One Ring to Rule Them All: Certifiably Robust Geometric Perception with Outliers (2006.06769v2)

Abstract: We propose the first 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 single rotation averaging, shape alignment, 3D point cloud and mesh registration, and 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.

• The paper’s main contribution is reformulating geometric perception using a Truncated Least Squares cost with Lasserre’s hierarchy to achieve global optimality.
• The approach exploits structural sparsity to significantly reduce computational complexity without sacrificing the tightness of the semidefinite relaxations.
• Dual certification via sums-of-squares relaxation enables scalable verification of solution optimality, enhancing real-time applicability in vision and robotics.

An Overview of Certifiably Robust Geometric Perception with Outliers

In the reviewed paper by Heng Yang and Luca Carlone, the authors present a comprehensive framework for designing certifiable algorithms aimed at tackling robust geometric perception challenges, particularly in the presence of substantial outliers. The paper is methodically structured, addressing both the theoretical underpinnings and the practical implementations of their proposed approach.

The crux of the problem addressed lies in geometric perception, which involves estimating unknown geometric models such as rotations, poses, and 3D structures from visual measurements. Traditional methods often struggle due to the non-convex nature of optimization problems and the prevalence of outliers in real-world data. As a solution, the authors propose using a Truncated Least Squares (TLS) cost function, reformed into an optimization problem over the ring of polynomials. They demonstrate that Lasserre's hierarchy of convex moment relaxations can achieve empirical tightness at the minimal relaxation order, providing a means to attain the global minimum of these non-convex problems.

The paper's contributions can be segmented into three distinct advancements:

1. Reformulation via TLS and Lasserre's Hierarchy: The authors reformulate geometric perception problems utilizing the TLS cost to handle outliers robustly. Their empirical findings confirm the tightness of Lasserre's hierarchy, even when applied at the minimum relaxation order, across various geometric perception tasks.
2. Exploiting Sparsity for Computational Reduction: By investigating the structural sparsity inherent in the objective and constraint polynomials, the authors leverage basis reduction techniques to substantially reduce the size of the semidefinite programs derived from moment relaxation. This approach does not compromise the tightness of the problem relaxation, thereby ensuring computational efficiency without a loss in solution quality.
3. Scalable Dual Certification via SOS Relaxation: The scalability of the framework is further enhanced through the development of dual optimality certifiers, utilizing sums-of-squares (SOS) relaxation. These certifiers can compute the suboptimality gap and possibly certify the global optimality of candidate solutions derived from fast heuristics like RANSAC or Graduated Non-Convexity (GNC).

The authors substantiate their claims through solid numerical experiments, evaluating their framework on diverse problems including single rotation averaging, 3D point cloud and mesh registration, shape alignment, and satellite pose estimation. These experiments not only demonstrate the robustness and correctness of their relaxations and certifications but also the scalability of the approach to scenarios beyond the capacity of existing solvers.

Implications and Future Directions

The implications of this work are profound for fields that rely on accurate geometric perception, such as robotics and computer vision. The authors suggest that their framework can enhance reliability in safety-critical systems, wherein accurate geometric estimations are paramount. The scalable nature of their dual certification method positions this framework as an attractive candidate for real-time applications where computational resources are constrained.

Looking forward, the exploration of further enhancing the scalability of the framework could be a significant area of development. While the proposed methods can handle larger instances than most available solvers, pushing these limits further would enhance applicability, particularly in complex real-time systems. Additionally, investigating the applicability and performance of this framework across other domains that involve high-dimensional data and require robust estimation amidst noise and outliers would be valuable.

The balanced approach of tackling both theoretical and practical aspects of a challenging problem domain makes this paper a significant contribution to the ongoing discourse on robust perception in AI and related fields.