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This paper introduces a set of numerical methods for Riemannian shapeanalysis of 3D surfaces within the setting of invariant (elastic) second-orderSobolev metrics. More specifically, we address the computation of geodesics andgeodesic distances between parametrized or unparametrized immersed surfacesrepresented as 3D meshes. Building on this, we develop tools for thestatistical shape analysis of sets of surfaces, including methods forestimating Karcher means and performing tangent PCA on shape populations, andfor computing parallel transport along paths of surfaces. Our proposed approachfundamentally relies on a relaxed variational formulation for the geodesicmatching problem via the use of varifold fidelity terms, which enable us toenforce reparametrization independence when computing geodesics betweenunparametrized surfaces, while also yielding versatile algorithms that allow usto compare surfaces with varying sampling or mesh structures. Importantly, wedemonstrate how our relaxed variational framework can be extended to tacklepartially observed data. The different benefits of our numerical pipeline areillustrated over various examples, synthetic and real.

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