PaperlayerView exact revision on arXiv

Source-linked AI summary

RING++: Roto-translation Invariant Gram for Global Localization on a Sparse Scan Map

Xuecheng Xu, Sha Lu, Jun Wu, Haojian Lu, Qiuguo Zhu, Yiyi Liao, Rong Xiong, Yue Wang

arXiv:2210.05984v1cs.ROcs.CV

TL;DR

Large viewpoint differences make LiDAR localization challenging on sparse maps because place recognition needs viewpoint-robust representations and pose estimation lacks reliable initialization. RING++ provides roto-translation-invariant recognition and globally convergent pose estimation; it outperforms state-of-the-art methods on benchmark datasets and is effective as a loop-closure module in real scenarios.

  • Problem

    Large differences between current and mapping-session viewpoints challenge LiDAR localization on sparse maps, where recognition needs similar representations and pose estimation lacks reliable initialization.

    Source: §I · p. 1, §III · p. 3

  • Method

    RING++ is a learning-free framework that uses roto-translation-invariant representations and globally convergent solvers for place recognition and pose estimation on sparse scan maps.

    Source: §I · p. 2

  • Results

    RING++ achieves superior performance on benchmark datasets, outperforming state-of-the-art methods.

    Source: §IX · p. 18

  • Takeaways & Limitations

    As a stand-alone loop-closure detection module, RING++ is effective in real scenarios without prior knowledge.

    Source: §IX · p. 18

Abstract

from arXiv · show

Global localization plays a critical role in many robot applications. LiDAR-based global localization draws the community's focus with its robustness against illumination and seasonal changes. To further improve the localization under large viewpoint differences, we propose RING++ which has roto-translation invariant representation for place recognition, and global convergence for both rotation and translation estimation. With the theoretical guarantee, RING++ is able to address the large viewpoint difference using a lightweight map with sparse scans. In addition, we derive sufficient conditions of feature extractors for the representation preserving the roto-translation invariance, making RING++ a framework applicable to generic multi-channel features. To the best of our knowledge, this is the first learning-free framework to address all subtasks of global localization in the sparse scan map. Validations on real-world datasets show that our approach demonstrates better performance than state-of-the-art learning-free methods, and competitive performance with learning-based methods. Finally, we integrate RING++ into a multi-robot/session SLAM system, performing its effectiveness in collaborative applications.

I. INTRODUCTION

Large viewpoint differences make LiDAR global localization challenging: they can confuse place recognition and leave pose estimation without a reliable initialization. RING++ addresses this on sparse scan maps with roto-translation-invariant place representations and globally convergent pose estimation.

I. INTRODUCTION

  • Large viewpoint differences can make scans from the same place appear different, complicating place recognition and requiring pose estimation with a large convergence basin.Dense pose discretization can reduce the viewpoint difference but requires large map memory; rotation invariance alone remains sensitive to translation differences.Source: §I · pp. 1–2
  • RING++ combines roto-translation-invariant place recognition with globally convergent rotation and translation estimation for localization on sparse scan maps.The framework also derives sufficient feature-extractor conditions for preserving invariance and aggregating multi-channel features.Source: §I · p. 2

II. RELATED WORKS

  • The related-work review organizes prior approaches around local point features and strategies for aggregating them into robust global representations.It notes that aggregation strategies seek compact representations with invariance properties.Source: §II · p. 2

A. Feature Extraction

  • Handcrafted geometric and shape features are widely used, but their instability in LiDAR place recognition is linked to unstructured, range-dependent point sparsity; learned methods and multi-channel features address this setting.The review describes CNN, PointNet-style, graph-based, kernel-based, semantic, and image-fusion approaches, motivating feature aggregation that preserves roto-translation invariance.Source: A. Feature Extraction · pp. 2–3

B. Aggregation and Invariance

  • Prior descriptors offer rotation invariance, but translation sensitivity can require dense discretization; RING++ extends the Radon-transform approach to multi-channel features and 3-DoF pose estimation with roto-translation invariance.The review contrasts this with Scan Context's translation augmentation under urban-road assumptions and DiSCO's sensitivity to large translation distortion.Source: B. Aggregation and Invariance · p. 3

III. OVERVIEW

  • Sparse-map localization requires representations similar across large relative-pose differences and pose estimators that converge globally without a reliable initial value.The pipeline retrieves an overlapping map scan before estimating the query's relative pose and applying it to the map pose.Source: §III · p. 3

A. Equivariance and Invariance

RING++ organizes global localization around representations with distinct invariance and equivariance properties. Its framework uses these representations to separate place recognition from rotation and translation estimation.

A. Equivariance and Invariance

B. Framework

1) BEV:

  • The scan is voxelized into occupancy features, then accumulated across height to produce a 2D BEV representation.Source: 1) BEV · p. 4

2) Sinogram:

  • Radon transform maps BEV features into a sinogram; rotation shifts it uniformly along θ, while translation shifts τ non-uniformly.Source: 2) Sinogram · pp. 4–5

3) TING:

TING removes translation effects while retaining rotation equivariance; correlation and pooling then produce RING, which is invariant to both rotation and translation.

3) TING:

  • Applying a row-wise 1D DFT along the sinogram’s τ axis and taking its magnitude yields TING, which is translation invariant and rotation equivariant.Source: 3) TING · p. 5

4) RING:

  • Circular cross-correlation of query and map TINGs followed by maximum pooling over rotation shifts forms RING, a roto-translation invariant representation.Source: 4) RING · pp. 5–6

B. Solving Pass

RING-based place recognition supports sparse-map localization, but finite scan-range occlusion limits its translation invariance and the full descriptor scheme can be costly to store and search.

1) Place Recognition:

  • RING remains invariant to arbitrary rotation changes but gradually degrades as translation grows relative to finite scan-range occlusion.Source: 1) Place Recognition · p. 6
  • Computing and storing RING descriptors for all map scans gives total storage and nearest-neighbor search complexity of O(|M|^2).Source: 1) Place Recognition · p. 6
  • The approximate retrieval method uses the maximum element of normalized RING to avoid computing and storing RINGs for every map scan.Source: 1) Place Recognition · p. 6

2) Pose Estimation:

After place recognition, RING++ estimates a reduced planar pose globally, then refines the full pose. Its aggregation framework can also use feature extractors beyond occupancy BEV.

2) Pose Estimation:

  • The pose pipeline first estimates rotation and planar translation globally, then uses that 3-DoF estimate to initialize 6-DoF refinement with ICP.Source: 2) Pose Estimation · p. 6
  • Rotation is estimated by the TING shift with maximum correlation; translation is estimated by BEV cross-correlation after compensating for rotation.Source: 2) Pose Estimation · p. 6

C. Perspective of Feature Aggregation

A. Feature Extraction

RING++ preserves roto-translation invariance when feature extraction is equivariant to both transformations or invariant to both. Its multi-channel representation carries six local features through BEV, SG, and TING, then combines channel-wise correlations for retrieval.

A. Feature Extraction

  • Theorem 2 guarantees that invariant aggregation preserves roto-translation invariance when the extractor is equivariant to both transformations or invariant to both.The proof treats the equivariant and invariant cases separately.Source: A. Feature Extraction · p. 7
  • RING++ uses six rotation-invariant, translation-equivariant local features, accumulated into a six-channel BEV representation by channel-wise max pooling.The features are extracted from each point’s 30-neighborhood; the BEV features encode height information such as building facades.Source: A. Feature Extraction · pp. 7–8

B. Multi-channel Representation Pass

  • The representation pass applies a Radon transform to each BEV channel to form six-channel SG, then row-wise DFT magnitudes to form six-channel TING.The DFT magnitude is used to achieve translation invariance.Source: B. Multi-channel Representation Pass · p. 8, Figure 5 · p. 8
  • For each query-map TING pair, RING++ computes circular cross-correlations per channel, sums them across channels, and takes the maximum as the final representation element.The resulting single-channel correlation map supplies the maximum used for the RING representation.Source: B. Multi-channel Representation Pass · p. 8

C. Multi-channel Solving Pass

The solving pass retrieves a map scan with the roto-translation-invariant RING descriptor, estimates relative rotation and translation from TING and BEV correlations, and refines the resulting pose with ICP. The evaluation uses multiple datasets and reports place-recognition performance with Recall@1.

C. Multi-channel Solving Pass

  • RING retrieves the closest map scan; TING correlation estimates relative rotation, and channel-wise BEV correlations yield the globally optimal translation after rotation compensation.The estimated 3-DoF transformation initializes ICP refinement to obtain a refined 6-DoF pose.Source: C. Multi-channel Solving Pass · p. 8

A. Dataset

2) MulRan Dataset:

  • MulRan combines Radar and LiDAR data across DCC, KAIST, Riverside, and Sejong City, providing temporal and structural diversity for place-recognition research.Source: 2) MulRan Dataset · p. 8

3) Oxford Radar RobotCar Dataset:

  • Oxford’s 32 traversals cover a central route in January 2019 under varied weather and lighting, with two vehicle-mounted LiDAR point clouds concatenated into one scan.Source: 3) Oxford Radar RobotCar Dataset · p. 9

1) Recall@1:

  • Recall@1 measures the ratio of top-1 true positives to total positives for evaluating place-recognition retrieval.Source: 1) Recall@1 · p. 9

2) Precision-Recall Curve:

The section defines retrieval and pose-estimation metrics, including curves that relate precision and recall and a success rate that combines retrieval with pose accuracy. It also specifies ATE for evaluating SLAM trajectories and identifies the methods used for comparison.

2) Precision-Recall Curve:

  • Precision and recall measure true positives relative to total matches and total positives, respectively; their curve plots values across thresholds.Each point corresponds to precision and recall at a specific threshold.Source: 2) Precision-Recall Curve · p. 9
  • F1 score is the harmonic mean of precision and recall, and its curve tracks that balance against recall.Source: 2) Precision-Recall Curve · p. 9

4) TE and RE:

  • Translation and rotation errors evaluate 2-DoF translation and 1-DoF rotation; pose errors are computed only for successfully matched scan pairs.Source: 4) TE and RE · p. 9

5) Success Rate:

  • Success Rate counts successfully aligned matches with translation error below 2m and rotation error below 5° as a percentage.Source: 5) Success Rate · p. 9

6) ATE:

  • Average ATE evaluates SLAM by measuring trajectory-point differences after aligning estimated and ground-truth poses; the study compares against learning-free and learning-based methods.Source: 6) ATE · p. 9, C. Comparative Methods · p. 9

D. Our Methods

The paper compares four RING variants that differ in occupancy versus multi-feature inputs and in their use of SG versus TING for rotation estimation. Experiments include invariance and localization checks, with a toycase illustrating successful place retrieval.

D. Our Methods

  • The four variants isolate the effects of translation-invariant TING and multi-feature extraction, varying occupancy or local features and SG or TING for rotation estimation.All use RING for place recognition; translation estimation uses single-channel BEV in RING variants and multi-channel BEV in RING++ variants.Source: D. Our Methods · p. 10

E. Implementation Details

  • For fair comparison, all methods use ground-removed scans filtered to the same range; Scan Context and RING use 120 × 120 BEV grids.This corresponds to 1.17m/pixel translation resolution and 3°/pixel rotation resolution.Source: E. Implementation Details · p. 10

VII. EXPERIMENTAL EVALUATION

  • The experiments assess invariance, large-pose-difference loop detection, 3-DoF pose initialization, computational efficiency, and integration into SLAM systems.Source: §VII · p. 10

1) Place Recognition Illustration:

2) Pose Estimation Illustration:

RING’s translation-invariant TING representation makes rotation estimation robust to translation perturbations. Across place-recognition and pose-estimation evaluations, RING++ performs strongly in sparse maps and across datasets.

2) Pose Estimation Illustration:

  • With translations up to 5m, TING’s rotation-estimation error remains nearly 0°, whereas Scan Context’s error rises to about 30° beyond 1m.SG and TING are more robust to translation perturbation than Scan Context in the illustrated comparison.Source: 2) Pose Estimation Illustration · pp. 10–11, Figure 7 · p. 11

B. Evaluation of Place Recognition

1) Single-session Scenarios:

  • RING-based methods are less affected by place density than other methods, and their advantage increases as density falls; RING and RING++ remain high-performing at 50m.RING++ variants outperform their RING counterparts; the authors attribute the improvement to local features benefiting representation discrimination.Source: 1) Single-session Scenarios · p. 11

2) Multi-session Scenarios:

  • At lower place densities, RING and RING++ outperform competing methods; across NCLT, MulRan, and Oxford at 20m, all four RING variants outperform the other approaches.RING++ variants outperform their RING counterparts, supporting the benefit of extracted local features for place recognition.Source: 2) Multi-session Scenarios · p. 12

3) Generalization:

  • Across generalized datasets, RING++ performs well; DiSCO and EgoNN retain much of their place-recognition performance, while polar transforms remain translation-variant.The learning-based comparison trains on MulRan and tests on NCLT and Oxford.Source: 3) Generalization · p. 13

C. Evaluation of Pose Estimation

  • RING++ performs best among the tested variants overall, while translation invariance benefits pose estimation and extracted local features do not obviously improve it.ICP with RING++ initial poses converges to global minima rather than local optima, and takes less time than without an initial pose.Source: C. Evaluation of Pose Estimation · p. 14

1) Comparison with Learning-free Methods:

RING++ estimates relative rotation during place recognition and 2-DoF translation during pose estimation, providing a 3-DoF initial pose. Its evaluations show strong results against learning-free and learning-based methods, while ablations examine resolution and feature choices.

1) Comparison with Learning-free Methods:

  • Compared with Scan Context-based methods, RING++ performs better on NCLT and MulRan, with higher global-localization success and lower translation and rotation errors at most quantiles.The authors report that its qualified 3-DoF initial guesses give ICP a better chance of reaching satisfactory alignment.Source: 1) Comparison with Learning-free Methods · p. 15, Table V · p. 15

2) Comparison with Learning-based Methods:

  • On MulRan, RING++ is competitive with EgoNN at 66.09% Success Rate versus 62.00%; on NCLT, it achieves top performance among the compared methods.For MulRan, RING++ has lower TE and RE than DiSCO and EgoNN; the authors note that differing LiDAR point distributions make NCLT generalization difficult.Source: 2) Comparison with Learning-based… · p. 15

D. Ablation Study

  • As resolution increases, Recall@1 rises quickly at first and then approaches a constant; translation error declines slightly, while rotation estimation benefits greatly.The ablation uses the NCLT multi-session pair and reports results in Tab. VI.Source: Table VI · p. 15, D. Ablation Study · p. 16
  • RING++ performs competitively with FPFH and SHOT features, supporting its use as a generic feature-aggregation method; the other experiments use six-channel features.SHOT uses two signature bins in this evaluation because memory consumption grows linearly with bin value.Source: D. Ablation Study · p. 16, Table VII · p. 16

E. Computational Cost

RING++ averages under 50 ms in the reported setup, while its SLAM evaluations show more loop closures and lower drift than Scan Context++ (PC). The authors also identify limits in evaluation performance and handcrafted feature representation.

E. Computational Cost

  • RING++ averaged less than 50 ms per computation in the reported hardware setup.Local-feature computation was the most time-consuming part, and time grew linearly with feature-channel count.Source: E. Computational Cost · p. 16

1) Limited Evaluation Performance:

  • RING++ retrieves places across large viewpoint differences, but some retrieved places may fall outside the evaluation’s revisited-place criteria.The authors report that its estimators can still provide satisfactory initial poses and improve overall global localization performance.Source: 1) Limited Evaluation Performance · p. 16

2) Leveraging Deep learning for Feature Extraction:

  • The multi-channel framework was validated with handcrafted local features, whose representation ability the authors describe as limited.They suggest learning-based local features with end-to-end training as a possible route to better performance.Source: 2) Leveraging Deep learning for… · p. 16

VIII. SYSTEM APPLICATION

  • The SLAM system combines stand-alone RING++ loop closure with a real-time back-end manager and a GTSAM pose-graph optimizer.FAST-LIO2 supplies front-end odometry in the reported system.Source: §VIII · p. 16

B. Multi-session SLAM System

A. Multi-robot SLAM System

IX. CONCLUSION

  • The conclusion reports superior benchmark performance and validates RING++ as a stand-alone loop-closure module in real scenarios without prior knowledge.The authors describe the system as effective in those scenarios.Source: §IX · pp. 17–18

APPENDIX A PROOF OF EQ. 19

The appendix proves that two formulations give the same ranking for a simple two-descriptor case, then describes an FFT procedure for computing circular cross-correlation efficiently.

APPENDIX A PROOF OF EQ. 19

  • The proof normalizes TING descriptors to zero mean and unit standard deviation before deriving the RING-element comparison.The prose defines μ(M) and σ(M) as the mean and standard deviation of M.Source: Appendix A · p. 18
  • For two map scans, the appendix reduces the ranking proof to showing that descriptor-distance ordering is equivalent to ordering the query descriptor’s two components.It concludes that Eq. 18 holds if and only if Eq. 19 holds.Source: Appendix A · p. 18

APPENDIX B PROOF OF FAST IMPLEMENTATION BY FFT

  • The FFT procedure transforms TING inputs along θ, multiplies one spectrum by the complex conjugate of the other, applies an inverse transform, then sums over kω to obtain a 1D circular correlation map.The intermediate inverse-transform output is a 2D correlation map.Source: Appendix B · p. 18
Loading 2210.05984v1…