Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework
This work addresses shape analysis challenges in fields like medical imaging or computer graphics, but it is incremental as it builds on existing Riemannian methods with new numerical implementations.
The paper tackled the problem of computing geodesics and distances between 3D surfaces, including unparametrized ones, by introducing a numerical framework based on second-order Sobolev metrics, and demonstrated its application to statistical shape analysis with tools like Karcher means and tangent PCA.
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.