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Regularization of Inverse Problems via Time Discrete Geodesics in Image Spaces

Banff International Research Station for Mathematical Innovation and Discovery

This talk addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model of Berkels, Effland and Rumpf with a variational model, actually the L 2 -T V model, for image restoration. We prove that the space continuous model has a minimizer and propose a minimization procedure which alternates over the involved sequences of deformations and images. The minimization with respect to the image sequence exploits recent algorithms from convex analysis to minimize the L 2 -T V functional. For the numerical computation we apply a finite difference approach on staggered grids together with a multilevel strategy. We present proof-of-the-concept numerical results for sparse and limited angle computerized tomography as well as for superresolution demonstrating the power of the method. Further we apply the morphing approach for image colorization. This is joint work with Sebastian Neumayer and Johannes Persch (TU Kaiserslautern).

Mathematics; Functional analysis; Calculus of variations and optimal control; optimization; Numerical analysis

video/mp4

Author affiliation: Technical University of Kaiserslautern

2019-12-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72914

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/