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

TL;DR

The paper tackles robust point-set registration under partial overlap by proposing RPM-BP, a globally optimal Branch-and-Bound method that branches only over low-dimensional transformation parameters. It derives a tight lower bound via bilinear convex envelope relaxation, decoupling the problem into a linear assignment for correspondences and a small convex quadratic program for the transformation, enabling efficient global optimization. Empirical results in 2D and 3D demonstrate strong robustness to non-rigid deformations, noise, and outliers, particularly when outliers are spatially distinct from inliers, while maintaining reasonable run times. The approach offers scalable global optimality for transformations with few parameters and provides a practical foundation for precise registration in applications with partial overlap, though it relies on appropriate selection of the overlap parameter $n_p$ and is less suited for high-DOF transformations.

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.

Figures (9)

• Figure 1: Derivation of a lower bound function of the RPM objective function via variable substitution and bilinear relaxation.
• Figure 2: Derivation of a lower bound function $E_l(\mathbf p,\boldsymbol\theta)$ of $E(\mathbf p,\boldsymbol\theta)$ via variable substitution and bilinear relaxation.
• Figure 3: Histograms depicting the smallest eigenvalue of $\mathbf H^0+\mathbf C$ across 100 registration instances.
• Figure 4: Upper (first row) and lower bounds (second row) generated in each iteration of RPM-BP. Our method is tested with the $n_p$ value chosen as the ground truth and with varying initial ranges of $\boldsymbol\theta$: $[\boldsymbol\theta_{gt}-\Delta, \boldsymbol\theta_{gt}+ \Delta]$, where $\boldsymbol\theta_{gt}$ represents the ground truth $\boldsymbol\theta$ solution. Here, the margin $\Delta$ takes values of $0.5$, $1$, and $1.5$ respectively.
• Figure 5: a) to (c): Model point sets and examples of scene point sets in the deformation and noise tests, respectively. (d) to (i): Examples of model and scene point sets in the mixed outliers and inliers test ((d), (e)), separate outliers and inliers test ((f), (g)), and occlusion+outlier test ((h), (i)), respectively. In all cases, model points are indicated by red circles, while scene points are represented by blue crosses.

Theorems & Definitions (2)

• Proposition 1
• proof