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Structural Similarity-Based Optimization Problems with \(L^1\)-Regularization: Smoothing Using Mollifiers

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Image Analysis and Recognition (ICIAR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9164))

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Abstract

In this paper we propose a new method of solving optimization problems involving the structural similarity image quality measure with \(L^1\)-regularization. The regularization term \(\Vert x \Vert _1\) is approximated by a sequence of smooth functions \(\Vert x \Vert _1^\varepsilon \) by means of \(C^\infty _0\) functions known as mollifiers. Because the functions \(\Vert x \Vert _1^\varepsilon \) epi-converge to \(\Vert x \Vert _1\), the sequence of minimizers of the smooth objective functions converges to a minimizer of the non-smooth problem. This approach permits the use of gradient-based methods to solve the minimization problems as opposed to methods based on subdifferentials.

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Acknowledgements

We gratefully acknowledge that this research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant (ERV).

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Daniel Otero & Edward R. Vrscay

  2. Department of Economics, Management, and Quantitative Methods, University of Milan, Milan, Italy

    Davide La Torre

  3. Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi, UAE

    Davide La Torre

Authors
  1. Daniel Otero
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  2. Davide La Torre
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  3. Edward R. Vrscay
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Corresponding author

Correspondence to Daniel Otero .

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Editors and Affiliations

  1. University of Waterloo, Waterloo, Ontario, Canada

    Mohamed Kamel

  2. University of Porto, Porto, Portugal

    Aurélio Campilho

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Otero, D., La Torre, D., Vrscay, E.R. (2015). Structural Similarity-Based Optimization Problems with \(L^1\)-Regularization: Smoothing Using Mollifiers. In: Kamel, M., Campilho, A. (eds) Image Analysis and Recognition. ICIAR 2015. Lecture Notes in Computer Science(), vol 9164. Springer, Cham. https://doi.org/10.1007/978-3-319-20801-5_4

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  • DOI: https://doi.org/10.1007/978-3-319-20801-5_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20800-8

  • Online ISBN: 978-3-319-20801-5

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