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A Smoothing Descent Method for Nonconvex TV\(^q\)-Models

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Efficient Algorithms for Global Optimization Methods in Computer Vision

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8293))

Abstract

A novel class of variational models with nonconvex \(\ell _q\)-norm-type regularizations (\(0<q<1\)) is considered, which typically outperforms popular models with convex regularizations in restoring sparse images. Due to the fact that the objective function is nonconvex and non-Lipschitz, such models are very challenging from an analytical as well as numerical point of view. In this work a smoothing descent method with provable convergence properties is proposed for computing stationary points of the underlying variational problem. Numerical experiments are reported to illustrate the effectiveness of the new method.

This research was supported by the Austrian Science Fund (FWF) through START project Y305 “Interfaces and Free Boundaries” and through SFB project F3204 “Mathematical Optimization and Applications in Biomedical Sciences”.

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Acknowledgement

The authors would like to thank Dr. Florian Knoll for contributing his data and codes to our experiments on MRI.

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Authors and Affiliations

  1. Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller

  2. Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Tao Wu

Authors
  1. Michael Hintermüller
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  2. Tao Wu
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Corresponding author

Correspondence to Michael Hintermüller .

Editor information

Editors and Affiliations

  1. University of Stuttgart, Stuttgart, Germany

    Andrés Bruhn

  2. Graz University of Technology, Graz, Austria

    Thomas Pock

  3. University of Bergen, Bergen, Norway

    Xue-Cheng Tai

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Hintermüller, M., Wu, T. (2014). A Smoothing Descent Method for Nonconvex TV\(^q\)-Models. In: Bruhn, A., Pock, T., Tai, XC. (eds) Efficient Algorithms for Global Optimization Methods in Computer Vision. Lecture Notes in Computer Science(), vol 8293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54774-4_6

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  • DOI: https://doi.org/10.1007/978-3-642-54774-4_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54773-7

  • Online ISBN: 978-3-642-54774-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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