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Adaptive Regularization for Image Reconstruction from Subsampled Data

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Imaging, Vision and Learning Based on Optimization and PDEs (IVLOPDE 2016)

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

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Abstract

Choices of regularization parameters are central to variational methods for image restoration . In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in Dong et al. (J Math Imaging Vis 40:82–104, 2011), which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.

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Acknowledgements

This research was supported by the Austrian Science Fund (FWF) through START-Project Y305 “Interfaces and Free Boundaries” and SFB-Project F3204 “Mathematical Optimization and Applications in Biomedical Sciences”, the German Research Foundation DFG through Project HI1466/7-1 “Free Boundary Problems and Level Set Methods”, as well as the Research Center MATHEON through Project C-SE15 “Optimal Network Sensor Placement for Energy Efficiency” supported by the Einstein Center for Mathematics Berlin.

Author information

Authors and Affiliations

  1. Weierstrass Institute, Berlin, Germany

    Michael Hintermüller

  2. Universität Stuttgart, Stuttgart, Germany

    Andreas Langer

  3. Humboldt-Universität zu Berlin, Berlin, Germany

    Carlos N. Rautenberg

  4. Technische Universität München, Garching, Germany

    Tao Wu

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

Correspondence to Michael Hintermüller .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Xue-Cheng Tai

  2. Norwegian Defence Research Establishment, Kjeller, Norway

    Egil Bae

  3. University of South-Eastern Norway, Porsgrunn, Norway

    Marius Lysaker

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Hintermüller, M., Langer, A., Rautenberg, C.N., Wu, T. (2018). Adaptive Regularization for Image Reconstruction from Subsampled Data. In: Tai, XC., Bae, E., Lysaker, M. (eds) Imaging, Vision and Learning Based on Optimization and PDEs. IVLOPDE 2016. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-91274-5_1

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