Matching with matrix norm minimization

Abstract

Given (r ₁,r ₂,...r _n ) ∈ R ⁿ, for any I=(I ₁,I ₂,...I _n ) ∈ Z ⁿ, let E _I =(e _ij ), where e _ij =(r _i −r _j )−(I _i −I _i ), find I ∈ Z ⁿ such that ∥E _I ∥ is minimized, where ∥·∥ is a matrix norm. This is a matching problem where, given a real-valued pattern, the goal is to find the best discrete pattern that matches the real-valued pattern. The criterion of the matching is based on the matrix norm minimization instead of simple pairwise distance minimization. This matching problem arises in optimal curve rasterization in computer graphics and in vector quantization of data compression. Until now, there has been no polynomial-time solution to this problem. We present a very simple O(nlgn) time algorithm to solve this problem under various matrix norms.

Research supported by the Natural Sciences and Engineering Research Council of Canada grant OGP0046373.