Approximating the Problem, not the Solution: An Alternative View of Point Set Matching

Abstract

This work discusses the issue of approximation in point set matching problems. In general, one may have two classes of approximations when tackling a matching problem: a representational approximation, which involves a simplified and suboptimal modeling for the original problem, and algorithmic approximation, which consists in using suboptimal algorithms to infer the assignment. Matching techniques in general have relied on the second approach: to keep a complete model of the original problem and use suboptimal techniques to solve it. In this paper, we show how a technique based on using exact inference in simple graphical models, which is an instance of the first class, can significantly outperform instances of techniques from the second class. We give theoretical insights of why this happens, and experimentally compare our approach with the well-known Shapiro and Brady and Christmas et al. methods, which are exemplars of the second class. We perform experiments with synthetic and real-world data sets, which reveal a significant accuracy improvement of the proposed technique both under point position jitter and size increasing of the point sets. The main conclusion is that techniques based on optimal algorithms with appropriate suboptimal representations may lead to better results than their counterparts which consist in using optimal representations, but approximate algorithms.