A Polynomial-time Solution for Robust Registration with Extreme Outlier Rates (1903.08588v2)

Abstract: We propose a robust approach for the registration of two sets of 3D points in the presence of a large amount of outliers. Our first contribution is to reformulate the registration problem using a Truncated Least Squares (TLS) cost that makes the estimation insensitive to a large fraction of spurious point-to-point correspondences. The second contribution is a general framework to decouple rotation, translation, and scale estimation, which allows solving in cascade for the three transformations. Since each subproblem (scale, rotation, and translation estimation) is still non-convex and combinatorial in nature, out third contribution is to show that (i) TLS scale and (component-wise) translation estimation can be solved exactly and in polynomial time via an adaptive voting scheme, (ii) TLS rotation estimation can be relaxed to a semidefinite program and the relaxation is tight in practice, even in the presence of an extreme amount of outliers. We validate the proposed algorithm, named TEASER (Truncated least squares Estimation And SEmidefinite Relaxation), in standard registration benchmarks showing that the algorithm outperforms RANSAC and robust local optimization techniques, and favorably compares with Branch-and-Bound methods, while being a polynomial-time algorithm. TEASER can tolerate up to 99% outliers and returns highly-accurate solutions.

• The paper presents TEASER, a novel approach that robustly registers 3D point clouds by discarding outliers with a truncated least squares formulation.
• It decomposes the challenging registration task into scale, rotation, and translation estimation subproblems, each solved in polynomial time.
• Empirical evaluations show that TEASER outperforms traditional methods like RANSAC, ensuring high efficiency and accuracy in real-world robotics applications.

A Polynomial-time Solution for Robust Registration with Extreme Outlier Rates: An Overview

The paper by Heng Yang and Luca Carlone presents a novel approach to the longstanding problem of point cloud registration in robotics and computer vision. This paper addresses the challenge of registering two sets of 3D points in environments where outliers are prevalent, proposing a method that is both robust to such perturbations and computationally feasible in polynomial time.

Point cloud registration involves determining the best transformation - including rotation, translation, and scaling - that aligns two point clouds. Traditional techniques often rely on the presumption of correct point correspondences or minimal noise, which are rarely the case in practical scenarios. The presence of significant outliers generally complicates the registration process, necessitating robust methodologies that can handle such discrepancies without compromising the accuracy of the estimate.

Methodological Contributions

The authors introduce the Truncated Least Squares (TLS) cost formulation, which effectively discards measurements with large residuals, thereby minimizing the influence of outliers on the registration outcome. This cost formulation is pivotal to the proposed method, ensuring that the estimation remains insensitive to noise and spurious data.

The paper details a framework that segregates the problem into distinct subproblems - scale, rotation, and translation estimation. Each of these components traditionally represents a non-convex challenge. However, through careful reformulation, the authors manage to address these subproblems in a manner that ensures both accuracy and computational feasibility. Specifically, their contributions include:

1. Scale and Translation Estimation: Utilizing an adaptive voting scheme, the authors show that TLS problems in these domains can be exactly solved in polynomial time. This is achieved by leveraging properties of scalar estimation to circumvent the typical combinatorial explosion.
2. Rotation Estimation: By formulating it as a semidefinite programming (SDP) problem, the authors present a relaxation that is empirically tight, even with an extreme number of outliers. This SDP approach ensures robustness to noise while maintaining computational tractability.

The culmination of these efforts is the algorithm TEASER (Truncated least squares Estimation And SEmidefinite Relaxation for Extreme Registration), which demonstrates superior performance in comparison to existing methods like RANSAC and Branch-and-Bound techniques. TEASER's notable capability to handle 99% outlier scenarios is validated through extensive benchmarks and practical robotics datasets.

Implications and Future Outlook

This paper's contributions have significant implications for fields involving 3D data processing, especially robotics and autonomous systems where environmental unpredictability is common. By providing a robust and efficient registration solution, it enables real-time applications in these areas even under challenging conditions with high outlier rates.

From a theoretical perspective, this work inspires further exploration into decoupling complicated optimization problems into more manageable subproblems. The methodologies employed, particularly the use of TLS and SDP in exploiting problem structure, offer a blueprint that can be adapted to other domains faced with similar robustness challenges.

In conclusion, Yang and Carlone provide a sophisticated yet practical framework for 3D point cloud registration, culminating in a solution that balances robustness and efficiency effectively. This research not only advances the understanding of algorithmic design for challenging registration tasks but also paves the way for future advancements in related areas of artificial intelligence and robotics. As the field progresses, it will be interesting to see how these ideas are refined and expanded to address even broader challenges in environmental modeling and machine perception.