Levy-Driven Correspondence Estimation for Registration

Qianliang Wu; Jiaqi Yang; Wankou Yang; Le Hui; Jin Xie; Jian Yang; Yaqing Ding

cs.CVSep 26, 2026

Levy-Driven Correspondence Estimation for Registration

Authors: Qianliang Wu, Jiaqi Yang, Wankou Yang, Le Hui, Jin Xie, Jian Yang, Yaqing Ding

Organizations: Nantong University · Northwestern Polytechnical University · Southeast University · Nanjing University · Nanjing University of Science and Technology

Abstract

Finding reliable point correspondences is difficult when point clouds have low overlap or undergo non-rigid deformation. Iterative refinement can correct uncertain matches, but costly network evaluations limit the number of updates. We present LevyMatch, a Lévy-driven method that uses random jumps to refine a soft matching matrix. At each step, a network uses the current matching state and geometric information to predict a target matching matrix. A Brownian reference bridge gives an explicit formula for the update toward this target. A Gamma random clock sets the time step for each update. The updated matches provide new geometric feedback for the next target prediction. We further propose a fixed front-loaded Gamma policy that assigns more expected clock time to early updates and less to later ones, without retraining or extra network evaluations. Reordering the same sampled Gamma increments shows that placing larger increments early gives higher accuracy than placing them late. On 4DMatch and 4DLoMatch, our method improves both non-rigid feature matching recall (NFMR) and inlier ratio (IR) over the compared methods. The front-loaded policy achieves 93.09% NFMR and 92.11% IR on 4DMatch, and 82.79% NFMR and 79.07% IR on 4DLoMatch.

Figures & tables

Figure 1 : Matching-matrix bridge and inference procedure. (a) Example paths for Random Gamma (blue) and Fixed front-loaded Gamma (orange), both starting from the lifted matrix $\mathbf{X}_{0}$ . Superscripts $r$ and $f$ identify the two clock policies, respectively, and $h$ denotes the current resolution. Subscript $k$ is the current state index. Numbers mark the first three updated states. Diamonds show endpoint predictions $\widehat{\mathbf{Y}}_{k}$ , and squares show final outputs $\widehat{\mathbf{Y}}^{h}$ . Bars show expected clock increments, not individual samples: $\mathbb{E}[\Delta G_{k}]=1/K$ for Random Gamma and $\mathbb{E}[\Delta G_{k}]=w_{k}$ for Fixed front-loaded Gamma. The normalized weights are defined in Eq. ( 9 ). The paths explain the intended effect of early time allocation; they are not measured trajectories. (b) The shared endpoint predictor $H_{\phi}$ estimates $\widehat{\mathbf{Y}}_{k}$ from the current state and geometric input. A clock-controlled update then gives $\mathbf{X}_{k+1}$ for the next prediction. Here $\alpha_{k}=\Delta G_{k+1}/R_{k}$ , and $R_{k}=T-U_{k}$ is the remaining operational time. The variance $V_{k+1}$ is given in Eq. ( 12 ), and $\mathbf{z}_{k}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ is newly sampled Gaussian noise. Stochastic updates use $k=0,\ldots,K-2$ . The last of the $K$ endpoint evaluations returns $\widehat{\mathbf{Y}}^{h}=\widehat{\mathbf{Y}}_{K-1}$ without another stochastic update.

	4DMatch		4DLoMatch
Method	NFMR $\uparrow$	IR $\uparrow$	NFMR $\uparrow$	IR $\uparrow$
PointPWC [ wu2019pointpwc ]	21.60	20.00	10.00	7.20
FLOT [ puy2020flot ]	27.10	24.90	15.20	10.70
D3Feat [ 3 ]	55.50	54.70	27.40	21.50
Predator [ huang2021predator ]	56.40	60.40	32.10	27.50
Lepard [ li2022lepard ]	83.60	82.64	66.63	55.55

Table 1 : Correspondence results on 4DMatch and 4DLoMatch (%). Baseline values are from the cited papers. Both LévyMatch policies use the same checkpoint. LévyMatch rows report mean

\pm

population standard deviation over three inference seeds. Fixed front-loaded Gamma uses

\beta=2

		4DMatch		4DLoMatch
Variant	Clock	NFMR $\uparrow$	IR $\uparrow$	NFMR $\uparrow$	IR $\uparrow$
Full checkpoint, Random/SDE	Gamma	$92.88{\pm}0.08$	$91.95{\pm}0.06$	$82.32{\pm}0.18$	$78.61{\pm}0.17$
Full checkpoint, uniform/SDE	uniform	$92.82{\pm}0.09$	$91.89{\pm}0.07$	$82.32{\pm}0.34$	$78.61{\pm}0.32$
Full checkpoint, uniform/ODE	uniform	$92.85{\pm}0.05$	$91.91{\pm}0.06$	$82.21{\pm}0.23$	$78.47{\pm}0.18$

Table 2 : Inference-clock and noise controls. All rows use the same checkpoint and endpoint-network evaluation budget. Only the inference clock or transition noise changes.

		4DMatch		4DLoMatch
Initialization	Lévy matrix bridge	NFMR $\uparrow$	IR $\uparrow$	NFMR $\uparrow$	IR $\uparrow$
Random valid matrix	✓	$92.30{\pm}0.12$	$91.46{\pm}0.09$	$\mathbf{82.94{\pm}0.09}$	$\mathbf{79.39{\pm}0.10}$
Lifted DDIM-5	–	48.54	36.06	34.79	18.13
Lifted DDIM-5	✓	$\mathbf{92.88{\pm}0.08}$	$\mathbf{91.95{\pm}0.06}$	$82.32{\pm}0.18$	$78.61{\pm}0.17$

Table 3 : Initial-matrix and bridge-bypass controls using the same checkpoint. Rows with the bridge report results over three inference seeds. The bypass row uses seed 0.

4DMatch
Clock order	NFMR $\uparrow$	IR $\uparrow$	$\Delta$ Entropy	$\Delta$ Geo. $\downarrow$
Uniform/ODE (seed 0)	92.79	91.85	-0.5294	-0.0318
Gamma, random	$92.88{\pm}0.08$	$91.95{\pm}0.06$	$-0.5291{\pm}0.0006$	$-0.0313{\pm}0.0002$
Gamma, large early	$\mathbf{93.23{\pm}0.09}$	$\mathbf{92.23{\pm}0.06}$	$-0.5402{\pm}0.0005$	$\mathbf{-0.0322{\pm}0.0001}$
Gamma, large late	$92.23{\pm}0.14$	$91.41{\pm}0.13$	$-0.5040{\pm}0.0007$	$-0.0296{\pm}0.0002$
4DLoMatch

Table 4 : Increment-order diagnostic. Gamma rows use the same sampled increment values for each seed and report three-seed results. The uniform row uses deterministic mean updates and seed 0.

\Delta

Entropy and

\Delta

Geo. are final-minus-initial changes in mean row entropy (nats) and weighted mean Euclidean alignment residual (point-coordinate units), respectively. Their definitions are given in Eqs. ( 13 )–( 15 ). Matching entropy does not measure accuracy.

		4DMatch		4DLoMatch
Clock	$\beta$	NFMR $\uparrow$	IR $\uparrow$	NFMR $\uparrow$	IR $\uparrow$
Random Gamma	0	$92.88{\pm}0.08$	$91.95{\pm}0.06$	$82.32{\pm}0.18$	$78.61{\pm}0.17$
Fixed front-loaded	0.5	$92.87{\pm}0.10$	$91.93{\pm}0.09$	$82.58{\pm}0.17$	$78.86{\pm}0.16$
Fixed front-loaded	1.0	$92.96{\pm}0.09$	$92.00{\pm}0.07$	$82.63{\pm}0.08$	$78.92{\pm}0.05$
Fixed front-loaded	2.0	$93.09{\pm}0.07$	$92.11{\pm}0.05$	$82.79{\pm}0.19$	$79.07{\pm}0.18$
Fixed front-loaded	4.0	$\mathbf{93.17{\pm}0.07}$	$\mathbf{92.19{\pm}0.05}$	$\mathbf{82.98{\pm}0.26}$	$\mathbf{79.21{\pm}0.23}$

Table 5 : Effect of front-loading strength over three inference seeds. All settings use the same checkpoint. Increments are sampled from the assigned Gamma distributions without sorting.

Dataset	Clock	Time (ms) $\downarrow$	Matches
4DMatch	Random Gamma	479.8	648.82
4DMatch	Fixed front-loaded ( $\beta=2$ )	500.9	650.33
4DLoMatch	Random Gamma	493.2	379.99
4DLoMatch	Fixed front-loaded ( $\beta=2$ )	481.8	380.18

Table 6 : Recursive inference cost and output coverage for the main comparison. Fixed front-loaded Gamma uses

\beta=2

. Both policies use 20 endpoint-network evaluations. Values are means over inference seeds 0, 1, and 2. Time includes the backbone, coarse DDIM source, and matrix-bridge solver, but excludes the downstream deformation solver. Matches is the number of retained correspondences per pair. Empty is the percentage of pairs with no retained matches.

Figure 2 : Selected qualitative comparisons on 4DMatch. LévyMatch uses Fixed front-loaded Gamma with $\beta=2$ and inference seed 0. For each pair, the left column shows the inputs before registration. The upper row shows interpolated test-point correspondences. The lower row shows the warped source (yellow) and target (blue). Green lines mark endpoint errors below 0.04 in dataset coordinate units; red lines mark errors at or above this threshold. The clouds are shown apart in the input and correspondence views. Camera views differ across panels.

Figure 3 : Selected qualitative comparisons on 4DLoMatch, using the layout and correctness threshold of Fig. 2 . LévyMatch uses Fixed front-loaded Gamma with $\beta=2$ and inference seed 0. The Pumpkinhulk example still has incorrect correspondences and visible warp errors around the extended limbs.

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Conditional Flow Matching Non-Rigid Shape Correspondence Point Cloud Registration

Open paper AskPDF arXiv

Reliable matching between partially observed, deforming point clouds requires global context and fine geometric detail. Coarse candidate selection can exclude correct fine-level correspondences. We present \paper, a unified conditional transport framework with the matching matrix itself as the evolving state. Coarse diffusion establishes global matching hypotheses; hierarchy-preserving lifting expands them into a structured high-resolution source. Geometry-conditioned ODE and SDE bridges continue refinement in the complete fine-level candidate space, allowing coarse errors to be corrected. The deterministic endpoint-parameterized conditional flow matching (CFM) design improves matching through iteration, while the SDE forward drift enables effective few-step refinement. Experiments on 4DMatch and 4DLoMatch demonstrate competitive matching and non-rigid registration, with transfer to CAPE and DeepDeform without target-domain training.

Abstract

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