Abstract
Morphing of Manifold-Valued Images inspired by Discrete Geodesics in Image Spaces
Sebastian Neumayer
Universität Kaiserslautern
Morphing of images is a fundamental challenge in image processing. The (time-continuous) metamorphosis model originally proposed by Trouvé, Younes, and coworkers is a special form of morphing, where the space of images is endowed with a Riemannian metric incorporating contributions due to transport and variations of image intensities along paths in the space of images.
A variational time discretization of the metamorphosis energy functional and the Mosco--convergence of the time-discrete to the time-continuous metamorphosis model for square-integrable images in Euclidean space are analyzed by Berkels, Effland, and Rumpf. This talk addresses the morphing of manifold-valued images based on a time-discrete geodesic path model.
We prove the existence of a minimizing sequence within the \( L^2 \) space of images taking values in a finite-dimensional Hadamard manifold, together with a minimizing sequence of admissible diffeomorphisms. We introduce a novel time-continuous metamorphosis energy functional for images on Hadamard manifolds, which coincides with the original energy functional in the Euclidean case, and prove Mosco--convergence of the energy functionals on Hadamard manifolds.
We propose a space-discrete model based on a finite difference approach on staggered grids, focusing on the linearized elastic potential in the regularizing term. The numerical minimization alternates between (i) the computation of a deformation sequence between given images via the parallel solution of certain registration problems for manifold-valued images, and (ii) the computation of an image sequence with fixed first (template) and last (reference) frame based on a given sequence of deformations via the solution of a system of equations arising from the corresponding Euler–Lagrange equation.
Numerical examples provide a proof of concept of our ideas.
Joint work with A. Effland (U Graz), J. Persch (Zeiss AG), G. Steidl (TU Kaiserslautern), and M. Rumpf (U Bonn).
A variational time discretization of the metamorphosis energy functional and the Mosco--convergence of the time-discrete to the time-continuous metamorphosis model for square-integrable images in Euclidean space are analyzed by Berkels, Effland, and Rumpf. This talk addresses the morphing of manifold-valued images based on a time-discrete geodesic path model.
We prove the existence of a minimizing sequence within the \( L^2 \) space of images taking values in a finite-dimensional Hadamard manifold, together with a minimizing sequence of admissible diffeomorphisms. We introduce a novel time-continuous metamorphosis energy functional for images on Hadamard manifolds, which coincides with the original energy functional in the Euclidean case, and prove Mosco--convergence of the energy functionals on Hadamard manifolds.
We propose a space-discrete model based on a finite difference approach on staggered grids, focusing on the linearized elastic potential in the regularizing term. The numerical minimization alternates between (i) the computation of a deformation sequence between given images via the parallel solution of certain registration problems for manifold-valued images, and (ii) the computation of an image sequence with fixed first (template) and last (reference) frame based on a given sequence of deformations via the solution of a system of equations arising from the corresponding Euler–Lagrange equation.
Numerical examples provide a proof of concept of our ideas.
Joint work with A. Effland (U Graz), J. Persch (Zeiss AG), G. Steidl (TU Kaiserslautern), and M. Rumpf (U Bonn).