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. Supplementary Information: The online version contains supplementary material available at 10.1007/s11263-022-01743-0.

This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration with partial matching constraints and topological inconsistencies. Weighted shape graphs are the union of an arbitrary number of component curves in Euclidean space with potential connectivity constraints between some of their boundary points, together with a weight function defined on each component curve. The framework of higher-order invariant Sobolev metrics is particularly well suited for constructing notions of distances and geodesics between unparametrized curves. The main difficulty in adapting this framework to the setting of shape graphs is the absence of topological consistency, which typically results in an inadequate search for an exact matching between two shape graphs. We overcome this hurdle by defining an inexact variational formulation of the matching problem between (weighted) shape graphs of any underlying topology, relying on the convenient measure representation given by varifolds to relax the exact matching constraint. We then prove the existence of minimizers to this variational problem when we choose Sobolev metrics of sufficient regularity and a total variation (TV) regularization on the weight function. We propose a numerical optimization approach which adapts the smoothed fast iterative shrinkage-thresholding algorithm (SFISTA) to deal with $TV$ norm minimization and allows us to reduce the matching problem to solving a sequence of smooth unconstrained minimization problems. We finally illustrate the capabilities of our new model through several examples showcasing its ability to tackle partially observed and topologically varying data.