NHO: A Neural Hamiltonian Operator for Anchor-based Region Localization and Dense Correspondence

Jing Li; Yawei Luo; Xiangze Meng; Ying Li; Tieru Wu; Rui Ma

cs.CVSep 29, 2026

NHO: A Neural Hamiltonian Operator for Anchor-based Region Localization and Dense Correspondence

Authors: Jing Li, Yawei Luo, Xiangze Meng, Ying Li, Tieru Wu, Rui Ma

Organizations: Jilin University · Zhejiang University · North China University of Technology

Abstract

Non-rigid partial-to-full shape correspondence from sparse anchors requires identifying the corresponding region on the full surface and recovering dense correspondences between the partial shape and that region. We present NHO, which combines sparse anchors with the intrinsic geometry of the partial shape to learn a neural Hamiltonian operator whose localized eigenspace encodes both the region support and intrinsic coordinates for dense correspondence. NHO parameterizes the Hamiltonian potential as an intrinsic neural field and optimizes it using anchor evidence together with spectral and geometric constraints. To resolve the spatial ambiguity left by sparse anchors, we introduce reciprocal refinement between operator estimation and correspondence recovery. At each round, the current eigenspace provides spectral coordinates and restricts matching to its induced support, while geometrically reliable correspondences provide additional evidence for updating the potential. After refinement, aggregated eigenfunction energy yields the final localization, and the recovered map initializes dense correspondence refinement. Experiments demonstrate competitive accuracy on both tasks and robustness to uniform scaling and rotation.

Figures & tables

Figure 1: NHO (Neural Hamiltonian Operator) models a neural eigenspace using spatial cues, supporting both region localization and dense correspondence estimation for non-rigid partial-to-full shape matching.

Figure 2: Overview of the NHO framework. Given a partial shape $\mathcal{M}$ , a full shape $\mathcal{N}$ , and sparse anchors $\mathcal{A}^{(0)}$ , a neural field parameterizes the Hamiltonian potential on $\mathcal{N}$ using $\bm{\Phi}_{\mathcal{N}}$ as an intrinsic positional encoding. At round $t$ , the Hamiltonian eigenspace induces the support $\widehat{\mathcal{R}}^{(t)}$ through aggregated energy and, together with $\bm{\Phi}_{\mathcal{M}}$ and $\mathcal{A}^{(t)}$ , determines the alignment $C^{(t)}$ . The support restricts the recovery of $T^{(t)}$ , while geometrically reliable matches update the anchors and operator. The final support localizes the corresponding region, while the final-round map $\widehat{T}$ initializes dense correspondence refinement.

	Default								Normalized
	Unrotated				Rotated				Unrotated				Rotated
Method	IoU	Precision	Recall	F1-score	IoU	Precision	Recall	F1-score	IoU	Precision	Recall	F1-score	IoU	Precision	Recall	F1-score
PFM	71.88	78.40	85.27	81.60	72.22	78.60	85.50	81.82	50.58	59.88	67.61	63.28	49.56	58.88	66.42	62.22
FSPM	51.59	57.17	80.02	66.28	51.56	57.21	79.84	66.25	50.52	56.38	79.32	65.47	50.39	56.35	79.30	65.46
Hamiltonian	50.73	60.34	62.37	61.02	50.38	60.02	61.58	60.48	52.43	62.04	64.40	62.85	52.96	62.69	65.37	63.63
DPFM	17.03	53.48	20.06	26.16	13.33	42.37	15.82	20.73	80.45	85.63	93.24	88.86	37.65	62.76	46.01	51.12

Table 1: Quantitative comparison of region localization on CUTS’24. All metrics are surface-area-weighted percentages; higher is better. Best in bold; second-best underlined.

Figure 3: IoU success curves for region localization on CUTS’24 under four input settings. Each point reports the percentage of shapes whose surface-area-weighted IoU exceeds the corresponding threshold.

Figure 5: Qualitative comparison at the default scale under $\mathrm{SO}(3)$ rotations. Green, orange, and purple denote correct, false-positive, and missed support, respectively.

Figure 6: Qualitative comparison of partial-to-full correspondence. Our method yields more accurate correspondences than competing methods, even for challenging non-isometric shape pairs.

Table 3: Component ablations on normalized, unrotated CUTS’24. Localization metrics are area-weighted percentages; correspondence is measured by mean geodesic error ( $\times 100$ ) before and after external NAM refinement. The best result is shown in bold.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Region-localizing operator	Explicit region localization	Dense correspondence	Uniform-scale and rotation robustness	Anchor-source agnostic
PFM ( Rodolà et al., 2017 )	✗	✓	✓	✗	✗
FSPM ( Litany et al., 2017 )	✗	✗	✓	✗	✗
Hamiltonian ( Rampini et al., 2019 )	✓	✓	✗	✗	✗
DPFM ( Attaiki et al., 2021 )	✗	✓	✓	✗	✗
Piecewise Smooth ( Bensaïd et al., 2023 )	✗	✓	✗	✗	✗
ULRSSM ( Cao et al., 2023 )	✗	✗	✓	✗	✗

Appendix

Table 4: Comparison with representative partial-to-full correspondence methods. “Anchor-source agnostic” indicates that a method can accept sufficiently reliable sparse anchors without prescribing how they are obtained.

Figure 8: Visualization of reciprocal refinement. The estimated support and correspondence progressively improve across rounds. Green, orange, and purple denote correct, false-positive, and missed support, respectively.

Figure 9: Evolution of the Hamiltonian eigenfunctions during reciprocal refinement. Top: the first four LBO eigenfunctions of the full shape (left) and Dirichlet eigenfunctions of the partial shape (right). Bottom: the anchor-aligned neural Hamiltonian eigenfunctions after initial operator estimation (left) and reciprocal refinement (right).

Panel of Table 5 — Table 5: Quantitative comparison of region localization on PFAUST-M, PFAUST-H, and PFARM under four input settings. All metrics are surface-area-weighted percentages; higher is better. Best in bold; second-best underlined.

Figure 11: Additional qualitative comparisons of region localization. Numbers report the surface-area-weighted IoU for each prediction. Green, orange, and purple denote correctly localized, false-positive, and missed regions, respectively.

Figure 12: PCK curves on four benchmarks under the three input settings not shown in Fig. 4 . Each point reports the percentage of correspondences whose geodesic error (

\times 100

) does not exceed the threshold.

Figure 13: Additional qualitative comparisons of partial-to-full dense correspondence. Numbers report the mean geodesic error ( $\times 100$ ) for each case; lower is better.

Table 6: Anchor sampling on normalized, unrotated CUTS’24. Localization scores are area-weighted percentages; Geo. is the final mean geodesic error (

\times 100

). Best values are bold.

Panel of Table 8 — Table 8: Robustness to mesh discretization on normalized, unrotated CUTS’24. Localization metrics are surface-area-weighted percentages; Geo. denotes the final mean geodesic error ( $\times 100$ ). Best values are shown in bold.

Figure 17: Qualitative results under severe discretization mismatch. The full meshes retain $10\%$ and $5\%$ of their original faces, while the partial meshes remain unchanged. Green, orange, and purple denote correct, false-positive, and missed support, respectively.

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