One Ring to Rule Them All: Certifiably Robust Geometric Perception with Outliers

Abstract

We propose a general and practical framework to design certifiable algorithms for robust geometric perception in the presence of a large amount of outliers. We investigate the use of a truncated least squares (TLS) cost function, which is known to be robust to outliers, but leads to hard, nonconvex, and nonsmooth optimization problems. Our first contribution is to show that -for a broad class of geometric perception problems-TLS estimation can be reformulated as an optimization over the ring of polynomials and Lasserre's hierarchy of convex moment relaxations is empirically tight at the minimum relaxation order (i.e., certifiably obtains the global minimum of the nonconvex TLS problem). Our second contribution is to exploit the structural sparsity of the objective and constraint polynomials and leverage basis reduction to significantly reduce the size of the semidefinite program (SDP) resulting from the moment relaxation, without compromising its tightness. Our third contribution is to develop scalable dual optimality certifiers from the lens of sums-of-squares (SOS) relaxation, that can compute the suboptimality gap and possibly certify global optimality of any candidate solution (e.g., returned by fast heuristics such as RANSAC or graduated non-convexity). Our dual certifiers leverage Douglas-Rachford Splitting to solve a convex feasibility SDP. Numerical experiments across different perception problems, including high-integrity satellite pose estimation, demonstrate the tightness of our relaxations, the correctness of the certification, and the scalability of the proposed dual certifiers to large problems, beyond the reach of current SDP solvers.Preprint. Under review.

Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications

“…17 The interested reader can find examples of failure modes in Appendix 1. Future work includes coupling the algorithms proposed in this paper with fast certifiers [65] that can detect and reject incorrect estimates.…”

“…Certifiably robust algorithms are a class of global solvers that have been recently shown to strike a good balance between computational complexity and global optimality [4], [65]. Certifiable algorithms relax non-convex robust estimation problems into convex semidefinite programs (SDP), whose solutions can be obtained in polynomial time and provide readily checkable a posteriori global optimality certificates [30], [31], [66], [67].…”

“…When the 3D model is far from the camera center, a weak perspective camera model can be adopted [41], which leads to efficient robust estimation using GNC [27]. Yang and Carlone [65] develop optimality certification algorithms for shape alignment with outliers, and demonstrate successful application to satellite pose estimation.…”

“…Yang and Carlone propose invariant measurements to decouple the rotation and translation estimation [67], and develop certifiably robust rotation search using semidefinite relaxation [31]. The joint use of fast heuristics (e.g., GNC) and optimality certification for both point cloud registration and mesh registration has been demonstrated in [4], [65]. The registration approach [4] has been shown to be robust to 99% outliers.…”

Certifying Global Optimality of AC-OPF Solutions via the CS-TSSOS Hierarchy

“…Background on polynomial optimization. Polynomial optimization considers optimization problems where both the cost function and constraints are defined by polynomials, which widely arises in numerous fields, such as optimal power flow [7], numerical analysis [14], computer vision [26], deep learning [5], discrete optimization [19], etc. Even though it is usually not hard to find a local optimal solution by a local solver (e.g., Ipopt [20]), the task of solving a polynomial optimization problem (POP) to global optimality is NP-hard in general.…”