Source-linked AI summary
Continuous and Orientation-preserving Correspondences via Functional Maps
Jing Ren, Adrien Poulenard, Peter Wonka, Maks Ovsjanikov
TL;DR
The paper addresses the challenge of computing orientation-preserving, approximately continuous, and bijective correspondences between non-rigid shapes without external information. It adds a spectral orientation-preservation term and a joint spectral-spatial refinement scheme to functional maps, yielding higher-quality correspondences. Extensive evaluations report improvements over state-of-the-art methods on challenging isometric and non-isometric benchmarks.
Problem
Intrinsic descriptors can leave symmetry ambiguities unresolved, while enforcing continuity during pointwise recovery can require difficult quadratic optimization problems.
Method
The method introduces a geometric orientation-preservation term for functional maps and BCICP refinement that jointly improves maps in spectral and spatial domains.
Results
The approach improves map quality, bijectivity, and continuity relative to recent state-of-the-art methods on well-known test datasets.
Takeaways & Limitations
The resulting functional-maps pipeline produces orientation-preserving, approximately continuous, and bijective pointwise correspondences without landmarks, symmetry knowledge, or training data.
Takeaways & Limitations
The method cannot enforce exact bijectivity or complete coverage, is not directly applicable to point clouds or partial shapes, and lacks a theoretical convergence-rate analysis for BCICP.
Abstract
from arXiv · showhide
We propose a method for efficiently computing orientation-preserving and approximately continuous correspondences between non-rigid shapes, using the functional maps framework. We first show how orientation preservation can be formulated directly in the functional (spectral) domain without using landmark or region correspondences and without relying on external symmetry information. This allows us to obtain functional maps that promote orientation preservation, even when using descriptors, that are invariant to orientation changes. We then show how higher quality, approximately continuous and bijective pointwise correspondences can be obtained from initial functional maps by introducing a novel refinement technique that aims to simultaneously improve the maps both in the spectral and spatial domains. This leads to a general pipeline for computing correspondences between shapes that results in high-quality maps, while admitting an efficient optimization scheme. We show through extensive evaluation that our approach improves upon state-of-the-art results on challenging isometric and non-isometric correspondence benchmarks according to both measures of continuity and coverage as well as producing semantically meaningful correspondences as measured by the distance to ground truth maps.
1 INTRODUCTION
The paper targets non-rigid shape correspondences that preserve orientation while remaining approximately continuous and bijective. It introduces functional-map terms and refinement procedures that improve map quality and outperform state-of-the-art methods on standard benchmarks.
- Non-rigid correspondence is difficult because intrinsic-isometry optimization is generally NP-hard and intrinsic symmetries create multiple plausible solutions.
- External landmarks or training data are commonly used to resolve symmetry ambiguity, reducing the practical utility of correspondence pipelines.
- The method adds a geometric orientation-preservation energy directly to functional-map estimation without landmarks, region correspondences, or external symmetry information.
- Its refinement scheme alternates spectral and spatial optimization to produce approximately continuous and bijective pointwise maps while promoting coverage and controlling outliers.
- The approach demonstrates significant improvement over state-of-the-art methods on standard benchmarks.
2 RELATED WORK
Related work spans point-based, parameterization-based, functional-map, optimal-transport, and orientation-preserving approaches. These methods trade off geometric fidelity, continuity, bijectivity, computational cost, and robustness to symmetries or non-isometries.
- Point-based methods: Point-based methods directly match vertices while minimizing energies such as approximate geodesic-distance distortion, often using landmarks to recover isometric maps.
- Bijections through parameterization: Parameterization methods map shapes to canonical domains such as spheres, enabling explicit continuous correspondences but potentially limiting topology and increasing distortion.
- Functional Maps: Functional-map methods estimate linear operators from function correspondences and offer efficient optimization, but their extensions require sufficiently high-quality initial maps and can fail under moderate non-isometries.
- Optimal Transport and Variance Minimization: Optimal-transport approaches seek bijective and continuous correspondences while leveraging computational advances such as entropic Sinkhorn optimization.
- Orientation Preserving Maps: Orientation-preserving methods commonly use parameterizing domains or multiscale analyses, which can induce distortion or detect only local orientation changes.
3 PROBLEM SETUP AND MOTIVATION
The paper seeks continuous, orientation-preserving, approximately bijective maps between arbitrarily discretized manifold triangle meshes without external information. It uses functional maps, an orientation-preserving spectral term, and BCICP refinement to improve maps in both spectral and spatial domains.
- Problem setup: The target is correspondence between manifold triangle meshes, possibly with boundaries, without requiring equal vertex counts or consistent triangulations.
- Problem setup: The primary deformation setting is approximately isometric, although experiments indicate the method can handle significant deviations from this model.
- Functional maps pipeline: Functional maps provide a general framework that handles different discretizations and incorporates descriptor or landmark constraints.
- Functional maps pipeline: The basic pipeline computes basis functions and descriptor coefficients, estimates a functional map, then converts it into a point-to-point map through spectral ICP refinement.
- Motivation: Descriptor preservation and Laplace-Beltrami commutativity constrain functional maps but may not resolve intrinsic symmetry ambiguities.
- Proposed approach: The proposed orientation-preserving term reduces symmetry ambiguity, while BCICP jointly refines functional and pointwise maps to promote continuity and bijectivity.
4 PROPOSED METHOD
The method combines a geometric orientation-preservation term in the functional-map framework with BCICP refinement to produce pointwise maps that promote orientation preservation, continuity, bijectivity, coverage, and outlier control.
- 4.1 Orientation preservation: The orientation term reduces symmetry ambiguity without external landmarks or training information, including when orientation-invariant WKS descriptors initialize the map.The symmetric operator can additionally recover self-symmetric maps by reversing the normal-field orientation.
- 4.1 Orientation preservation: Directly enforcing orientation preservation through the map differential is difficult on discrete surfaces and can produce non-convex optimization energies.The direct formulation also places the functional map multiple times in terms and requires constraints over many function pairs.
- 4.1 Orientation preservation: The pipeline first estimates orientation-preserving functional maps using a purely geometric energy formulated through gradients and functional-map operators.For isometries, gradient commutation translates orientation preservation into the functional language, avoiding direct optimization over map differentials.
- 4.2 Bijective and Continuous ICP (BCICP): BCICP jointly refines opposite-direction functional maps to promote invertibility and bijectivity while improving maps in both spectral and spatial domains.The refinement is designed to improve overall map quality and promote continuity without sacrificing computational efficiency.
- 4.2 Bijective and Continuous ICP (BCICP): BCICP modifies pointwise correspondences to rectify discontinuities, remove outliers, and improve coverage after functional-map estimation.Outliers are detected from large edge-length distortions and disconnected components; coverage is defined by the target surface area reached by the map.
- 4.2 Bijective and Continuous ICP (BCICP): The coverage-improvement step better conditions the linear system for functional-map recovery, enabling more accurate vertex-to-vertex maps.The procedure uses the opposite-direction map to assign previously unmapped target vertices and improve each map's coverage.
5 RESULTS
The evaluation compares the complete directOp + BCICP framework with functional-map, ICP, BIM, and PMF alternatives on remeshed FAUST, TOSCA, and SHREC benchmarks. Across automatic matching and refinement studies, the method improves accuracy, coverage, and bijectivity, while continuity gains are more nuanced and convergence lacks a theoretical guarantee.
- 5.1 Evaluation for automatic shape matching: The experiments compare directOp + BCICP against BIM, ICP, PMF, and functional-map initializations using WKS or SEG descriptors.The datasets are independently remeshed to approximately 5k vertices, and hierarchical aspects are omitted to make comparisons more interpretable.
- 5.1 Evaluation for automatic shape matching: 17.5% improvement on 200 FAUST isometric pairs in direct error marks one of the reported gains over BIM and SEG + ICP.The corresponding direct-error improvements are 18.4% on FAUST non-isometric, 38.8% on TOSCA isometric, and 43.5% on TOSCA non-isometric pairs.
- 5.1 Evaluation for automatic shape matching: 17.8% improvement in per-vertex accuracy on FAUST isometric pairs accompanies gains of 24.1%, 31.4%, and 38.3% on the other three benchmark settings.These values are reported for FAUST non-isometric, TOSCA isometric, and TOSCA non-isometric pairs, respectively.
- 5.1 Evaluation for automatic shape matching: The orientation-preservation term improves results beyond BCICP alone for both WKS and SEG initialization.The comparison is reported as “+directOp + BCICP” performing consistently better than “+ BCICP.”
- 5.2 Evaluation of BCICP w.r.t each component: Each BCICP component contributes relative to regular ICP, while combining them produces a substantially better result.The tested variants remove bijectivity-related, coverage, or continuity-improving steps, including outlier removal and continuity improvement.
- 5.3 Improving coverage and smoothness using BCICP: BCICP drastically improves coverage and bijectivity over BIM and ICP, whereas its continuity is comparable to those methods.The apparently good smoothness of BIM and ICP is attributed to low coverage and many-to-one mappings, so continuity must be interpreted with coverage.
- 5.4 Parameters & Performance: Map quality improves substantially within the first 3 iterations for most tested pairs, although the energy curves are not monotonically decreasing.The convergence caption tracks energy, uncovered vertices or areas, bijection error, and outlier ratios normalized by initialization.
- 5.4 Parameters & Performance: Alternative functional-map coupling changes results negligibly except for TOSCA non-isometric perMap and direct error measures.Other coupling approaches are left as future work.
6 LIMITATIONS AND FUTURE WORK
The authors identify limits in exact bijectivity, complete coverage, input types, and scalability, and propose extensions for future work.
- Limitations: The method trades off matching quality, bijectivity, coverage, and smoothness rather than enforcing exact bijectivity or complete coverage.Experiments with PMF indicate that strong bijectivity can produce undesirable outliers.
- Limitations: The approach is not directly applicable to point clouds or partial shapes.Future work aims to process point clouds from scanned data.
- Limitations: Shape collections are typically limited to meshes with up to 10k vertices, motivating faster algorithms for much larger shapes.The authors suggest parallelizing map-refinement steps as one possible direction.
7 CONCLUSIONS
The paper concludes that its functional-maps algorithm computes orientation-preserving correspondences while improving map quality, bijectivity, and continuity against recent methods.
- The algorithm computes orientation-preserving maps through the functional maps pipeline.
- A novel spectral-domain term promotes orientation preservation when estimating functional maps.
- The extended post-processing pipeline improves maps in both spectral and spatial domains.
- Comparisons on established datasets show improved quality relative to several recent state-of-the-art methods, including bijectivity and continuity.
8 APPENDIX
The appendix proves an integral identity for smooth functions on compact oriented surfaces and shows that the same identity holds for closed triangle meshes.
- Theorem 4.1: The proof extends surface functions into the enclosed volume and applies the divergence theorem to establish the identity.The extensions are chosen so their gradients agree with the original functions on the surface.
- Theorem 4.1: The final equality follows from vector-product properties and the curl-free nature of gradients.
- Discrete setting: The identity also holds discretely for triangle meshes with face normals and gradients obtained by linear interpolation of vertex values.
- Discrete setting: For any closed manifold mesh, the area-weighted sum of the face terms ⟨∇f(p)×∇д(p), n(p)⟩A(t) equals zero.