Sparse Bounded Degree Sum of Squares Optimization for Certifiably Globally Optimal Rotation Averaging

TL;DR

This paper introduces a novel convex relaxation approach for rotation averaging that guarantees globally optimal solutions for any problem instance, improving reliability over traditional local methods.

Contribution

It formulates rotation averaging as a polynomial optimization problem over quaternions and solves it using a sparse SOS relaxation with formal global optimality guarantees.

Findings

First globally optimal rotation averaging method for all instances.

Open source implementation available for practical use.

Demonstrates improved accuracy and reliability over local optimization methods.

Abstract

Estimating unknown rotations from noisy measurements is an important step in SfM and other 3D vision tasks. Typically, local optimization methods susceptible to returning suboptimal local minima are used to solve the rotation averaging problem. A new wave of approaches that leverage convex relaxations have provided the first formal guarantees of global optimality for state estimation techniques involving SO(3). However, most of these guarantees are only applicable when the measurement error introduced by noise is within a certain bound that depends on the problem instance's structure. In this paper, we cast rotation averaging as a polynomial optimization problem over unit quaternions to produce the first rotation averaging method that is formally guaranteed to provide a certifiably globally optimal solution for \textit{any} problem instance. This is achieved by formulating and solving a sparse convex sum of squares (SOS) relaxation of the problem. We provide an open source implementation of our algorithm and experiments, demonstrating the benefits of our globally optimal approach.