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Non-smooth Variational Regularization for Processing Manifold-Valued Data

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

Many methods for processing scalar and vector valued images, volumes and other data in the context of inverse problems are based on variational formulations. Such formulations require appropriate regularization functionals that model expected properties of the object to reconstruct. Prominent examples of regularization functionals in a vector-space context are the total variation (TV) and the Mumford-Shah functional, as well as higher-order schemes such as total generalized variation models. Driven by applications where the signals or data live in nonlinear manifolds, there has been quite some interest in developing analogous methods for nonlinear, manifold-valued data recently. In this chapter, we consider various variational regularization methods for manifold-valued data. In particular, we consider TV minimization as well as higher order models such as total generalized variation (TGV). Also, we discuss (discrete) Mumford-Shah models and related methods for piecewise constant data. We develop discrete energies for denoising and report on algorithmic approaches to minimize them. Further, we also deal with the extension of such methods to incorporate indirect measurement terms, thus addressing the inverse problem setup. Finally, we discuss wavelet sparse regularization for manifold-valued data.

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Acknowledgements

MH acknowledges support by the Austrian Science Fund (FWF) (Grant J 4112). AW acknowledges support by the DFG Grants WE 5886/3-1 and WE 5886/4-1.

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Authors and Affiliations

  1. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  2. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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  2. Andreas Weinmann
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Correspondence to Martin Holler .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Holler, M., Weinmann, A. (2020). Non-smooth Variational Regularization for Processing Manifold-Valued Data. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_2

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