vs Parallel Warps, Technical Memo 134, Computer Graphics Department, Computer Division, Lucasfilm Ltd., June, 1985 (now Technical Memo 134, Pixar).
11. Smith, Alvy Ray. Digital Filtering Tutorial for Computer Graphics Technical Memo 27, Computer Graphics Department, Computer Division, Lucasfilm Ltd., March, 1983 (now Technical Memo 27, Pixar). Also in Tutorial Notes for Siggraph 1983 and Siggraph 1984.
12. Smith, Airy Ray. Digital Filtering Tutorial, Part H Technical Memo 44, Computer Graphics Department, Computer Division, Lucasfilm Ltd., May, 1983 (now Technical Memo 44, Pixar). Also in Tutorial Notes for Siggraph 1983 and Siggraph 1984.