A Quaternion-based Certifiably Optimal Solution to the Wahba Problem with Outliers

TL;DR

This paper introduces QUASAR, a novel quaternion-based convex relaxation method that efficiently and reliably finds globally optimal rotations in the Wahba problem even with up to 95% outliers, outperforming existing techniques.

Contribution

It formulates a robust, certifiably optimal solution to the Wahba problem using a novel TLS cost, quaternion representation, and SDP relaxation, handling high outlier ratios.

Findings

QUASAR outperforms RANSAC and other methods in accuracy and robustness.

The SDP relaxation is tight even with large noise and outliers.

The approach can handle up to 95% outliers in data.

Abstract

The Wahba problem, also known as rotation search, seeks to find the best rotation to align two sets of vector observations given putative correspondences, and is a fundamental routine in many computer vision and robotics applications. This work proposes the first polynomial-time certifiably optimal approach for solving the Wahba problem when a large number of vector observations are outliers. Our first contribution is to formulate the Wahba problem using a Truncated Least Squares (TLS) cost that is insensitive to a large fraction of spurious correspondences. The second contribution is to rewrite the problem using unit quaternions and show that the TLS cost can be framed as a Quadratically-Constrained Quadratic Program (QCQP). Since the resulting optimization is still highly non-convex and hard to solve globally, our third contribution is to develop a convex Semidefinite Programming (SDP) relaxation. We show that while a naive relaxation performs poorly in general, our relaxation is tight even in the presence of large noise and outliers. We validate the proposed algorithm, named QUASAR (QUAternion-based Semidefinite relAxation for Robust alignment), in both synthetic and real datasets showing that the algorithm outperforms RANSAC, robust local optimization techniques, global outlier-removal procedures, and Branch-and-Bound methods. QUASAR is able to compute certifiably optimal solutions (i.e. the relaxation is exact) even in the case when 95% of the correspondences are outliers.