Global Truncated Loss Minimization for Robust and Threshold-Resilient Geometric Estimation

Abstract

To achieve outlier-robust geometric estimation, robust objective functions are generally employed to mitigate the influence of outliers. The widely used consensus maximization(CM) is highly robust when paired with global branch-and-bound(BnB) search. However, CM relies solely on inlier counts and is sensitive to the inlier threshold. Besides, the discrete nature of CM leads to loose bounds, necessitating extensive BnB iterations and computation cost. Truncated losses(TL), another continuous alternative, leverage residual information more effectively and could potentially overcome these issues. But to our knowledge, no prior work has systematically explored globally minimizing TL with BnB and its potential for enhanced threshold resilience or search efficiency. In this work, we propose GTM, the first unified BnB-based framework for globally-optimal TL loss minimization across diverse geometric problems. GTM involves a hybrid solving design: given an n-dimensional problem, it performs BnB search over an (n-1)-dimensional subspace while the remaining 1D variable is solved by bounding the objective function. Our hybrid design not only reduces the search space, but also enables us to derive Lipschitz-continuous bounding functions that are general, tight, and can be efficiently solved by a classic global Lipschitz solver named DIRECT, which brings further acceleration. We conduct a systematic evaluation on various BnB-based methods for CM and TL on the robust linear regression problem, showing that GTM enjoys remarkable threshold resilience and the highest efficiency compared to baseline methods. Furthermore, we apply GTM on different geometric estimation problems with diverse residual forms. Extensive experiments demonstrate that GTM achieves state-of-the-art outlier-robustness and threshold-resilience while maintaining high efficiency across these estimation tasks.

Figures (10)

• Figure 1: Comparison of the CM and TL objective functions in a 2D robust estimation problem (c.f. Section \ref{['sec:2dest']}) under various thresholds $\xi$. Stars mark the objective function values of ground truth (GT) and dots mark the global optimums. As $\xi$ increases, the GT loses its status as the global maximum in CM while maintaining the global minimum in TL.
• Figure 2: Experiments on robust linear regression (c.f. Section \ref{['subsection:compare']}). (a) Increasing the problem dimension with fixed threshold $\xi = 0.02$; (b) Increasing $\xi$ on the 3-dimensional problem. TL-based methods exhibit higher accuracy and threshold robustness than CM-based methods. Among TL-based methods, our GTM achieves a $3\times$-$45\times$ speed-up over standalone BnB or DIRECT. ($M = 500$, 90$\%$ outliers, 50 trials)
• Figure 3: Illustration of the planar motion constraint in the relative pose estimation problem (c.f. Section \ref{['sec:2dest']}). The camera motion is constrained to 1-dimensional rotation parameterized by an angle $\theta$, and 2-dimensional translation parameterized by the length $\rho$ (usually set to unit due to monocular ambiguity) and angle $\phi$. Given a set of 2D-2D matches consisting of both inliers and outliers, we aim to recover the $\theta$ and $\phi$.
• Figure 4: Evaluation results of application 1 on simulated datasets (c.f. Section \ref{['sec:2dest-sim']}). (a) Increasing outlier ratio with fixed $\xi = 0.1$; (b) Increasing $\xi$ with fixed $90\%$ outlier ratio. Only our GTM and the DIRECT-2D can survive all cases, with GTM being several times faster; Moreover, while both relying on BnB, GTM usually runs several times faster than ACM as GTM requires much fewer branching iterations. ($M = 2000$, 50 trials)
• Figure 5: Illustration of the outlier-robust point cloud registration problem (c.f. Section \ref{['sec:3dest']}). Given a set of outlier-corrupted 3D-3D correspondences between the source model and the target point set, we aim to estimate a rigid transformation that best aligns them.

Theorems & Definitions (1)

• Proposition 1