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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {D}^m\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {D}^m\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> preserve any higher order Sobolev regularity of their initial velocity.</p>

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Convergence of spectral discretization for the flow of diffeomorphisms

  • 期刊论文

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  • 发表日期：

    2026年3月28日

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  • Benedikt Wirth

摘要

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 \(\mathcal {D}^m\) 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 \(\mathcal {D}^m\) D m preserve any higher order Sobolev regularity of their initial velocity.