GOGMA: Globally-Optimal Gaussian Mixture Alignment

TL;DR

GOGMA introduces a globally-optimal algorithm for 3D Gaussian mixture alignment using branch-and-bound, ensuring accuracy regardless of initial conditions and outperforming existing methods on challenging datasets.

Contribution

First globally-optimal solution for 3D Gaussian mixture alignment under L2 distance, combining branch-and-bound with local optimization for efficiency.

Findings

Outperforms existing globally-optimal registration methods on challenging datasets

Provides guaranteed global optimality regardless of initialisation

Employs novel bounds based on SE(3) geometry

Abstract

Gaussian mixture alignment is a family of approaches that are frequently used for robustly solving the point-set registration problem. However, since they use local optimisation, they are susceptible to local minima and can only guarantee local optimality. Consequently, their accuracy is strongly dependent on the quality of the initialisation. This paper presents the first globally-optimal solution to the 3D rigid Gaussian mixture alignment problem under the L2 distance between mixtures. The algorithm, named GOGMA, employs a branch-and-bound approach to search the space of 3D rigid motions SE(3), guaranteeing global optimality regardless of the initialisation. The geometry of SE(3) was used to find novel upper and lower bounds for the objective function and local optimisation was integrated into the scheme to accelerate convergence without voiding the optimality guarantee. The evaluation empirically supported the optimality proof and showed that the method performed much more robustly on two challenging datasets than an existing globally-optimal registration solution.