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A Shearlet-Based Fast Thresholded Landweber Algorithm for Deconvolution

by P. Grohs and U. Wiesmann and Z. Kereta

(Report number 2014-09)

Abstract
Image deconvolution is an important problem which has seen plenty of progress in the last decades. Due to its ill-posedness, a common approach is to formulate the reconstruction as an optimisation problem, regularised by an additional sparsity-enforcing term. This term is often modeled as an \(\ell_1\) norm measured in the domain of a suitable signal transform. The resulting optimisation problem can be solved by an iterative approach via Landweber iterations with soft thresholding of the transform coefficients. Previous approaches focused on thresholding in the wavelet-domain. In particular, recent work [1] has shown that the use of Shannon wavelets results in particularly efficient reconstruction algorithms. The present paper extends this approach to Shannon shearlets, which we also introduce in this work. We show that for anisotropic blurring filters, such as the motion blur, the novel shearlet-based approach allows for further improvement in efficiency. In particular, we observe that for such kernels using shearlets instead of wavelets improves the quality of image restoration and SERG when compared after the same number of iterations.

Keywords: Shannon Shearlets, Image Deconvolution, Fast Thresholded Landweber Method

@Techreport{GWK14_559,
  author = {P. Grohs and U. Wiesmann and Z. Kereta},
  title = {A Shearlet-Based Fast Thresholded Landweber Algorithm for Deconvolution},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-09.pdf },
  year = {2014}
}

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