Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

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 surf...

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Bibliographic Details
Published in:International journal of computer vision 2023-05, Vol.131 (5), p.1183-1209
Main Authors: Hartman, Emmanuel, Sukurdeep, Yashil, Klassen, Eric, Charon, Nicolas, Bauer, Martin
Format: Article
Language:English
Subjects:
Algorithms
Applied mathematics
Artificial Intelligence
Computation
Computer Imaging
Computer Science
Elastic analysis
Geodesy
Image Processing and Computer Vision
Numerical methods
Pattern Recognition
Pattern Recognition and Graphics
Pipelining (computers)
Registration
Statistical analysis
Vision
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Summary: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.
ISSN:0920-5691
1573-1405
DOI:10.1007/s11263-022-01743-0