Representing matrices as quadtrees for parallel processors☆

The quadtree representation of matrices is a uniform representation for both sparse and dense matrices which can facilitate shared manipulation on multiprocessors. This paper presents worst-case and average-case resource requirements for storing and retrieving familiar families of patterned matrices: packed, symmetric, triangular, Toeplitz, and banded. Using this representation it compares resource requirements of three kinds of permutation matrices, as examples of nondense, unpatterned matrices. Exact values for the shuffle and bit-reversal permutations (as in the fast Fourier transform) and tight bounds on the expected values from purely random permutations are derived. Two different measures, density and sparsity, are proposed from these values. Analysis of quadtree matrix addition relates density of addends to space bounds on their sum and relates their sparsity to time bounds for computing that sum.

This paper presents a bibliography of nearly 1100 references related to the computer processing of pictorial information, arranged by subject matter. Coverage is restricted, for the most part, to a selected set of U.S. journals and proceedings of specialized meetings. The topics covered include digitization, approximation, and compression; transforms, filtering, enhancement, restoration, and reconstruction; architectures, systems, software, and techniques; pictorial pattern recognition; feature detection, segmentation, and image analysis; matching and time-varying imagery; shape and pattern; geometry; texture; and three-dimensional scene analysis. No attempt is made to evaluate or summarize the items cited; the purpose is simply to provide a convenient compendium of references.