Journal of Scientific Computing

, Volume 74, Issue 2, pp 851–871 | Cite as

A Domain Decomposition Fourier Continuation Method for Enhanced \(L_1\) Regularization Using Sparsity of Edges in Reconstructing Fourier Data

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Abstract

\(L_1\) regularization is widely used in various applications for sparsifying transform. In Wasserman et al. (J Sci Comput 65(2):533–552, 2015) the reconstruction of Fourier data with \(L_1\) minimization using sparsity of edges was proposed—the sparse PA method. With the sparse PA method, the given Fourier data are reconstructed on a uniform grid through the convex optimization based on the \(L_1\) regularization of the jump function. In this paper, based on the method proposed by Wasserman et al. (J Sci Comput 65(2):533–552, 2015) we propose to use the domain decomposition method to further enhance the quality of the sparse PA method. The main motivation of this paper is to minimize the global effect of strong edges in \(L_1\) regularization that the reconstructed function near weak edges does not benefit from the sparse PA method. For this, we split the given domain into several subdomains and apply \(L_1\) regularization in each subdomain separately. The split function is not necessarily periodic, so we adopt the Fourier continuation method in each subdomain to find the Fourier coefficients defined in the subdomain that are consistent to the given global Fourier data. The numerical results show that the proposed domain decomposition method yields sharp reconstructions near both strong and weak edges. The proposed method is suitable when the reconstruction is required only locally.

Keywords

Fourier reconstruction Gibbs phenomenon Convex optimization with \(L_1\) minimization Sparsity Domain decomposition Fourier continuation 

Notes

Acknowledgements

The authors thank Anne Gelb for introducing the sparse PA method and programming and her helpful comments on the manuscript. The authors also thank Wai-Sun Don for helping them to understand and implement the Fourier continuation method.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity at Buffalo, The State University of New YorkBuffaloUSA

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