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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 2736306, 17 pages
Research Article

Sparse Signal Inversion with Impulsive Noise by Dual Spectral Projected Gradient Method

1Department of Mathematics, Northeast Forestry University, No. 26 Hexing Street, Xiangfang District, Harbin, China
2School of Economics and Finance, Harbin University of Commerce, No. 1 Xuehai Street, Songbei District, Harbin, China

Correspondence should be addressed to Liang Ding

Received 20 April 2017; Accepted 24 July 2017; Published 14 September 2017

Academic Editor: Haranath Kar

Copyright © 2017 Liang Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider sparse signal inversion with impulsive noise. There are three major ingredients. The first is regularizing properties; we discuss convergence rate of regularized solutions. The second is devoted to the numerical solutions. It is challenging due to the fact that both fidelity and regularization term lack differentiability. Moreover, for ill-conditioned problems, sparsity regularization is often unstable. We propose a novel dual spectral projected gradient (DSPG) method which combines the dual problem of multiparameter regularization with spectral projection gradient method to solve the nonsmooth optimization functional. We show that one can overcome the nondifferentiability and instability by adding a smooth regularization term to the original optimization functional. The advantage of the proposed functional is that its convex duality reduced to a constraint smooth functional. Moreover, it is stable even for ill-conditioned problems. Spectral projected gradient algorithm is used to compute the minimizers and we prove the convergence. The third is numerical simulation. Some experiments are performed, using compressed sensing and image inpainting, to demonstrate the efficiency of the proposed approach.