## Orthogonal Pyramid Transforms for Image Coding

### Edward H. Adelson, Eero Simoncelli, and Rajesh Hingorani

Published in
SPIE Proceedings on Visual Communications and Image Processing II, pp. 50-58
(1987).

We describe a set of pyramid transforms that decompose an image into a
set of basis functions that are (a) spatial-frequency tuned, (b)
orientation tuned, (c) spatially localized, and (d) self-similar. For
computational reasons the set is also (e) orthogonal and lends itself to
(f) rapid computation. The systems are derived from concepts in matrix
algebra, but are closely connected to decompositions based on
quadrature mirror filters. Our computations take place hierarchically,
leading to a pyramid representation in which all of the basis functions
have the same basic shape, and appear at many scales. By placing the
high-pass and low-pass kernels on staggered grids, we can derive odd-tap
QMF kernels that are quite compact. We have developed pyramids using
separable, quincunx, and hexagonal kernels. Image data compression with
the pyramids gives excellent results, both in terms of MSE and visual
appearance. A non-orthogonal variant allows good performance with 3-tap
basis kernels and the appropriate inverse sampling kernels.