Convergence of spectral discretization for the flow of diffeomorphisms

TL;DR

This paper proves the convergence of a Fourier-type spectral discretization method for the geodesic equations in the flow of diffeomorphisms, a key mathematical framework in shape analysis and computational anatomy.

Contribution

It provides a rigorous convergence proof for a commonly used spectral discretization of the geodesic equation in the Sobolev diffeomorphism group, including a new regularity estimate.

Findings

Spectral discretization converges for the geodesic equation in Sobolev diffeomorphisms

Higher order Sobolev regularity of geodesics is preserved

New proof of regularity preservation in diffeomorphism flows

Abstract

The Large Deformation Diffeomorphic Metric Mapping (LDDMM) or flow of diffeomorphism is a classical framework in the field of shape spaces and is widely applied in mathematical imaging and computational anatomy. Essentially, it equips a group of diffeomorphisms with a right-invariant Riemannian metric, which allows to compute (Riemannian) distances or interpolations between different deformations. The associated Euler--Lagrange equation of shortest interpolation paths is one of the standard examples of a partial differential equation that can be approached with Lie group theory (by interpreting it as a geodesic ordinary differential equation on the Lie group of diffeomorphisms). The particular group $\mathcal D^m$ of Sobolev diffeomorphisms is by now sufficiently understood to allow the analysis of geodesics and their numerical approximation. We prove convergence of a widely used Fourier-type space discretization of the geodesic equation. It is based on a regularity estimate, for which we also provide a new proof: Geodesics in $\mathcal D^m$ preserve any higher order Sobolev regularity of their initial velocity.