Graduated Non-Convexity for Robust Spatial Perception: From Non-Minimal Solvers to Global Outlier Rejection (1909.08605v4)

Abstract: Semidefinite Programming (SDP) and Sums-of-Squares (SOS) relaxations have led to certifiably optimal non-minimal solvers for several robotics and computer vision problems. However, most non-minimal solvers rely on least-squares formulations, and, as a result, are brittle against outliers. While a standard approach to regain robustness against outliers is to use robust cost functions, the latter typically introduce other non-convexities, preventing the use of existing non-minimal solvers. In this paper, we enable the simultaneous use of non-minimal solvers and robust estimation by providing a general-purpose approach for robust global estimation, which can be applied to any problem where a non-minimal solver is available for the outlier-free case. To this end, we leverage the Black-Rangarajan duality between robust estimation and outlier processes (which has been traditionally applied to early vision problems), and show that graduated non-convexity (GNC) can be used in conjunction with non-minimal solvers to compute robust solutions, without requiring an initial guess. Although GNC's global optimality cannot be guaranteed, we demonstrate the empirical robustness of the resulting robust non-minimal solvers in applications, including point cloud and mesh registration, pose graph optimization, and image-based object pose estimation (also called shape alignment). Our solvers are robust to 70-80% of outliers, outperform RANSAC, are more accurate than specialized local solvers, and faster than specialized global solvers. We also propose the first certifiably optimal non-minimal solver for shape alignment using SOS relaxation.

• The paper introduces a robust method that integrates graduated non-convexity with non-minimal solvers to effectively mitigate outliers in spatial perception.
• It gradually transitions cost functions from convex approximations to their non-convex forms, ensuring accurate and efficient optimization.
• Empirical results demonstrate superior performance over RANSAC, handling up to 70-80% outliers in tasks like point cloud registration and pose estimation.

Graduated Non-Convexity for Robust Spatial Perception

The paper "Graduated Non-Convexity for Robust Spatial Perception: From Non-Minimal Solvers to Global Outlier Rejection" introduces a methodology to enhance the robustness of spatial perception algorithms in handling outliers, based on the concept of Graduated Non-Convexity (GNC). This approach seeks to combine non-minimal solvers with robust estimation techniques, providing a framework capable of handling significant percentages of outliers across various applications in robotics and computer vision, such as point cloud registration, pose graph optimization, and shape alignment.

Key Contributions

1. Graduated Non-Convexity Integration: The paper proposes the integration of GNC with existing non-minimal solvers, enhancing their robustness against outliers without requiring an initial guess. This is a significant stride as it provides a solution to leverage robust estimation methods while maintaining the certifiably optimal characteristics of non-minimal solvers.
2. Robustness Through GNC: By applying GNC, the methodology adapts robust cost functions gradually from convex approximations to their original non-convex form, which improves computational feasibility and solutions' resilience to outliers.
3. Empirical Validation: Demonstrations across applications show that the proposed GNC methods are robust to 70-80% outliers, outperforming RANSAC in terms of accuracy and speed in several scenarios.

Technical Insights

• Handling Non-Convexity: The approach addresses the challenge of non-convexity introduced by robust cost functions which typically complicate the use of non-minimal solvers. By employing GNC, the transition from a convex problem to a non-convex one is carefully managed, thus facilitating a more robust optimization process.
• Certifiably Optimal Solutions: The ability to provide globally optimal solutions despite the presence of high outlier ratios is emphasized. The paper presents an SOS relaxation for the shape alignment problem, ensuring certifiable optimal solutions in more complex scenarios.
• Algorithm Performance: In practical implementation, the algorithm provides rapid convergence and computational efficiency, significantly reducing the iterations needed compared to traditional methods such as RANSAC.

Practical Implications and Future Directions

The methodology's capability to handle large outlier ratios efficiently opens up new possibilities for spatial perception tasks crucial in various fields, including autonomous navigation, augmented reality, and mapping technologies. As spatial perception tools are integral to many emerging technologies, improvements in robustness and efficiency will undoubtedly impact sectors from autonomous vehicles to virtual reality setups.

Future developments could focus on further optimizing GNC parameters to ensure even faster convergence or implementing more complex models beyond current applications, broadening the applicability of this robust estimation methodology. Additionally, exploring theoretical guarantees on the convergence of GNC solutions could provide deeper insights and enhancements to the methodology.

In summary, the paper provides a well-structured approach to improve the robustness of non-minimal solvers in spatial perception tasks, with extensive empirical validation that showcases its practical value and potential for future research developments in robust optimization and estimation.