Convergence of spectral discretization for the flow of diffeomorphisms
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918356879147008 |
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| author | Wirth, Benedikt |
| author_facet | Wirth, Benedikt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of spectral discretization for the flow of diffeomorphisms Wirth, Benedikt Numerical Analysis 65M12 (Primary) 35Q35, 53Z50, 58D25 (Secondary) 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. |
| title | Convergence of spectral discretization for the flow of diffeomorphisms |
| topic | Numerical Analysis 65M12 (Primary) 35Q35, 53Z50, 58D25 (Secondary) |
| url | https://arxiv.org/abs/2410.06788 |