A Concave Optimization Algorithm for Matching Partially Overlapping Point Sets

TL;DR

This paper introduces a novel concave optimization algorithm for matching partially overlapping point sets, leveraging a linear transformation of the objective function and a branch-and-bound approach for improved robustness and accuracy.

Contribution

It models point matching as a concave optimization problem and develops an efficient branch-and-bound algorithm with a new lower bounding scheme for partial point set matching.

Findings

Outperforms state-of-the-art methods in robustness and accuracy

Uses a concave optimization formulation with few nonlinear terms

Employs an efficient lower bounding scheme in branch-and-bound

Abstract

Point matching refers to the process of finding spatial transformation and correspondences between two sets of points. In this paper, we focus on the case that there is only partial overlap between two point sets. Following the approach of the robust point matching method, we model point matching as a mixed linear assignment-least square problem and show that after eliminating the transformation variable, the resulting problem of minimization with respect to point correspondence is a concave optimization problem. Furthermore, this problem has the property that the objective function can be converted into a form with few nonlinear terms via a linear transformation. Based on these properties, we employ the branch-and-bound (BnB) algorithm to optimize the resulting problem where the dimension of the search space is small. To further improve efficiency of the BnB algorithm where computation of the lower bound is the bottleneck, we propose a new lower bounding scheme which has a k-cardinality linear assignment formulation and can be efficiently solved. Experimental results show that the proposed algorithm outperforms state-of-the-art methods in terms of robustness to disturbances and point matching accuracy.