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Robust Single-Parameter Point Cloud Registration using Truncated Lp Functions

Seong Hun Lee, Patrick Vandewalle, Javier Civera

arXiv:2610.34271v1cs.CVcs.RO

TL;DR

Robust point-cloud registration must identify geometrically consistent correspondences despite outliers, while multi-parameter pipelines can require dataset-specific tuning. TLP uses truncated Lp functions across correspondence scoring, hypothesis selection, and refinement with one user-tuned distance threshold. It achieves the highest mAA in 10 of 12 evaluated settings and ranks among the three fastest methods in all four dataset–descriptor configurations.

  • Problem

    Point-cloud registration must identify geometrically consistent correspondences when outliers can greatly outnumber inliers.

    Source: A. Problem Definition · p. 2

  • Method

    TLP uses truncated Lp functions with different exponents for correspondence scoring, hypothesis selection, and staged refinement, with one user-tuned distance threshold.

    Source: §I · p. 1, A. Overview of TLP · p. 3

  • Results

    TLP achieves the highest mAA in 10 of the 12 dataset–descriptor–cutoff combinations and ranks among the three fastest methods in all four dataset–descriptor configurations.

    Source: C. Registration Accuracy · p. 7, §VI · p. 8

  • Takeaways & Limitations

    Across the evaluated 3DMatch and KITTI settings with FPFH and FCGF, TLP combines high rotation-based mAA with runtime rankings among the three fastest methods.

    Source: §VI · p. 8

Abstract

from arXiv · show

We propose a novel method for robust point cloud registration with only one tunable parameter. Our method, dubbed TLP, uses truncated $L_p$ functions with different values of $p$ to assign priority scores to correspondences, select the hypothesis, and refine the final model. By leveraging the robustness of truncated functions, TLP effectively handles the high outlier rates commonly encountered in real-world 3D feature matching. We also exploit GPU parallelization to evaluate hypotheses efficiently, enabling registration from thousands of correspondences in under a second on average. With only one tunable parameter, our method offers greater practicality than multi-parameter methods that require dataset-specific tuning of several parameters. Evaluations on public benchmark datasets demonstrate that TLP achieves higher accuracy than state-of-the-art single-parameter methods. Furthermore, it outperforms two widely used multi-parameter methods on three of the four evaluated datasets while maintaining competitive runtime.

I. INTRODUCTION

TLP addresses outlier-heavy point-cloud registration with a coordinated pipeline using one user-tuned parameter. Its scoring, hypothesis-generation, and refinement stages draw on truncated Lp functions while serving different objectives.

I. INTRODUCTION

  • TLP coordinates correspondence scoring, hypothesis selection, and refinement with truncated Lp functions and only one user-tuned parameter.The distance parameter is dataset-dependent; other settings remain fixed.Source: §I · p. 1

A. Compatibility-Based Correspondence Scoring

B. Hypothesis Generation and Consensus Selection

C. Robust Objectives and Refinement

III. PRELIMINARIES

The registration problem is to estimate a rigid transformation from putative correspondences when the inlier set is unknown. The paper defines normalized, truncated, smoothed Lp costs that saturate at a threshold and whose complementary scores support inlier-count selection.

A. Problem Definition

B. Normalized, Truncated, Smoothed Lp Cost

  • For residual r and threshold δ, the normalized, truncated, smoothed Lp cost is nondecreasing, equals zero at r = 0, and saturates at one for r ≥ δ.The smoothing parameter ϵ is fixed to 10^-12 in this work.Source: B. Normalized, Truncated, Smoothed Lp… · p. 2
  • Maximizing the total complementary score is equivalent to maximizing the number of inliers.Source: B. Normalized, Truncated, Smoothed Lp… · p. 2
  • In the illustrated example, support aggregation raises the mean inlier-to-outlier ratio from 1.9 to 2.28 while both scores rank correct matches above incorrect ones.Source: Figure 2 · p. 3

IV. METHOD

TLP uses different truncated Lp costs across correspondence ranking, hypothesis selection, and refinement, with the distance threshold τ as its only dataset-dependent parameter. Its correspondence scores combine continuous compatibility with support aggregation to produce rankings for hypothesis generation.

A. Overview of TLP

  • TLP ranks correspondences with truncated L3 compatibility and support aggregation, then selects hypotheses by inlier count and refines them with truncated L2, L0+, and L0.4 costs.The distance threshold τ is the only dataset-dependent parameter; other settings remain fixed.Source: A. Overview of TLP · p. 3

B. Correspondence Scoring

  • Correspondences with identical pairwise distances receive compatibility one; discrepancies at or above σq receive zero, while intermediate discrepancies get continuous rather than binary weights, for q ∈ {0.1, 0.2, 0.3, 0.4}.Source: B. Correspondence Scoring · p. 4
  • Support aggregation first sums each correspondence’s compatibility, then weights compatible neighbors by their own support to favor well-supported correspondences.The method replaces the binary compatibility matrix in [1] with a continuous one.Source: B. Correspondence Scoring · p. 4
  • Sorting the four fixed-q scores yields four correspondence orderings, allowing hypothesis generation to explore different compatibility neighborhoods without tuning four separate thresholds.The scoring scheme ranks correct correspondences above incorrect ones even when some incorrect correspondences have high pairwise compatibility.Source: B. Correspondence Scoring · p. 4

C. Batched Hypothesis Generation

TLP generates hypotheses deterministically from ranked correspondence triplets, prioritizing triplets with lower rank-index sums. It estimates and evaluates distinct samples in parallel batches, stopping when consensus-based conditions are met.

C. Batched Hypothesis Generation

  • Rank triplets are enumerated in increasing index-sum order, so combinations of highly ranked correspondences are visited first; the enumeration follows the method in.For each q ordering, the rank triplets map to correspondence samples; repeated samples with the same original indices are evaluated only once.Source: C. Batched Hypothesis Generation · p. 4
  • Each batch contains 256 distinct samples whose transformations are estimated and evaluated in parallel.For each sample, local frames built from ordered source and target triplets yield a rigid transformation for nondegenerate triplets.Source: C. Batched Hypothesis Generation · pp. 4–5
  • Generation stops when the largest consensus reaches n_suff, or after at least 1,500 visited rank triplets when it reaches n_acc; any partial batch is processed before final refinement.The largest consensus found across batches is retained.Source: C. Batched Hypothesis Generation · p. 5

D. Truncated Lp Cost Minimization

TLP minimizes truncated Lp costs by alternating model fitting on current inliers with residual-threshold updates. Its IRLS steps solve weighted rigid least-squares problems, with constant weights making p = 2 a single-fit case.

D. Truncated Lp Cost Minimization

  • Refinement alternates fitting the model on the current inlier set with thresholding all residuals to update that set.Residuals above the threshold incur a constant cost of one, so fitting uses only the untruncated cost of current inliers.Source: D. Truncated Lp Cost Minimization · p. 5
  • Each IRLS step solves a weighted rigid least-squares problem and recomputes weights from the resulting residuals; for p = 2, constant weights require only one fit.The weights follow by differentiating the untruncated cost with respect to squared residuals, with common factors dropped.Source: D. Truncated Lp Cost Minimization · p. 5
  • Batch evaluation uses p = 2 followed by p = 0+ with truncation threshold 1.2τ, while final refinement uses p = 0.4 with threshold τ.Algorithm 3 is the stated implementation of truncated Lp cost minimization.Source: D. Truncated Lp Cost Minimization · p. 5

A. Experimental Setup

The evaluation compares TLP with single- and multi-parameter registration methods on 3DMatch and KITTI using FPFH and FCGF descriptors. Single-parameter thresholds are tuned separately for each dataset–descriptor configuration, while multi-parameter baselines use their authors’ recommended settings.

A. Experimental Setup

  • TLP is compared with five single-parameter methods and two multi-parameter methods, including SC2-PCR and MAC.HERE is evaluated both with and without global point-cloud scaling.Source: A. Experimental Setup · p. 5
  • The evaluation uses 1,623 3DMatch pairs and 555 KITTI pairs per descriptor setting, with FPFH and FCGF correspondences prepared using the SC2-PCR pipeline.Registration runs on precomputed correspondences.Source: A. Experimental Setup · p. 5
  • Multi-parameter baselines use author-recommended settings, while single-parameter thresholds are tuned separately for each dataset–descriptor configuration by maximizing mAA at 15°.Table I lists the selected thresholds; N/A denotes an unavailable setting.Source: A. Experimental Setup · p. 5, Table I · p. 5

B. Evaluation Metric

The section evaluates registration accuracy with rotation-based mean average accuracy (mAA). The score averages accuracy over angular thresholds, providing a scale-independent proxy for registration quality while not measuring translation accuracy.

B. Evaluation Metric

  • Rotation-based mAA is the evaluation metric for registration accuracy.The metric uses estimated rotation errors relative to ground truth.Source: B. Evaluation Metric · p. 5
  • mAA averages accuracy over integer angular thresholds from 1° to K°, rewarding smaller errors within that range.Its indicator function counts errors below a threshold as accurate.Source: B. Evaluation Metric · p. 7
  • Rotation-based mAA is scale-independent but does not measure translation accuracy, so a high score alone does not guarantee correct alignment.The passage describes it as a useful proxy for overall registration quality because rotation and translation estimates tend to be strongly correlated.Source: B. Evaluation Metric · p. 7

C. Registration Accuracy

Among available results, TLP has the highest mAA in 10 of 12 dataset–descriptor–cutoff combinations and consistently beats the evaluated single-parameter baselines. It also beats the evaluated multi-parameter baselines except at two 3DMatch FPFH cutoffs, while some baseline results were unavailable.

C. Registration Accuracy

  • MAC results are unavailable for both FPFH configurations, and TEASER exceeded two hours on some 3DMatch FCGF pairs.TEASER also frequently exceeded half an hour on KITTI FCGF pairs.Source: C. Registration Accuracy · p. 7
  • 10 of 12 available dataset–descriptor–cutoff combinations have their highest mAA with TLP.TLP consistently outperforms evaluated single-parameter baselines and all evaluated multi-parameter baselines except SC2-PCR on 3DMatch with FPFH at the 10° and 15° cutoffs.Source: C. Registration Accuracy · p. 7
  • For five examples from 3DMatch using FPFH, Fig. 3 shows that TLP achieves more accurate alignment than the evaluated single-parameter baselines.Source: C. Registration Accuracy · p. 7

D. Ablation Study

TLP’s default configuration achieves the highest or joint-highest mAA in most ablation settings, while the full method ranks among the three fastest and leads in most evaluated settings overall.

D. Ablation Study

  • The default configuration has the highest or joint-highest mAA in 9 of 12 ablation settings; component removal generally reduces 3DMatch/FPFH accuracy but has small effects on KITTI.The dataset-specific pattern qualifies the ablation result.Source: D. Ablation Study · p. 8

E. Runtime

  • TLP ranks third, first, third, and third in runtime across the four dataset–descriptor combinations, placing it among the three fastest methods throughout.The runtime comparison covers nine methods.Source: E. Runtime · p. 8

VI. CONCLUSION

  • TLP achieves the highest rotation-based mAA in 10 of 12 settings among available baselines and ranks among the three fastest in all four configurations.The conclusion attributes these results to accurate and efficient registration with limited parameter tuning.Source: §VI · p. 8

APPENDIX: ENDPOINT LIMITS OF THE TRUNCATED Lp

The appendix outlines proof steps for endpoint limits of truncated Lp functions, including ratio behavior as p approaches infinity and boundary cases at r = δ.

APPENDIX: ENDPOINT LIMITS OF THE TRUNCATED Lp

  • For the p →0+ limit, the appendix invokes ex = 1 + x + O(x2), then substitutes into the bound using log B > 0 to prove equation (3).The proof first establishes A ≥ 1 and B > 1, then divides by ϵp/2 in equation (2).Source: APPENDIX: ENDPOINT LIMITS OF THE… · p. 8
  • For p →∞ with r < δ, the appendix states that A/B < 1.The passage introduces this case before giving the next proof step.Source: APPENDIX: ENDPOINT LIMITS OF THE… · p. 8
  • At r = δ, the ratio is exactly one for every p > 0; at r > δ, it exceeds one and truncation makes the cost exactly one.These cases are stated to prove equation (4).Source: APPENDIX: ENDPOINT LIMITS OF THE… · p. 8

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