Graph rigidity, Cyclic Belief Propagation and Point Pattern Matching

TL;DR

This paper introduces a new globally rigid graph for point pattern matching that enables more efficient inference via loopy belief propagation, maintaining optimality in noiseless scenarios and comparable accuracy with noise.

Contribution

It presents a smaller, globally rigid graph that allows efficient inference with loopy belief propagation, improving over previous chordal graph methods.

Findings

Inference with the new graph is more efficient due to smaller maximal clique size.

Loopy belief propagation converges to the optimal solution in the proposed graph.

Accuracy remains comparable to previous methods under noisy conditions.

Abstract

A recent paper \cite{CaeCaeSchBar06} proposed a provably optimal, polynomial time method for performing near-isometric point pattern matching by means of exact probabilistic inference in a chordal graphical model. Their fundamental result is that the chordal graph in question is shown to be globally rigid, implying that exact inference provides the same matching solution as exact inference in a complete graphical model. This implies that the algorithm is optimal when there is no noise in the point patterns. In this paper, we present a new graph which is also globally rigid but has an advantage over the graph proposed in \cite{CaeCaeSchBar06}: its maximal clique size is smaller, rendering inference significantly more efficient. However, our graph is not chordal and thus standard Junction Tree algorithms cannot be directly applied. Nevertheless, we show that loopy belief propagation in such a graph converges to the optimal solution. This allows us to retain the optimality guarantee in the noiseless case, while substantially reducing both memory requirements and processing time. Our experimental results show that the accuracy of the proposed solution is indistinguishable from that of \cite{CaeCaeSchBar06} when there is noise in the point patterns.