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ReMatching: Low-Resolution Representations for Scalable Shape Correspondence

Filippo Maggioli, Daniele Baieri, Emanuele Rodolà, Simone Melzi

TL;DR

ReMatching addresses scalable dense shape correspondence by replacing the original dense meshes with a carefully crafted low-resolution representation that preserves topology and metric properties, enabling efficient functional-map optimization. A novel remeshing scheme creates an intrinsic Delaunay triangulation on a uniform sampling of the surface, allowing the functional map to be computed on the reduced mesh and then extended back to the original dense shapes via a fast barycentric-based extension. Across SHREC19, TOSCA, and the challenging BadTOSCA dataset, the approach demonstrates superior time-accuracy trade-offs compared to baselines such as SFM, IRM, and IEM, while maintaining robustness to non-isometric deformations. The work thus provides a scalable pathway for high-quality shape matching on meshes containing millions of vertices, with practical implications for large-scale 3D shape analysis and applications needing fast, reliable correspondences.

Abstract

We introduce \emph{ReMatching}, a novel shape correspondence solution based on the functional maps framework. Our method, by exploiting a new and appropriate \emph{re}-meshing paradigm, can target shape-\emph{matching} tasks even on meshes counting millions of vertices, where the original functional maps does not apply or requires a massive computational cost. The core of our procedure is a time-efficient remeshing algorithm which constructs a low-resolution geometry while acting conservatively on the original topology and metric. These properties allow translating the functional maps optimization problem on the resulting low-resolution representation, thus enabling efficient computation of correspondences with functional map approaches. Finally, we propose an efficient technique for extending the estimated correspondence to the original meshes. We show that our method is more efficient and effective through quantitative and qualitative comparisons, outperforming state-of-the-art pipelines in quality and computational cost.

ReMatching: Low-Resolution Representations for Scalable Shape Correspondence

TL;DR

ReMatching addresses scalable dense shape correspondence by replacing the original dense meshes with a carefully crafted low-resolution representation that preserves topology and metric properties, enabling efficient functional-map optimization. A novel remeshing scheme creates an intrinsic Delaunay triangulation on a uniform sampling of the surface, allowing the functional map to be computed on the reduced mesh and then extended back to the original dense shapes via a fast barycentric-based extension. Across SHREC19, TOSCA, and the challenging BadTOSCA dataset, the approach demonstrates superior time-accuracy trade-offs compared to baselines such as SFM, IRM, and IEM, while maintaining robustness to non-isometric deformations. The work thus provides a scalable pathway for high-quality shape matching on meshes containing millions of vertices, with practical implications for large-scale 3D shape analysis and applications needing fast, reliable correspondences.

Abstract

We introduce \emph{ReMatching}, a novel shape correspondence solution based on the functional maps framework. Our method, by exploiting a new and appropriate \emph{re}-meshing paradigm, can target shape-\emph{matching} tasks even on meshes counting millions of vertices, where the original functional maps does not apply or requires a massive computational cost. The core of our procedure is a time-efficient remeshing algorithm which constructs a low-resolution geometry while acting conservatively on the original topology and metric. These properties allow translating the functional maps optimization problem on the resulting low-resolution representation, thus enabling efficient computation of correspondences with functional map approaches. Finally, we propose an efficient technique for extending the estimated correspondence to the original meshes. We show that our method is more efficient and effective through quantitative and qualitative comparisons, outperforming state-of-the-art pipelines in quality and computational cost.
Paper Structure (10 sections, 1 equation, 10 figures, 2 tables)

This paper contains 10 sections, 1 equation, 10 figures, 2 tables.

Table of Contents

  1. Results
  2. Quality of matching
  3. Scalable performance
  4. Benefits of remeshing
  5. Conclusions
  6. Proof of Theorem \ref{['thm:flatunion']}
  7. Proof of Proposition \ref{['prp:2ball']}
  8. Handling large triangles
  9. BadTOSCA dataset
  10. Closest surface point mapping

Figures (10)

  • Figure 4: Left: Accuracy curves on the SHREC19 challenge pairs for the tested methods. We also consider ideal approaches where the initial functional map for ZoomOut is given ground truth dashed lines). Right: Cumulative curves of the execution time for evaluated methods. For all methods, we also consider the time required to run the remeshing or resampling step(s).
  • Figure 5: Accuracy curves and average geodesic error on the TOSCA dataset (left) and the BadTOSCA dataset (right) for the tested methods.
  • Figure 6: Coordinate transfer between non-isometric shapes. The coordinates are used to generate highly complex patterns, whose sensitivity to the input enhances the transfer errors.
  • Figure 7: Inter-class coordinate transfer from the wolf model to a cat and a dog. Our pipeline is not constrained by the standard Laplacian basis and can be used with other approaches as well.
  • Figure 8: Comparison of function transfer between very high-resolution models with SFM and our approach.
  • ...and 5 more figures

Theorems & Definitions (2)

  • proof
  • proof