# Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework arXiv: https://arxiv.org/abs/2204.04238 ยท HTML: https://mlsanity.com/paper/2204.04238/ ## Key facts - Cited by 19 papers in the map (data through 14 Oct 2026). - Builds on 23 of its 88 references that are in the map. - First posted on arXiv 8 Apr 2022. ## Builds on (most cited first, 23 of 23) - [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021) - [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018) - [3D Hand Shape and Pose from Images in the Wild](https://mlsanity.com/paper/1902.03451/) (2019) - [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020) - [Continuous and Orientation-preserving Correspondences via Functional Maps](https://mlsanity.com/paper/1806.04455/) (2018) - [Partial Functional Correspondence](https://mlsanity.com/paper/1506.05274/) (2015) - [LIMP: Learning Latent Shape Representations with Metric Preservation Priors](https://mlsanity.com/paper/2003.12283/) (2020) - [Shape registration in the time of transformers](https://mlsanity.com/paper/2106.13679/) (2021) - [ARAPReg: An As-Rigid-As Possible Regularization Loss for Learning Deformable Shape Generators](https://mlsanity.com/paper/2108.09432/) (2021) - [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017) - [The varifold representation of non-oriented shapes for diffeomorphic registration](https://mlsanity.com/paper/1304.6108/) (2013) - [Hamiltonian Dynamics for Real-World Shape Interpolation](https://mlsanity.com/paper/2004.05199/) (2020) - [Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces](https://mlsanity.com/paper/1610.04531/) (2016) - [Shape Analysis of Surfaces Using General Elastic Metrics](https://mlsanity.com/paper/1910.02045/) (2019) - [A numerical framework for elastic surface matching, comparison, and interpolation](https://mlsanity.com/paper/2006.11652/) (2020) - [4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data](https://mlsanity.com/paper/2101.09403/) (2021) - [A Riemannian Framework for Analysis of Human Body Surface](https://mlsanity.com/paper/2108.11449/) (2021) - [A new variational model for shape graph registration with partial matching constraints](https://mlsanity.com/paper/2105.00678/) (2021) - [Shape Aware Matching of Implicit Surfaces based on Thin Shell Energies](https://mlsanity.com/paper/1509.06559/) (2015) - [Supervised deep learning of elastic SRV distances on the shape space of curves](https://mlsanity.com/paper/2101.04929/) (2021) - [Metric Registration of Curves and Surfaces using Optimal Control](https://mlsanity.com/paper/1809.06422/) (2018) - [An inexact matching approach for the comparison of plane curves with general elastic metrics](https://mlsanity.com/paper/2001.02858/) (2020) - [Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics](https://mlsanity.com/paper/1402.6504/) (2014) ## Built on by (most cited first, 19 of 19) - [DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains](https://mlsanity.com/paper/2402.07250/) (2024) - [BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes](https://mlsanity.com/paper/2211.13185/) (2022) - [Toward Mesh-Invariant 3D Generative Deep Learning with Geometric Measures](https://mlsanity.com/paper/2306.15762/) (2023) - [Basis restricted elastic shape analysis on the space of unregistered surfaces](https://mlsanity.com/paper/2311.04382/) (2023) - [VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Data](https://mlsanity.com/paper/2307.03553/) (2023) - [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024) - [GenAnalysis: Joint Shape Analysis by Learning Man-Made Shape Generators with Deformation Regularizations](https://mlsanity.com/paper/2503.00807/) (2025) - [Stochastics of shapes and Kunita flows](https://mlsanity.com/paper/2512.11676/) (2025) - [A function space perspective on stochastic shape evolution](https://mlsanity.com/paper/2302.05382/) (2023) - [Explainable Binary Classification of Separable Shape Ensembles](https://mlsanity.com/paper/2410.12994/) (2024) - [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) - [ScanMove: Motion Prediction and Transfer for Unregistered Body Meshes](https://mlsanity.com/paper/2508.21095/) (2025) - [Automatic classification of deformable shapes](https://mlsanity.com/paper/2211.02530/) (2022) - [Planar Curve Registration using Bayesian Inversion](https://mlsanity.com/paper/2307.04909/) (2023) - [Geodesic Regression Characterizes 3D Shape Changes in the Female Brain During Menstruation](https://mlsanity.com/paper/2309.16662/) (2023) - [VPREG: An Optimal Control Formulation for Diffeomorphic Image Registration Based on the Variational Principle Grid Generation Method](https://mlsanity.com/paper/2510.13109/) (2025) - [ARAPDiffusion: ARAP Regularization for Diffusion-Based Deformable Shape Space Learning](https://mlsanity.com/paper/2606.06887/) (2026) - [Verifier-Guided Synthetic Augmentation for 3D Human Shape Generation](https://mlsanity.com/paper/2610.04006/) (2026) ## 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.