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Total Generalized Variation for Manifold-valued Data

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VMVW01 - Variational methods, new optimisation techniques and new fast numerical algorithms

Co-authors: Kristian Bredies (University of Graz), Martin Storath (University of Heidelberg), Andreas Weinmann (Darmstadt University of Applied Sciences)

Introduced in 2010, the total generalized variation (TGV) functional is nowadays amongst the most successful regularization functionals for variational image reconstruction. It is defined for an arbitrary order of differentiation and provides a convex model for piecewise smooth vector-space data. On the other hand, variational models for manifold-valued data have become popular recently and many successful approaches, such as first- and second-order TV regularization, have been successfully generalized to this setting. Despite the fact that TGV regularization is, generally, considered to be preferable to such approaches, an appropriate extension for manifold-valued data was still missing. In this talk we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present concrete instances that fulfill the proposed axioms. We prove well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data.

This talk is part of the Isaac Newton Institute Seminar Series series.

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