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Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty

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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

We study and extend the recently introduced total generalized variation (TGV) functional for multichannel images. This functional has already been established to constitute a well-suited convex model for piecewise smooth scalar images. It comprises exactly the functions of bounded variation but is, unlike purely total-variation based functionals, also aware of higher-order smoothness. For the multichannel version which is developed in this paper, basic properties and existence of minimizers for associated variational problems regularized with second-order TGV is shown. Furthermore, we address the design of numerical solution methods for the minimization of functionals with TGV\(^2\) penalty and present, in particular, a class of primal-dual algorithms. Finally, the concrete realization for various image processing problems, such as image denoising, deblurring, zooming, dequantization and compressive imaging, are discussed and numerical experiments are presented.

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Acknowledgements

Support by the Austrian Science Fund (FWF) under grant SFB F32 (SFB “Mathematical Optimization and Applications in Biomedical Sciences”) is gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010, Graz, Austria

    Kristian Bredies

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  1. Kristian Bredies
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Correspondence to Kristian Bredies .

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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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Bredies, K. (2014). Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty. 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_3

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  • DOI: https://doi.org/10.1007/978-3-642-54774-4_3

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  • Print ISBN: 978-3-642-54773-7

  • Online ISBN: 978-3-642-54774-4

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