Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

TL;DR

This paper develops a metamorphosis framework for varifolds, extending geometric measure registration by combining diffeomorphic deformations with weight transformations, and introduces a Riemannian metric with practical computation methods.

Contribution

It introduces a novel metamorphosis model for varifolds with a new Riemannian metric and optimal control formulations, enhancing geometric measure registration techniques.

Findings

Well-defined Riemannian metric on varifolds with geodesics

Existence of solutions for the optimal control problems

Numerical methods for computing metamorphoses between varifolds

Abstract

This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.