Voronoi-based Second-order Descriptor with Whitened Metric in LiDAR Place Recognition

Jaein Kim; Hee Bin Yoo; Dong-Sig Han; Byoung-Tak Zhang

cs.CVMar 16, 2026

Voronoi-based Second-order Descriptor with Whitened Metric in LiDAR Place Recognition

Authors: Jaein Kim, Hee Bin Yoo, Dong-Sig Han, Byoung-Tak Zhang

Organizations: Interdisciplinary Program in Neuroscience, Seoul National University · Département d’Informatique, École Normale Supérieure (ENS) · Department of Computing, Imperial College London · Dept. of Computer Science and Engineering, Seoul National University

Abstract

The pooling layer plays a vital role in aggregating local descriptors into the metrizable global descriptor in the LiDAR Place Recognition (LPR). In particular, the second-order pooling is capable of capturing higher-order interactions among local descriptors. However, its existing methods in the LPR adhere to conventional implementations and post-normalization, and incur the descriptor unsuitable for Euclidean distancing. Based on the recent interpretation that associates NetVLAD with the second-order statistics, we propose to integrate second-order pooling with the inductive bias from Voronoi cells. Our novel pooling method aggregates local descriptors to form the second-order matrix and whitens the global descriptor to implicitly measure the Mahalanobis distance while conserving the cluster property from Voronoi cells, addressing its numerical instability during learning with diverse techniques. We demonstrate its performance gains through the experiments conducted on the Oxford Robotcar and Wild-Places benchmarks and analyze the numerical effect of the proposed whitening algorithm.

Figures & tables

Fig. 1: The illustration of feature space of each Voronoi cell learned by our method. We select the query place and its top-3 closest neighbors metrized by our method from the Oxford [ 32 ] and visualize each cell of their descriptors reduced by ICA. White dots and colored cross marks denote query and neighbors. Our whitening transforms each Voronoi cell more homogeneous and suitable for Euclidean distancing.

Fig. 2: The overall architecture. First, the input local descriptors $\mathbf{X}$ are mapped by networks $\mathcal{F}_{\text{proj}}$ and $\mathcal{F}_{\text{score}}$ , multiplied into the global descriptor $\widetilde{\mathbf{X}}$ . Our whitening module implemented with ZCA whitening [ 21 ] computes the standard deviation matrix $\mathbf{S}$ from $\widetilde{\mathbf{X}}$ and normalizes each cluster’s feature by multiplying $\mathbf{S}^{-1}$ over $\widetilde{\mathbf{X}}$ . Finally, the global descriptor $\mathbf{Z}$ is vectorized and scaled down to $\frac{1}{\sigma}\mathbf{Z}$ .

Methods	Oxford		U.S.		R.A.		B.D		Average
	R@1	R@1%	R@1	R@1%	R@1	R@1%	R@1	R@1%	R@1	R@1%
PointNetVLAD [ 42 ]	74.88	87.89	68.09	81.37	64.05	74.66	63.42	70.63	67.61	78.64
TransLoc3D [ 47 ]	93.99	98.13	83.50	93.23	80.11	90.39	80.01	87.05	84.40	92.20
MinkLoc3D [ 24 ]	93.76	97.91	86.01	95.04	81.11	91.19	82.66	88.48	85.89	93.16
MinkLoc3Dv2 [ 25 ]	96.26	98.87	90.85	96.65	86.49	93.75	86.26	91.15	89.97	95.11
CASSPR [ 46 ]	95.84	98.75	92.91	98.00	89.44	94.69	87.25	92.36	91.36	95.95

TABLE I: The average recall at top-1 and top-1% retrievals in the Oxford [ 32 ] . †: the results quoted from the original paper.

Fig. 3: Mean Average Precision at K (MAP@K) plotted up to 10 predictions in the Oxford and In-house datasets.

Methods	Descriptor Dimensions	Intra-sequence								Inter-sequence
Methods	Descriptor Dimensions	V-03		V-04		K-03		K-04		Venman		Karawatha
		F1	R@1	F1	R@1	F1	R@1	F1	R@1	R@1	MRR	R@1	MRR
ScanContext [ 22 ]	1200	05.54	12.79	39.28	43.33	31.33	43.55	58.09	61.72	43.15	45.23	52.79	56.40
TransLoc3D [ 47 ]	256	22.92	35.50	75.00	64.03	43.50	38.95	75.80	75.53	62.85	74.92	54.32	69.08
MinkLoc3Dv2 [ 25 ]	256	49.85	49.94	82.19	71.61	51.39	50.97	80.00	71.18	75.77	84.86	67.81	79.20
LoGG3D-Net [ 44 ]	1024	54.03	62.40	80.37	72.47	64.26	64.05	84.54	80.26	79.84	86.55	74.67	82.66

TABLE II: The intra-sequence and inter-sequence results in the Wild-Places [ 23 ] . †: the results quoted from the original paper.

Fig. 4: The ROC curves by place recognition models in the Wild-Places intra-sequence evaluations. False positive rate (FPR) and true positive rate (TPR) were measured with the decision threshold in range $[0,1]$ metrized in the descriptor space.

Methods	Oxford		U.S.		R.A.		B.D
Methods	R@1	R@1%	R@1	R@1%	R@1	R@1%	R@1	R@1%
RBLW ✗, SVDPI ✗	93.25	97.60	82.86	93.23	70.83	83.33	74.72	82.50
RBLW ✗, SVDPI ✗	(-3.38)	(-1.27)	(-8.18)	(-4.13)	(-16.13)	(-10.56)	(-12.83)	(-9.26)
RBLW ✓, SVDPI ✗	————————————– Diverged ————————————–
RBLW ✗, SVDPI ✓	95.82	98.57	89.18	95.75	84.95	91.47	85.12	89.94
RBLW ✗, SVDPI ✓	(-0.81)	(-0.30)	(-1.86)	(-1.61)	(-2.01)	(-2.42)	(-2.42)	(-1.82)

TABLE III: Ablation studies in the Oxford and In-house datasets. (Top) Ablations on the ZCA whitening. (Bottom) Ablations on the backbone architectures.

Fig. 5: We measured the matrix rank and effective rank [ 39 ] of covariances before the whitening ( $\hat{\mathbf{\Sigma}}$ ), after RBLW shrinkage ( $\hat{\mathbf{\Sigma}}_{\text{RBLW}}$ ), and after the whitening ( $\hat{\mathbf{Q}}\hat{\mathbf{\Lambda}}\hat{\mathbf{Q}^{\top}}$ ) using our default $(C\!=\!16,\;M\!=\!16)$ method. Red and blue lines denote the medians of matrix ranks and effective ranks each.

Fig. 6: We measured the empirical Wasserstein-2 distances between the Voronoi cells of descriptors acquired by ablation models in Table III . The brighter color denotes the farther distributional distance between cells.

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Mar 6, 2026J/K move · Enter open · S save

PROBE: Probabilistic Occupancy BEV Encoding with Analytical Translation Robustness for 3D Place Recognition

Jinseop Lee, Byoungho Lee, Gichul Yoo

SK Intellix, Seoul, Republic of Korea

LiDAR Place Recognition Bird's-Eye-View Representation LiDAR Localization Robot Localization

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We present PROBE (PRobabilistic Occupancy BEV Encoding), a learning-free LiDAR place recognition descriptor that models each BEV cell's occupancy as a Bernoulli random variable. Rather than relying on discrete point-cloud perturbations, PROBE analytically marginalizes over continuous Cartesian translations via the polar Jacobian, yielding a distance-adaptive angular uncertainty

σ_θ= σ_t / r

\mathcal{O}(R{\cdot}S)

time. The primary parameter

σ_t

represents the expected translational uncertainty in meters, a sensor-independent physical quantity that enhances cross-sensor generalization while reducing the need for extensive per-dataset tuning. Pairwise similarity combines a Bernoulli-KL Jaccard with exponential uncertainty gating and FFT-based height cosine similarity for rotation alignment. Evaluated on four datasets spanning four diverse LiDAR types, PROBE achieves the highest accuracy among handcrafted descriptors in multi-session evaluation and competitive single-session performance relative to both handcrafted and supervised baselines. The source code and supplementary materials are available at https://sites.google.com/view/probe-pr.

Abstract

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