Efficient Colored Point Set Matching Under Noise

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two colored point sets in $\mathcal{R}^2$, with $|\mathcal{A}| \le |\mathcal{B}|$. We propose a process for determining matches, in terms of the bottleneck distance, between $\mathcal{A}$ and subsets of $\mathcal{B}$ under color preserving rigid motion, assuming that the position of all colored points in both sets contains a certain amount of ”noise”. The process consists of two main stages: a lossless filtering algorithm and a matching algorithm. The first algorithm determines a number of candidate zones which are regions that contain a subset $\mathcal{S}$ of $\mathcal{B}$ such that $\mathcal{A}$ may match one or more subsets $\mathcal{B}'$ of $\mathcal{S}$. We use a compressed quadtree to have easy access to the subsets of $\mathcal{B}$ related to candidate zones and store geometric information that is used by the lossless filtering algorithm in each quadtree node. The second algorithm solves the colored point set matching problem: we generate all, up to a certain equivalence, possible motions that bring $\mathcal{A}$ close to some subset $\mathcal{B'}$ of every $\mathcal{S}$ and seek for a matching between sets $\mathcal{A}$ and $\mathcal{B}'$. To detect these possible matchings we use a bipartite matching algorithm that uses Skip Quadtrees for neighborhood queries. We have implemented the proposed algorithms and report results that show the efficiency of our approach.