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The Wahba problem, also known as rotation search, seeks to find the bestrotation to align two sets of vector observations given putativecorrespondences, and is a fundamental routine in many computer vision androbotics applications. This work proposes the first polynomial-time certifiablyoptimal approach for solving the Wahba problem when a large number of vectorobservations are outliers. Our first contribution is to formulate the Wahbaproblem using a Truncated Least Squares (TLS) cost that is insensitive to alarge fraction of spurious correspondences. The second contribution is torewrite the problem using unit quaternions and show that the TLS cost can beframed as a Quadratically-Constrained Quadratic Program (QCQP). Since theresulting 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. Wevalidate the proposed algorithm, named QUASAR (QUAternion-based SemidefiniterelAxation for Robust alignment), in both synthetic and real datasets showingthat the algorithm outperforms RANSAC, robust local optimization techniques,global outlier-removal procedures, and Branch-and-Bound methods. QUASAR is ableto compute certifiably optimal solutions (i.e. the relaxation is exact) even inthe case when 95% of the correspondences are outliers.

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