Decomposed Global Optimization for Robust Point Matching with Low-Dimensional Branching

TL;DR

This paper presents a novel global optimization approach for robust point matching that transforms the RPM objective into a decomposable form, enabling efficient and robust alignment of point sets under various transformations.

Contribution

It introduces a new quadratic reformulation of RPM, a tight lower bound via convex envelope, and a specialized Branch-and-Bound algorithm focusing on transformation parameters.

Findings

Outperforms state-of-the-art methods in robustness to noise and outliers.

Efficiently solves the alignment problem with polynomial-time subproblems.

Demonstrates effectiveness on both 2D and 3D real-world data.

Abstract

Numerous applications require algorithms that can align partially overlapping point sets while maintaining invariance to geometric transformations (e.g., similarity, affine, rigid). This paper introduces a novel global optimization method for this task by minimizing the objective function of the Robust Point Matching (RPM) algorithm. We first reveal that the original RPM objective is a cubic polynomial. Through a concise variable substitution, we transform this objective into a quadratic function. By leveraging the convex envelope of bilinear monomials, we derive a tight lower bound for this quadratic function. This lower bound problem conveniently and efficiently decomposes into two parts: a standard linear assignment problem (solvable in polynomial time) and a low-dimensional convex quadratic program. Furthermore, we devise a specialized Branch-and-Bound (BnB) algorithm that branches exclusively on the transformation parameters, which significantly accelerates convergence by confining the search space. Experiments on 2D and 3D synthetic and real-world data demonstrate that our method, compared to state-of-the-art approaches, exhibits superior robustness to non-rigid deformations, positional noise, and outliers, particularly in scenarios where outliers are distinct from inliers.