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Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests

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Abstract

Based on the weighted total variation model and its analysis pursued in Hintermüller and Rautenberg 2016, in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.

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Author information

Authors and Affiliations

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

    Michael Hintermüller, Carlos N. Rautenberg & Tao Wu

  2. Department of Mathematics, University of Stuttgart, Pfaffenwaldring 57/8.345, 70569, Stuttgart, Germany

    Andreas Langer

Authors
  1. Michael Hintermüller
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  2. Carlos N. Rautenberg
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  3. Tao Wu
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  4. Andreas Langer
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Corresponding author

Correspondence to Michael Hintermüller.

Additional information

This research was carried out in the framework of Matheon supported by the Einstein Foundation Berlin within the ECMath projects OT1, SE5 and SE15 as well as by the DFG under Grant No. HI 1466/7-1 “Free Boundary Problems and Level Set Methods”.

A. Langer is listed as a co-author as he was involved in early numerical tests prior to writing this paper. In particular, he found the discretization of the \(\nabla \circ {\text {div}}\)-operator of [15] suitable for the present context, performed numerical tests concerning the choice of the upper level objective and the initial choice of \(\alpha =2.5~\times ~10^{-3}\) when solving the bilevel problem. He also provided the original source images used in Figs. 6 and 8.

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Hintermüller, M., Rautenberg, C.N., Wu, T. et al. Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests. J Math Imaging Vis 59, 515–533 (2017). https://doi.org/10.1007/s10851-017-0736-2

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