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

Tianyu Huang; Liangzu Peng; Xinyue Zhang; Tongfan Guan; Jinhu Dong; Haoang Li; Laurent Kneip; Yun-Hui Liu

cs.CVMar 16, 2026

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

Authors: Tianyu Huang, Liangzu Peng, Xinyue Zhang, Tongfan Guan, Jinhu Dong, Haoang Li, Laurent Kneip, Yun-Hui Liu

Organizations: Hong Kong Centre For Logistics Robotics, The Chinese University of Hong Kong. · Center for Innovation in Data Engineering and Science (IDEAS), University of Pennsylvania. · Mobile Perception Lab of the School of Information and Technology, ShanghaiTech University. · Thrust of Robotics and Autonomous Systems, The Hong Kong University of Science and Technology (Guangzhou).

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 & tables

Panel of Figure 1 — Fig. 1: Comparison of the CM and TL objective functions in a 2D robust estimation problem (c.f. Section V ) 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.

Panel of Figure 2 — Fig. 2: Experiments on robust linear regression (c.f. Section IV-D ). (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)

Method	Dimension ( $\xi=0.02$ )				$\xi$ (Dimension: 3)
Method	2	3	4	5	0.12	0.22	0.32	0.42
1D BnB	0.256	0.307	0.388	0.453	0.312	0.309	0.321	0.320
1D DIRECT	0.053	0.058	0.064	0.077	0.048	0.056	0.061	0.057

TABLE I: Average time (ms) of the inner solvers for lower bound computation at each iteration on the experiments of Fig. 2 ; the inner solvers are 1D BnB in Nested BnB (TL) and 1D DIRECT in GTM (c.f. Section IV-D ).

Fig. 3: Illustration of the planar motion constraint in the relative pose estimation problem (c.f. Section V ). 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 $\phi$ .

Fig. 4: Evaluation results of application 1 on simulated datasets (c.f. Section V-C1 ). (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)

Threshold $\xi$	Method \ Seq.	00-05				06-10
Threshold $\xi$	Method \ Seq.	SR ( $\%$ ) $\uparrow$	Max. Angular Error (deg.) $\downarrow$	Time (s) $\downarrow$	# of Iterations $\downarrow$	SR ( $\%$ ) $\uparrow$	Max. Angular Error (deg.) $\downarrow$	Time (s) $\downarrow$	# of Iterations $\downarrow$
$10^{-4}$	ACM [ 21 ]	79.17	4.13	0.86	3779	78.91	3.11	0.82	4049
	DIRECT-2D [ 26 ]	79.11	3.07	1.32	-	80.20	3.15	1.41	-
	GTM	80.49	3.11	0.45	467	80.11	3.01	0.46	459
$10^{-3}$	ACM [ 21 ]	71.10	4.77	0.88	3829	74.07	3.41	0.79	3751
	DIRECT-2D [ 26 ]	84.61	2.95	1.27	-	84.22	2.96	1.21	-

TABLE II: Evaluation results for planar motion estimation on the real-world KITTI dataset (c.f. Section V-C2 ). Max. angular errors are averaged results over all successful image pairs of each method. Our GTM is more robust threshold

\xi

than ACM and is several times faster than DIRECT-2D .

Fig. 5: Illustration of the outlier-robust point cloud registration problem (c.f. Section VI ). 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.

Fig. 6: Evaluation results for 3-DoF translation estimation on simulated data (c.f. Section VI-D1 ). (a) Varying outlier rates with fixed

\xi=2.5

; (b) Varying

\xi

with fixed

90\%

outlier rates. In various setups, only GTM maintains high robustness. Moreover, GTM usually runs 3-15 times faster than ACM and DIRECT-3D . (

M=1000

, 50 trials)

Threshold $\xi$	Method \ Scene	Arch & Facade			Courtyard			Office & Trees
Threshold $\xi$	Method \ Scene	SR $\uparrow$	$\textit{E}_{\mathbf{R}}/\textit{E}_{\mathbf{t}}$ ( ∘ /cm) $\downarrow$	Time (s) $\downarrow$	SR $\uparrow$	$\textit{E}_{\mathbf{R}}/\textit{E}_{\mathbf{t}}$ ( ∘ /cm) $\downarrow$	Time (s) $\downarrow$	SR $\uparrow$	$\textit{E}_{\mathbf{R}}/\textit{E}_{\mathbf{t}}$ ( ∘ /cm) $\downarrow$	Time (s) $\downarrow$
$0.04$	RANSAC [ 9 ]	11/31	1.04/15.37	0.19	13/28	0.47/13.26	0.41	2/25	0.64/10.11	0.17
	MLESAC [ 13 ]	14/31	1.10/21.96	0.24	9/28	0.55/10.84	0.53	2/25	0.61/9.96	0.32
	FGR [ 72 ]	9/31	0.95/19.98	0.21	2/28	1.40/30.01	0.46	0/25	-/-	0.23
	TEASER++ [ 59 ]	29/31	0.25/ 5.87	0.42	28/28	0.09/3.87	1.43	23/25	0.52/7.23	2.08
	MAC [ 73 ]	27/31	0.17 /6.28	4.66	28/28	0.11/4.01	18.29	20/25	0.31 / 5.62	25.30

TABLE III: Evaluation results for 6-DoF rigid pose estimation on the real-world ETH dataset (c.f. Section VI-D2 ). The best results are boldfaced . The second best are underlined . Running times are marked as gray for methods with SR lower than

50\%

. GTM is the most robust to threshold variations.

Panel of Figure 10 — Fig. 7: Two representative visual cases of 6-DoF rigid pose estimation on the real-world ETH dataset (c.f. Section VI-D2 ). The first column presents the input point pairs and the other columns show the outputs of various methods. Only MAC and our GTM succeed in all cases, and GTM is $10\times$ faster than MAC with comparable accuracy.

Fig. 8: Illustration of the rotational homography estimation problem for panoramic image stitching (c.f. Section VII ). Given a set of outlier -corrupted 2D-2D pixel matches between two images captured by a camera rotating around its optical centre, we aim to estimate the 3-DoF relative rotation and the 1-DoF focal length of the camera.

Fig. 9: Evaluation resluts of application 2 on simulated datasets (c.f. Section VII-C1 ). (a) Increasing outlier ratio with fixed

\xi=2.5

; (b) Increasing

\xi

with fixed

80\%

outlier ratio. Across various setups, only GTM and ACM maintains high robustness, meanwhile GTM usually runs 10-60 times faster than ACM . (

M=200

, 50 trials)

Method	$\xi=2$	$\xi=4$	$\xi=6$	Time (s) $\downarrow$
Method	SR ( $\%$ ) $\uparrow$	SR ( $\%$ ) $\uparrow$	SR ( $\%$ ) $\uparrow$	Time (s) $\downarrow$
RANSAC [ 9 ]	91	81	74	0.03
MLESAC [ 13 ]	90	83	71	0.05
ACM [ 21 ]	94	85	80	7.24
DIRECT-4D [ 26 ]	0	0	0	10.69
GTM	94	91	90	1.38

TABLE IV: Evaluation results of application 3 on the real-world PASSTA dataset (c.f. Section VII-C2 ). GTM is the most robust to thresholds

\xi

Panel of Figure 14 — Fig. 10: Two representative visual cases of rotational homography estimation on the real-world PASSTA dataset (c.f. Section VII-C2 ). The first column presents the input 2D-2D image matches and the other columns show the outputs of various methods. Only GTM succeeds in all presented cases.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Scene	Arch	Courtyard	Facade	Office	Trees
Overlap	30 $\%$ -40 $\%$	40 $\%$ -70 $\%$	60 $\%$ -70 $\%$	$>$ 80 $\%$	$\approx$ 50 $\%$
$\#$ of Pairs	10	28	21	10	15
Average of $M$	8892	9877	1356	912	14987
Average of Outlier Ratio	99.44 $\%$	96.23 $\%$	95.86 $\%$	97.92 $\%$	99.70 $\%$

Appendix

TABLE V: Detailed information of the processed ETH dataset [ 74 ] .

M

in the table represent the number of correspondences.

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