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Scalable 3D Registration via Truncated Entry-wise Absolute Residuals

Tianyu Huang, Liangzu Peng, René Vidal, Yun-Hui Liu

TL;DR

This paper addresses robust 3D registration under extreme outlier rates and massive data scales by introducing TEAR, a loss based on truncated entry-wise $\ell_1$ residuals. TEAR decomposes the 6-DoF problem into TEAR-1 and TEAR-2, then solves each via a custom, highly efficient branch-and-bound scheme with tight $O(N\log N)$ bounds, enabling global optimization for over $10^7$ point pairs with up to $\sim 99.8\%$ outliers. Empirical results on synthetic and real data show TEAR matches or surpasses state-of-the-art accuracy while outperforming alternative scalable methods in memory and speed, including at huge scales where consistency-graph approaches fail. The approach demonstrates that branch-and-bound can be practical for scalable, outlier-robust 3D registration and points to extensions to related geometric estimation problems.

Abstract

Given an input set of $3$D point pairs, the goal of outlier-robust $3$D registration is to compute some rotation and translation that align as many point pairs as possible. This is an important problem in computer vision, for which many highly accurate approaches have been recently proposed. Despite their impressive performance, these approaches lack scalability, often overflowing the $16$GB of memory of a standard laptop to handle roughly $30,000$ point pairs. In this paper, we propose a $3$D registration approach that can process more than ten million ($10^7$) point pairs with over $99\%$ random outliers. Moreover, our method is efficient, entails low memory costs, and maintains high accuracy at the same time. We call our method TEAR, as it involves minimizing an outlier-robust loss that computes Truncated Entry-wise Absolute Residuals. To minimize this loss, we decompose the original $6$-dimensional problem into two subproblems of dimensions $3$ and $2$, respectively, solved in succession to global optimality via a customized branch-and-bound method. While branch-and-bound is often slow and unscalable, this does not apply to TEAR as we propose novel bounding functions that are tight and computationally efficient. Experiments on various datasets are conducted to validate the scalability and efficiency of our method.

Scalable 3D Registration via Truncated Entry-wise Absolute Residuals

TL;DR

This paper addresses robust 3D registration under extreme outlier rates and massive data scales by introducing TEAR, a loss based on truncated entry-wise residuals. TEAR decomposes the 6-DoF problem into TEAR-1 and TEAR-2, then solves each via a custom, highly efficient branch-and-bound scheme with tight bounds, enabling global optimization for over point pairs with up to outliers. Empirical results on synthetic and real data show TEAR matches or surpasses state-of-the-art accuracy while outperforming alternative scalable methods in memory and speed, including at huge scales where consistency-graph approaches fail. The approach demonstrates that branch-and-bound can be practical for scalable, outlier-robust 3D registration and points to extensions to related geometric estimation problems.

Abstract

Given an input set of D point pairs, the goal of outlier-robust D registration is to compute some rotation and translation that align as many point pairs as possible. This is an important problem in computer vision, for which many highly accurate approaches have been recently proposed. Despite their impressive performance, these approaches lack scalability, often overflowing the GB of memory of a standard laptop to handle roughly point pairs. In this paper, we propose a D registration approach that can process more than ten million () point pairs with over random outliers. Moreover, our method is efficient, entails low memory costs, and maintains high accuracy at the same time. We call our method TEAR, as it involves minimizing an outlier-robust loss that computes Truncated Entry-wise Absolute Residuals. To minimize this loss, we decompose the original -dimensional problem into two subproblems of dimensions and , respectively, solved in succession to global optimality via a customized branch-and-bound method. While branch-and-bound is often slow and unscalable, this does not apply to TEAR as we propose novel bounding functions that are tight and computationally efficient. Experiments on various datasets are conducted to validate the scalability and efficiency of our method.
Paper Structure (26 sections, 9 theorems, 42 equations, 12 figures, 4 tables, 6 algorithms)

This paper contains 26 sections, 9 theorems, 42 equations, 12 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Prior Art
  3. Our Contribution: TEAR
  4. The Design of TEAR
  5. Problem Formulation: TEAR
  6. Tear Off: Decomposition of TEAR
  7. Branch and Bound with Tears
  8. TEAR Versus CM and TLS
  9. Standard Experiments
  10. Experiments on Synthetic Data
  11. Experiments on Real Data
  12. Huge-Scale Experiments
  13. Conclusion and Future Work
  14. The General Branch-and-bound Algorithm
  15. Bounds for TEAR-1
  16. ...and 11 more sections

Key Result

Theorem 1

We can solve eq:tear-1-UB in $O(N\log N)$ time and compute a "tight" lower bound of eq:tear-1-LB also in $O(N\log N)$ time.

Figures (12)

  • Figure 1: Comparisons of $\mathop{\mathrm{\texttt{TEAR}}}\nolimits$ (ours) to prior methods on random, synthetic, noisy data (20 trials). The outlier ratio is set to $95\%$ and all presented methods find accurate solutions. \ref{['fig:storage_time']}a: $\mathop{\mathrm{\texttt{TEAR}}}\nolimits$ is $10^3$ times more scalable than methods based on consistency graphs including TEASER++ Yang-T-R2021, SC$^2$-PCR Chen-CVPR2022b, MAC Zhang-CVPR2023 and than deep learning methods including PointDSC Bai-CVPR2021 and VBReg Jiang-CVPR2023. \ref{['fig:storage_time']}b: $\mathop{\mathrm{\texttt{TEAR}}}\nolimits$ is $100$ times faster than TR-DE Chen-CVPR2022c, a recent branch-and-bound method.
  • Figure 2: The pipeline of $\mathop{\mathrm{\texttt{TEAR}}}\nolimits$ visualized (cf.\ref{['subsection:tear-off']}). Green (resp. red) values denote the numbers of inliers (resp. outliers), and green (resp. red) lines denote inlier (resp. outlier) point pairs. Top left: Input point pairs; top right: point pairs indexed by $\hat{\mathcal{I}}_1$\ref{['eq:I1-after-tear1']} after solving \ref{['eq:tear-1']}; bottom right: point pairs indexed by $\hat{\mathcal{I}}_2$\ref{['eq:I2-after-tear2']} after solving \ref{['eq:tear-2']}; bottom left: the final output.
  • Figure 3: Solve \ref{['eq:tear-1']} and \ref{['eq:CM-1']} via branch-and-bound on random, synthetic, noisy data (\ref{['subsection:TEAR-CM-TLS']}). TR-DE Chen-CVPR2022c, a recent branch-and-bound method, is also compared. Outlier ratio: 95$\%$ (\ref{['fig:TEAR-CM']}a, \ref{['fig:TEAR-CM']}b); $N=10000$ (\ref{['fig:TEAR-CM']}c, \ref{['fig:TEAR-CM']}d). 30 trials.
  • Figure 4: Solve \ref{['eq:tear-1']} and \ref{['eq:TLS-1']} via branch-and-bound on random, synthetic, noisy data (\ref{['subsection:TEAR-CM-TLS']}). Outlier ratio: 99$\%$. 30 trials.
  • Figure 5: Synthetic Experiments (\ref{['sec:syn_exper']}). $N=10000$, 30 trials.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Remark 1: Truncated Least Unsquared Deviations
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • ...and 5 more