## TR#99: Orthogonal pyramid transforms for image coding

### E.H. Adelson, E. Simoncelli, R. Hingorani

Published in pages 50-58 of:
Proceedings of SPIE, Vol. 845

Visual Communication and Image Processing II

27-29 October 1987

Cambridge, MA

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 pyrmaids 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.