Nonsmooth Optimization on Riemannian Manifolds and Manifold-Valued Data Processing

Ronny Bergmann
Technische Universität Chemnitz

In many real life scenarios measured data appears as values from a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on a equispaced grid like usual signals and image but they also suffer from the same measurement errors like presence of noise or incompleteness. Hence there is a need to perform data processing tasks like denoising, inpainting or even interpolation on these manifold-valued data.

In this talk we present variational methods and nonsmooth optimization
algorithms for processing manifold-valued data. For the first, we present first and second order difference based priors and their application to the tasks from image processing. To compute minimizers of the variational methods, we present algorithms to efficiently solve nonsmooth optimization tasks on manifolds.

Presentation (PDF File)

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