Metamorphoses of functional shapes in Sobolev spaces

TL;DR

This paper develops a comprehensive model for geometric-functional shape variability using Sobolev spaces, extending previous models to higher regularity, and provides theoretical, computational, and practical insights into shape matching and metamorphosis.

Contribution

The paper extends the original $L^2$ model to Sobolev spaces, introduces a bundle structure for fshape spaces, and develops a Hamiltonian framework for shape matching with both theoretical and numerical analysis.

Findings

Successful implementation of Sobolev-based shape matching

Demonstration of the model on synthetic and real data

Insights into geodesic equations and conservation laws

Abstract

In this paper, we describe in detail a model of geometric-functional variability between fshapes. These objects were introduced for the first time by the authors in [Charlier et al. 2015] and are basically the combination of classical deformable manifolds with additional scalar signal map. Building on the aforementioned work, this paper's contributions are several. We first extend the original $L^2$ model in order to represent signals of higher regularity on their geometrical support with more regular Hilbert norms (typically Sobolev). We describe the bundle structure of such fshape spaces with their adequate geodesic distances, encompassing in one common framework usual shape comparison and image metamorphoses. We then propose a formulation of matching between any two fshapes from the optimal control perspective, study existence of optimal controls and derive Hamiltonian equations and conservation laws describing the dynamics of geodesics. Secondly, we tackle the discrete counterpart of these problems and equations through appropriate finite elements interpolation schemes on triangular meshes. At last, we show a few results of metamorphosis matchings on synthetic and several real data examples in order to highlight the key specificities of the approach.