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Convergence of spectral discretization for the flow of diffeomorphisms

Benedikt Wirth

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.

Convergence of spectral discretization for the flow of diffeomorphisms

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 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 preserve any higher order Sobolev regularity of their initial velocity.

Paper Structure

This paper contains 6 sections, 16 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The Riemannian manifold of Sobolev diffeomorphisms
  3. Refined and new regularity estimates
  4. Spectral space discretization and its properties
  5. Convergence of spectral discretization
  6. Acknowledgements.

Key Result

Theorem 1

Under standard conditions on $\mathcal{D}^m$ and its Riemannian metric, if the initial velocity $v_0$ has Sobolev regularity $m+k$ for $k\geq1$, the numerical approximation of bandlimiting the right-hand sides in eqn:EPDiffIntro converges to the true solution as the bandlimit $R$ tends to infinity.

Figures (1)

  • Figure 1: Numerical validation of the convergence result \ref{['thm:convergence']} in $d=2$ space dimensions for $m=3$. The numerically estimated error of the discretized solution is shown as a function of the truncation cutoff $R$ for experiments with different Sobolev regularity of the initial condition $v_0$. The grey lines indicate the rates $R^0$, $R^{-1}$, $R^{-2}$, and $R^{-3}$.

Theorems & Definitions (37)

  • Theorem 1: Convergence of bandlimited EPDiff equation
  • Theorem 2: Sobolev regularity preservation along geodesics
  • Definition 3: Group of Sobolev diffeomorphisms
  • Definition 4: Right-invariant Sobolev metric
  • Definition 5: Path energy and Riemannian distance on $\mathcal{D}_{\mathrm{id}}^m$
  • Definition 6: Eulerian velocity and flow
  • Lemma 7: Regularity of composition
  • proof
  • Proposition 8: Norm and composition estimates for the flow
  • proof
  • ...and 27 more