TY - JOUR

T1 - Convex feasibility modeling and projection methods for sparse signal recovery

AU - Carmi, Avishy

AU - Censor, Yair

AU - Gurfil, Pini

N1 - Funding Information:
The work of Y. Censor is supported by Grant No. 2009012 from the United States–Israel Binational Science Foundation (BSF) and by US Department of Army award number W81XWH-10-1-0170.

PY - 2012/11/1

Y1 - 2012/11/1

N2 - A computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibility-seeking sequential and simultaneous subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonly-used CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFP-based projection methods are superior to some of the state-of-the-art methods in recovering the signal's support. Numerical experiments show that the CFP-based projection methods are viable for solving large-scale CS problems with compressible signals.

AB - A computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibility-seeking sequential and simultaneous subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonly-used CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFP-based projection methods are superior to some of the state-of-the-art methods in recovering the signal's support. Numerical experiments show that the CFP-based projection methods are viable for solving large-scale CS problems with compressible signals.

KW - Compressed sensing

KW - Convex feasibility problems

KW - Signal processing

KW - Subgradient projection methods

UR - http://www.scopus.com/inward/record.url?scp=84862871914&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2012.03.021

DO - 10.1016/j.cam.2012.03.021

M3 - Article

AN - SCOPUS:84862871914

SN - 0377-0427

VL - 236

SP - 4318

EP - 4335

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 17

ER -