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

Liang Ding,Hui Zhao,Yixin Dou

doi:10.1155/2017/2736306

Abstract

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 l1+l1 optimization functional. We show that one can overcome the nondifferentiability and instability by adding a smooth l2 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.

Highlights

In the present manuscript we are concerned with ill-posed linear operator equation: Ax = y, (1)where x is sparse with respect to an orthonormal basis and A : D(A) ⊂ X → Y is a bounded linear operator
Though the optimal relative error of ADMl1 is better than dual spectral projected gradient (DSPG) method, the corresponding optimal iteration number or stopping tolerance of alternating direction method (ADM)-l1 is difficult to estimate in practice
With source and finite basis injectivity (FBI) conditions, we have proved that l1 + l1 regularization method yields convergence rates of order δ1−휖 and δ

Summary

Introduction

Reference [21] discussed the prior sparse representation and the data-fidelity term and proposed the two-phase approach. In [19], Yang and Zhang proposed a Primal Dual-Interior Point Methods (PD-IPM) for EIT problem, which is efficient at dealing with the nondifferentiability. They did not give the convergence proof. Yang and Zhang reformulated the l1 + l1 problem into the basis pursuit model which can be solved effectively by ADM method [18]. It is a competitive method compared with other algorithms for compressive sensing. Numerical experiments involving compressed sensing and image inpainting are presented in Section 6, showing that our proposed approaches are robust and efficient

Regularization Properties

Dual Problem

Computation of Minimizers

Choice of Parameter α and β

Numerical Simulations

Conclusion