Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Emmanuel Hartman, Yashil Sukurdeep, Eric Klassen, Nicolas Charon, Martin Bauer

TL;DR

The paper develops an open-source framework for elastic shape analysis of 3D surfaces using invariant second-order Sobolev metrics ($H^2$) on both parametrized and unparametrized surfaces. It combines a discrete $H^2$ energy with a varifold-based relaxed matching term to compute geodesics and distances without explicit reparametrization, enabling robust handling of sampling differences and partial data. A complete statistical pipeline is provided, including Karcher means, tangent PCA, and parallel transport, alongside a weighted, partial-matching extension that can erase regions to accommodate topology differences or missing data. The approach offers practical robustness for biomedical and geometric applications, with an emphasis on mesh-independence, partial matching, and a scalable numerical strategy, all released as open source.

Abstract

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.

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

TL;DR

The paper develops an open-source framework for elastic shape analysis of 3D surfaces using invariant second-order Sobolev metrics () on both parametrized and unparametrized surfaces. It combines a discrete energy with a varifold-based relaxed matching term to compute geodesics and distances without explicit reparametrization, enabling robust handling of sampling differences and partial data. A complete statistical pipeline is provided, including Karcher means, tangent PCA, and parallel transport, alongside a weighted, partial-matching extension that can erase regions to accommodate topology differences or missing data. The approach offers practical robustness for biomedical and geometric applications, with an emphasis on mesh-independence, partial matching, and a scalable numerical strategy, all released as open source.

Abstract

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.
Paper Structure (35 sections, 1 theorem, 53 equations, 16 figures, 2 tables, 7 algorithms)

This paper contains 35 sections, 1 theorem, 53 equations, 16 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work in Riemannian shape analysis
  4. Contributions
  5. Outline
  6. Sobolev metrics on surfaces
  7. Metrics on spaces of parametrized surfaces
  8. Geodesic boundary value problem:
  9. Geodesic initial value problem:
  10. Metrics on unparametrized surfaces
  11. Relaxed matching problem
  12. Varifold representation and distance
  13. Relaxed surface matching
  14. Numerical optimization approach
  15. The $H^2$-metric on the space of triangular meshes
  16. ...and 20 more sections

Key Result

lemma thmcounterlemma

The family of $H^2$-metrics $G$ is invariant under the action of the group of reparametrizations $\mathcal{D}$, the group of rotations $\operatorname{Rot}({\mathbb R}^3)$ and the group of translations ${\mathbb R}^3$, i.e., for any $q \in \mathcal{I}$, $h,k \in T_q \mathcal{I}$, $R \in \operatorname It follows that geodesic distance is also preserved by these transformations.

Figures (16)

  • Figure 1: Examples of optimal deformations (geodesics) between different types of data with unknown point correspondences: genus zero surfaces (line 1 and 2), higher genus surfaces with boundaries and inconsistent topologies (line 3 and 4), shape complexes (line 5 and 6), partial matching (line 7). Animations of the obtained surface deformations can be found in the supplementary material and on the github repository.
  • Figure 2: Point correspondences obtained after matching two unparametrized surfaces: the coloring of the two surfaces encode the obtained point correspondences. In addition, we highlight the obtained matching for selected points by displaying connecting lines.
  • Figure 3: The induced pullback metric on $M$ of an immersion $q: M\to \mathbb R^3$.
  • Figure 4: Defining $H^2$-metrics using discrete differential geometry. The cell dual to the vertex $v$ is shown in blue.
  • Figure 5: Solution to a parametrized BVP (top) and to the corresponding IVP (middle), i.e., after solving the BVP, we calculated the corresponding initial velocity of the solution and used this as the initial condition to solve the IVP. The results are overlaid (bottom) to illustrate the small discrepancy in the solutions.
  • ...and 11 more figures

Theorems & Definitions (7)

  • remark thmcounterremark: Su et. al. su2020shape
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark