# Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization

arXiv: https://arxiv.org/abs/2210.02870 · HTML: https://mlsanity.com/paper/2210.02870/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 55 references that are in the map.
- First posted on arXiv 5 Oct 2022.

## Builds on (most cited first, 14 of 14)

- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [4DComplete: Non-Rigid Motion Estimation Beyond the Observable Surface](https://mlsanity.com/paper/2105.01905/) (2021)
- [Continuous and Orientation-preserving Correspondences via Functional Maps](https://mlsanity.com/paper/1806.04455/) (2018)
- [Smooth Shells: Multi-Scale Shape Registration with Functional Maps](https://mlsanity.com/paper/1905.12512/) (2019)
- [A Survey of Non-Rigid 3D Registration](https://mlsanity.com/paper/2203.07858/) (2022)
- [REALY: Rethinking the Evaluation of 3D Face Reconstruction](https://mlsanity.com/paper/2203.09729/) (2022)
- [Product Manifold Filter: Non-Rigid Shape Correspondence via Kernel Density Estimation in the Product Space](https://mlsanity.com/paper/1701.00669/) (2017)
- [ZoomOut: Spectral Upsampling for Efficient Shape Correspondence](https://mlsanity.com/paper/1904.07865/) (2019)
- [Efficient Deformable Shape Correspondence via Kernel Matching](https://mlsanity.com/paper/1707.08991/) (2017)
- [Reversible Harmonic Maps between Discrete Surfaces](https://mlsanity.com/paper/1801.02453/) (2018)
- [Convex Relaxations for Permutation Problems](https://mlsanity.com/paper/1306.4805/) (2013)
- [Intuitive and Efficient Roof Modeling for Reconstruction and Synthesis](https://mlsanity.com/paper/2109.07683/) (2021)
- [MapTree: Recovering Multiple Solutions in the Space of Maps](https://mlsanity.com/paper/2006.02532/) (2020)
- [Point-wise Map Recovery and Refinement from Functional Correspondence](https://mlsanity.com/paper/1506.05603/) (2015)

## Built on by (most cited first, 30 of 37)

- [PARTFIELD: Learning 3D Feature Fields for Part Segmentation and Beyond](https://mlsanity.com/paper/2504.11451/) (2025)
- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Learning Multi-resolution Functional Maps with Spectral Attention for Robust Shape Matching](https://mlsanity.com/paper/2210.06373/) (2022)
- [Spatially and Spectrally Consistent Deep Functional Maps](https://mlsanity.com/paper/2308.08871/) (2023)
- [Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation](https://mlsanity.com/paper/2402.18920/) (2024)
- [Partial-to-Partial Shape Matching with Geometric Consistency](https://mlsanity.com/paper/2404.12209/) (2024)
- [Memory-Scalable and Simplified Functional Map Learning](https://mlsanity.com/paper/2404.00330/) (2024)
- [Denoising Functional Maps: Diffusion Models for Shape Correspondence](https://mlsanity.com/paper/2503.01845/) (2025)
- [Non-Rigid Shape Registration via Deep Functional Maps Prior](https://mlsanity.com/paper/2311.04494/) (2023)
- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [Beyond Complete Shapes: A Benchmark for Quantitative Evaluation of 3D Shape Surface Matching Algorithms](https://mlsanity.com/paper/2411.03511/) (2024)
- [DV-Matcher: Deformation-based Non-Rigid Point Cloud Matching Guided by Pre-trained Visual Features](https://mlsanity.com/paper/2408.08568/) (2024)
- [DiffuMatch: Category-Agnostic Spectral Diffusion Priors for Robust Non-rigid Shape Matching](https://mlsanity.com/paper/2507.23715/) (2025)
- [Revisiting Map Relations for Unsupervised Non-Rigid Shape Matching](https://mlsanity.com/paper/2310.11420/) (2023)
- [DiscoMatch: Fast Discrete Optimisation for Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2310.08230/) (2023)
- [Synchronous Diffusion for Unsupervised Smooth Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2407.08244/) (2024)
- [Learning Part-Aware Dense 3D Feature Field for Generalizable Articulated Object Manipulation](https://mlsanity.com/paper/2602.14193/) (2026)
- [SIGMA: Scale-Invariant Global Sparse Shape Matching](https://mlsanity.com/paper/2308.08393/) (2023)
- [Fast Globally Optimal and Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2504.06385/) (2025)
- [FRIDU: Functional Map Refinement with Guided Image Diffusion](https://mlsanity.com/paper/2506.14322/) (2025)
- [Spatial and Surface Correspondence Field for Interaction Transfer](https://mlsanity.com/paper/2405.03221/) (2024)
- [TwoSquared: 4D Generation from 2D Image Pairs](https://mlsanity.com/paper/2504.12825/) (2025)
- [Symmetry Understanding of 3D Shapes via Chirality Disentanglement](https://mlsanity.com/paper/2508.05505/) (2025)
- [FUSE: A Flow-based Mapping Between Shapes](https://mlsanity.com/paper/2511.13431/) (2025)
- [Symmetry Informative and Agnostic Feature Disentanglement for 3D Shapes](https://mlsanity.com/paper/2601.14804/) (2026)
- [SGMatch: Semantic-Guided Non-Rigid Shape Matching with Flow Regularization](https://mlsanity.com/paper/2603.12937/) (2026)
- [RINO: Rotation-Invariant Non-Rigid Correspondences](https://mlsanity.com/paper/2603.27773/) (2026)
- [PaNDaS: Learnable Deformation Modeling with Localized Control](https://mlsanity.com/paper/2412.02306/) (2024)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)

## Abstract (arXiv)

We introduce pointwise map smoothness via the Dirichlet energy into the functional map pipeline, and propose an algorithm for optimizing it efficiently, which leads to high-quality results in challenging settings. Specifically, we first formulate the Dirichlet energy of the pulled-back shape coordinates, as a way to evaluate smoothness of a pointwise map across discrete surfaces. We then extend the recently proposed discrete solver and show how a strategy based on auxiliary variable reformulation allows us to optimize pointwise map smoothness alongside desirable functional map properties such as bijectivity. This leads to an efficient map refinement strategy that simultaneously improves functional and point-to-point correspondences, obtaining smooth maps even on non-isometric shape pairs. Moreover, we demonstrate that several previously proposed methods for computing smooth maps can be reformulated as variants of our approach, which allows us to compare different formulations in a consistent framework. Finally, we compare these methods both on existing benchmarks and on a new rich dataset that we introduce, which contains non-rigid, non-isometric shape pairs with inter-category and cross-category correspondences. Our work leads to a general framework for optimizing and analyzing map smoothness both conceptually and in challenging practical settings.