Shape analysis via gradient flows on diffeomorphism groups

TL;DR

This paper introduces a gradient flow approach on Sobolev diffeomorphisms for image registration, ensuring well-posedness and providing a geometric interpretation of the gradient in terms of the momentum map.

Contribution

It establishes the well-posedness of the gradient flow on Sobolev diffeomorphisms and offers a geometric description of the gradient via the momentum map.

Findings

Proves well-posedness of the gradient flow.

Provides a geometric interpretation of the gradient.

Enhances understanding of deformation penalization in image registration.

Abstract

We study a gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing deformation of the metric tensor. The main result is well-posedness of the flow. We also give a geometric description of the gradient in terms of the momentum map.