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A Guide to the TV Zoo

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Level Set and PDE Based Reconstruction Methods in Imaging

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2090))

Abstract

Total variation methods and similar approaches based on regularizations with ℓ 1-type norms (and seminorms) have become a very popular tool in image processing and inverse problems due to peculiar features that cannot be realized with smooth regularizations. In particular total variation techniques had particular success due to their ability to realize cartoon-type reconstructions with sharp edges. Due to an explosion of new developments in this field within the last decade it is a difficult task to keep an overview of the major results in analysis, the computational schemes, and the application fields. With these lectures we attempt to provide such an overview, of course biased by our major lines of research. We are focusing on the basic analysis of total variation methods and the extension of the original ROF-denoising model due various application fields. Furthermore we provide a brief discussion of state-of-the art computational methods and give an outlook to applications in different disciplines.

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Acknowledgements

The work of MB has been supported by the German Science Foundation DFG through the project Regularization with Singular Energies and SFB 656 Molecular Cardiovascular Imaging, and by the German Ministery of Education and Research (BMBF) through the project INVERS: Deconvolution problems with sparsity constraints in nanoscopy and mass spectrometry.

The work of SO has been supported by NSF grants DMS0835863 and DMS0914561, ONR grant N000140910360 and ARO MURI subs from Rice University and the University of North Carolina.

The authors thank Alex Sawatzky (WWU Münster) for careful proofreading and comments to improve the manuscript.

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  1. Institute for Computational and Applied Mathematics, University of Münster, Münster, Germany

    Martin Burger

  2. Department of Mathematics, UCLA, Los Angeles, USA

    Stanley Osher

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Burger, M., Osher, S. (2013). A Guide to the TV Zoo. In: Level Set and PDE Based Reconstruction Methods in Imaging. Lecture Notes in Mathematics(), vol 2090. Springer, Cham. https://doi.org/10.1007/978-3-319-01712-9_1

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