Rotation Coordinate Descent for Fast Globally Optimal Rotation Averaging

TL;DR

This paper introduces Rotation Coordinate Descent (RCD), a fast, globally optimal algorithm for rotation averaging that outperforms existing SDP-based methods in efficiency and is easy to implement.

Contribution

The paper proposes RCD, a novel algorithm that directly updates rotations, avoiding large matrix storage, and proves its convergence, significantly improving speed over prior SDP relaxation methods.

Findings

RCD converges mathematically and empirically.

RCD outperforms state-of-the-art methods in speed.

RCD is simple to implement and adaptable.

Abstract

Under mild conditions on the noise level of the measurements, rotation averaging satisfies strong duality, which enables global solutions to be obtained via semidefinite programming (SDP) relaxation. However, generic solvers for SDP are rather slow in practice, even on rotation averaging instances of moderate size, thus developing specialised algorithms is vital. In this paper, we present a fast algorithm that achieves global optimality called rotation coordinate descent (RCD). Unlike block coordinate descent (BCD) which solves SDP by updating the semidefinite matrix in a row-by-row fashion, RCD directly maintains and updates all valid rotations throughout the iterations. This obviates the need to store a large dense semidefinite matrix. We mathematically prove the convergence of our algorithm and empirically show its superior efficiency over state-of-the-art global methods on a variety of problem configurations. Maintaining valid rotations also facilitates incorporating local optimisation routines for further speed-ups. Moreover, our algorithm is simple to implement; see supplementary material for a demonstration program.