Abstract
The flow of diffeomorphisms, aka LDDMM, is a framework to define a group G of diffeomorphisms of chosen regularity with a Riemannian structure. If these diffeomorphisms are used to deform a template shape or image, they generate a space of shapes or images to which the Riemannian structure descends. For this reason LDDMM lies at the centre of computational anatomy, shape space theory, and image metamorphosis. Typically, to obtain a geodesic equation on G one formally applies the geodesic equation of Riemannian Lie groups, to which G has structural similarity. Then interpolation tasks between two given deformations, needed for all kinds of statistical analyses, are solved by shooting discretized geodesics within an optimal control approach. If G is chosen with Sobolev regularity, it is known to be a veritable infinite-dimensional Riemannian manifold. In this setting we derive the weak geodesic PDE, which in its strong form coincides with the formally derived one, and present a time discretization to compute a geodesic between given deformations by minimizing a time-discrete path energy, which Mosco-converges to the continuous path energy. This variational ansatz is a more natural alternative to shooting and to our knowledge the first numerical approach to LDDMM by minimization.
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Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 211504053 – Collaborative Research Center 1060 and via project 431460824 – Collaborative Research Center 1450 and via Germany’s Excellence Strategy project 390685813 – Hausdorff Center for Mathematics and project 390685587 – Mathematics Münster: Dynamics-Geometry-Structure.
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Guastini, M., Rajković, M., Rumpf, M., Wirth, B. (2023). The Variational Approach to the Flow of Sobolev-Diffeomorphisms Model. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_42
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