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Author: Hintermüller, Michael × Start date: 2019 × Start date: 2011 × DDC: 510 × Subject: Image restoration ×

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Now showing 1 - 4 of 4
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    Adaptive regularization for image reconstruction from subsampled data
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hintermüller, Michael; Langer, Andreas; Rautenberg, Carlos N.; Wu, Tao
    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 [15], 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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    Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Wu, Tao; Langer, Andreas
    Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), 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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    Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.; Sun, Hongpeng
    Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.
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    Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.
    A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.