A domain decomposition Fourier continuation method for enhanced L_1 regularization using the sparsity of edges in reconstructing Fourier data

Abstract

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 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 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 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 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.