Convergence of spectral discretization for the flow of diffeomorphisms

2024/10/09 by Wirth, Benedikt
#53Z50 #58D25 (Secondary) #65M12 (Primary) 35Q35 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2410.06788

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 new regularity estimate: We prove that geodesics in \mathcal D^m preserve any higher order Sobolev regularity of their initial velocity.

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