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From Feature Learning to Spectral Basis Learning: A Unifying and Flexible Framework for Efficient and Robust Shape Matching

Feifan Luo, Hongyang Chen

Abstract

Shape matching is a fundamental task in computer graphics and vision, with deep functional maps becoming a prominent paradigm. However, existing methods primarily focus on learning informative feature representations by constraining pointwise and functional maps, while neglecting the optimization of the spectral basis-a critical component of the functional map pipeline. This oversight often leads to suboptimal matching results. Furthermore, many current approaches rely on conventional, time-consuming functional map solvers, incurring significant computational overhead. To bridge these gaps, we introduce Advanced Functional Maps, a framework that generalizes standard functional maps by replacing fixed basis functions with learnable ones, supported by rigorous theoretical guarantees. Specifically, the spectral basis is optimized through a set of learned inhibition functions. Building on this, we propose the first unsupervised spectral basis learning method for robust non-rigid 3D shape matching, enabling the joint, end-to-end optimization of feature extraction and basis functions. Our approach incorporates a novel heat diffusion module and an unsupervised loss function, alongside a streamlined architecture that bypasses expensive solvers and auxiliary losses. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art feature-learning approaches, particularly in challenging non-isometric and topological noise scenarios, while maintaining high efficiency. Finally, we reveal that optimizing basis functions is equivalent to spectral convolution, where inhibition functions act as filters. This insight enables enhanced representations inspired by spectral graph networks, opening new avenues for future research. Our code is available at https://github.com/LuoFeifan77/Unsupervised-Spectral-Basis-Learning.

From Feature Learning to Spectral Basis Learning: A Unifying and Flexible Framework for Efficient and Robust Shape Matching

Abstract

Shape matching is a fundamental task in computer graphics and vision, with deep functional maps becoming a prominent paradigm. However, existing methods primarily focus on learning informative feature representations by constraining pointwise and functional maps, while neglecting the optimization of the spectral basis-a critical component of the functional map pipeline. This oversight often leads to suboptimal matching results. Furthermore, many current approaches rely on conventional, time-consuming functional map solvers, incurring significant computational overhead. To bridge these gaps, we introduce Advanced Functional Maps, a framework that generalizes standard functional maps by replacing fixed basis functions with learnable ones, supported by rigorous theoretical guarantees. Specifically, the spectral basis is optimized through a set of learned inhibition functions. Building on this, we propose the first unsupervised spectral basis learning method for robust non-rigid 3D shape matching, enabling the joint, end-to-end optimization of feature extraction and basis functions. Our approach incorporates a novel heat diffusion module and an unsupervised loss function, alongside a streamlined architecture that bypasses expensive solvers and auxiliary losses. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art feature-learning approaches, particularly in challenging non-isometric and topological noise scenarios, while maintaining high efficiency. Finally, we reveal that optimizing basis functions is equivalent to spectral convolution, where inhibition functions act as filters. This insight enables enhanced representations inspired by spectral graph networks, opening new avenues for future research. Our code is available at https://github.com/LuoFeifan77/Unsupervised-Spectral-Basis-Learning.
Paper Structure (40 sections, 6 theorems, 40 equations, 6 figures, 9 tables, 1 algorithm)

This paper contains 40 sections, 6 theorems, 40 equations, 6 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Functional map methods
  4. Deep functional map methods
  5. Spectral Basis functions
  6. Background and notation
  7. Advanced Functional Maps
  8. Learnable Spectral Basis Representation
  9. Advanced Functional Map
  10. Map Recovery
  11. Unsupervised Spectral Basis Learning for Efficient and Robust Shape Matching
  12. Feature Extraction
  13. Basis Learning
  14. Multi-scale Eigenvalue-Agnostic Heat Diffusion.
  15. Maps computation
  16. ...and 25 more sections

Key Result

Theorem 4.2

Advanced functional maps. Consider the first $k$ adaptable basis functions $\Psi_{\mathcal{X},k} = \Phi_{\mathcal{X},k} G_{\mathcal{X}} \in \mathbb{R}^{|V_\mathcal{X}|\times k}$ and $\Psi_{\mathcal{Y},k} = \Phi_{\mathcal{Y},k} G_{\mathcal{Y}}\in \mathbb{R}^{|V_\mathcal{Y}|\times k}$, defined on shap where $E_{reg} = \left\| C^{\mathrm{A}}_{\mathcal{XY}}\Lambda_{\mathcal{X}} - \Lambda_{\mathcal{Y}}

Figures (6)

  • Figure 1: Existing deep functional map architectures. Dual-branch: Integrates both a functional map solver and a spectral projector to compute functional maps, supervised by unsupervised losses (e.g., orthogonality, bijectivity) and coupled with a consistency loss between the two functional representations. Single-branch: Employs either a functional map solver or a spectral projector alone, regulated by corresponding unsupervised constraints.
  • Figure 2: An overview of our method. (1) Feature Extraction: Learned features ${F}_{\mathcal{X}}$ and ${F}_{\mathcal{Y}}$ are extracted from shapes $\mathcal{X}$ and $\mathcal{Y}$, respectively. (2) Basis Learning: The optimized basis functions, $\Psi_{\mathcal{X}}$ and $\Psi_{\mathcal{Y}}$, are generated by the proposed heat diffusion network. (3) Map Estimation: The differentiable pointwise map ${\Pi}_\mathcal{YX}$ is computed using the softmax operator \ref{['eq: compute soft map']}, and the advanced functional map ${{C}}^{\mathrm{A}}_\mathcal{XY}$ is calculated via spectral basis projection \ref{['equ: compute C by Pi adj']}. (4) An unsupervised loss \ref{['equ:multi_spectral_loss']} is constructed to supervise basis functions, pointwise maps, and functional maps. (5) The pointwise map ${\Pi}^{end}_\mathcal{YX}$ is recovered during inference using \ref{['equ: nnsearch1 in test']} and Eq.\ref{['equ: nnsearch2 in test']}.
  • Figure 3: The quantitative results of learned inhibition functions across different datasets.
  • Figure 4: Comparisons of cross-dataset generalization performance with other methods. Top row: training on FAUST ren2018continuous and testing on SHREC'19 melzi2019shrec. Bottom row: Training on SCAPE ren2018continuous and testing on SHREC'19. Our method exhibits fewer errors and less color distortion compared to other approaches, highlighting its robust performance.
  • Figure 5: Comparisons with other methods on the SMAL zuffi2017 (top row) and DT4D-H 2022SmoothNonRigidShapeMatchingviaEffectiveDirichletEnergyOptimization (bottom row) datasets. Our approach results in fewer errors and less texture distortion than other methods, demonstrating its superior performance for non-isometric shape matching.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 4.1
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 5.1
  • Theorem
  • proof
  • Theorem
  • proof
  • ...and 2 more