Maximizing robustness of point-set registration by leveraging non-convexity

Point-set registration is a classical image processing problem that looks for the optimal transformation between two sets of points. In this work, we analyze the impact of outliers when finding the optimal rotation between two point clouds. The presence of outliers motivates the use of least unsquared deviation, which is a non-smooth minimization problem over non-convex domain. We compare approaches based on non-convex optimization over special orthogonal group and convex relaxations. We show that if the fraction of outliers is larger than a certain threshold, any naive convex relaxation fails to recover the ground truth rotation regardless of the sample size and dimension. In contrast, minimizing the least unsquared deviation directly over the special orthogonal group exactly recovers the ground truth rotation for any level of corruption as long as the sample size is large enough. These theoretical findings are supported by numerical simulations.