mlsanityData to 7 Oct 2026

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

arXiv 2204.04238 ↗First posted 8 Apr 2022cs.CVcited by 18

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Abstract (arXiv)

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 23 of its 88 references that are in the map. First posted on arXiv 8 Apr 2022.

Builds on

23 of 88 references.

References · 23

  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2018Generating 3D faces using Convolutional Mesh Autoencoderscited by 267
  3. 20193D Hand Shape and Pose from Images in the Wildcited by 190
  4. 2020Kernel Operations on the GPU, with Autodiff, without Memory Overflowscited by 101
  5. 2018Continuous and Orientation-preserving Correspondences via Functional Mapscited by 77
  6. 2015Partial Functional Correspondencecited by 61
  7. 2020LIMP: Learning Latent Shape Representations with Metric Preservation Priorscited by 54
  8. 2021Shape registration in the time of transformerscited by 51
  9. 2021ARAPReg: An As-Rigid-As Possible Regularization Loss for Learning Deformable Shape Generatorscited by 40
  10. 2017Optimal Transport for Diffeomorphic Registrationcited by 39
  11. 2013The varifold representation of non-oriented shapes for diffeomorphic registrationcited by 24
  12. 2020Hamiltonian Dynamics for Real-World Shape Interpolationcited by 13
11 more
  1. 2016Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfacescited by 12
  2. 2019Shape Analysis of Surfaces Using General Elastic Metricscited by 8
  3. 2020A numerical framework for elastic surface matching, comparison, and interpolationcited by 8
  4. 20214D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Datacited by 8
  5. 2021A Riemannian Framework for Analysis of Human Body Surfacecited by 7
  6. 2021A new variational model for shape graph registration with partial matching constraintscited by 5
  7. 2015Shape Aware Matching of Implicit Surfaces based on Thin Shell Energiescited by 4
  8. 2021Supervised deep learning of elastic SRV distances on the shape space of curvescited by 4
  9. 2018Metric Registration of Curves and Surfaces using Optimal Controlcited by 2
  10. 2020An inexact matching approach for the comparison of plane curves with general elastic metricscited by 2
  11. 2014Geodesics on Shape Spaces with Bounded Variation and Sobolev Metricscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 6; 2024: 4; 2025: 5; 2026: 1.

18 papers.

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  1. 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 14
  2. 2022BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapescited by 7
  3. 2023Toward Mesh-Invariant 3D Generative Deep Learning with Geometric Measurescited by 6
  4. 2023Basis restricted elastic shape analysis on the space of unregistered surfacescited by 5
  5. 2023VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Datacited by 4
  6. 2024Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motionscited by 2
  7. 2025GenAnalysis: Joint Shape Analysis by Learning Man-Made Shape Generators with Deformation Regularizationscited by 2
  8. 2025Stochastics of shapes and Kunita flowscited by 2
  9. 2023A function space perspective on stochastic shape evolutioncited by 1
  10. 2024Explainable Binary Classification of Separable Shape Ensemblescited by 1
  11. 2024PaNDaS: Learnable Deformation Modeling with Localized Controlcited by 1
  12. 2025ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fieldscited by 1
6 more
  1. 2022Automatic classification of deformable shapescited by 0
  2. 2023Planar Curve Registration using Bayesian Inversioncited by 0
  3. 2023Geodesic Regression Characterizes 3D Shape Changes in the Female Brain During Menstruationcited by 0
  4. 2025ScanMove: Motion Prediction and Transfer for Unregistered Body Meshescited by 0
  5. 2025VPREG: An Optimal Control Formulation for Diffeomorphic Image Registration Based on the Variational Principle Grid Generation Methodcited by 0
  6. 2026ARAPDiffusion: ARAP Regularization for Diffusion-Based Deformable Shape Space Learningcited by 0