Certifiably Optimal Outlier-Robust Geometric Perception: Semidefinite Relaxations and Scalable Global Optimization

We propose the first general and scalable framework to design
certifiable
algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated least squares (TLS), maximum consensus, Geman-McClure, Tukey's biweight, among others, can be reformulated as
polynomial optimization problems
(POPs). By focusing on the TLS cost, our second contribution is to exploit
sparsity
in the POP and propose a sparse
semidefinite programming
(SDP) relaxation that is much smaller than the standard Lasserre's hierarchy while preserving empirical
exactness
, i.e., the SDP recovers the optimizer of the nonconvex POP with an
optimality certificate
. Our third contribution is to solve the SDP relaxations at an unprecedented scale and accuracy by presenting
${{\sf STRIDE}}$
, a solver that blends
global descent
on the convex SDP with fast
local search
on the nonconvex POP. Our fourth contribution is an evaluation of the proposed framework on six geometric perception problems including single and multiple rotation averaging, point cloud and mesh registration, absolute pose estimation, and category-level object pose and shape estimation. Our experiments demonstrate that (i) our sparse SDP relaxation is empirically exact with up to
$60\%$
–
$90\%$
outliers across applications; (ii) while still being far from real-time,
${{\sf STRIDE}}$
is up to 100 times faster than existing SDP solvers on medium-scale problems, and is the only solver that can solve large-scale SDPs with hundreds of thousands of constraints to high accuracy; (iii)
${{\sf STRIDE}}$
safeguards existing fast heuristics for robust estimation (e.g.,
${{\sf RANSAC}}$
or Graduated Non-Convexity), i.e., it certifies global optimality if the heuristic estimates are optimal, or detects and allows escaping local optima when the heuristic estimates are suboptimal.