Signal recovery method for compressive sensing using relaxation and second-order cone programming

Abstract

A signal recovery method for compressive sensing under noisy measurements is proposed. The problem is formulated as a nonconvex nonsmooth constrained optimization problem that uses the smoothly clipped absolute deviation (SCAD) function to promote sparsity. Relaxation is employed by means of a series of local linear approximations (LLAs) of the SCAD in a constrained formulation. The relaxation is shown to converge to a minimum of the original nonconvex constrained optimization problem. In order to solve each nonsmooth convex relaxation problem, a second-order cone programming (SOCP) formulation is used, which can be applied by using standard state-of-the-art SOCP solvers such as SeDuMi. Experimental results demonstrate that signals recovered using the proposed method exhibit reduced ℓ_∞ reconstruction error when compared with competing methods such as ℓ₁ -Magic. Simulations demonstrate that significant reduction in the reconstruction error can be achieved with computational cost that is comparable to that required by the ℓ₁ -Magic algorithm.

Full text

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Compressive SensingSparse-Signal RecoveryProposed MethodNumerical SimulationsConclusions
Signal Recovery Method for Compressive Sensing Using
Relaxation and Second-Order Cone Programming
Fl´ avio C. A. TeixeiraStuart W. A. BergenAndreas Antoniou
Department of Electrical and Computer Engineering
University of Victoria
Victoria, BC, Canada
2011 IEEE International Symposium on Circuits and Systems
F. Teixeira, S. Bergen, A. Antoniou (UVic)A Signal Recovery Method for CSIEEE ISCAS 20111 / 16

Page 2

Compressive SensingSparse-Signal RecoveryProposed MethodNumerical SimulationsConclusions
Main idea
Introduction
Compressive sensing (CS) is a process of representing a large signal by
a small number of measurements.
The price that must be paid for compact signal representation is a
nontrivial signal recovery process.
The recovery process can be formulated as an undetermined
least-squares problem where the solution is known to be sparse.
The solution sparsity assumption is based on the fact that most
practical signals can be represented concisely in a transform
domain.
F. Teixeira, S. Bergen, A. Antoniou (UVic)A Signal Recovery Method for CSIEEE ISCAS 20112 / 16

Page 3

Compressive SensingSparse-Signal RecoveryProposed MethodNumerical SimulationsConclusions
Main idea
Introduction
Compressive sensing (CS) is a process of representing a large signal by
a small number of measurements.
The price that must be paid for compact signal representation is a
nontrivial signal recovery process.
The recovery process can be formulated as an undetermined
least-squares problem where the solution is known to be sparse.
The solution sparsity assumption is based on the fact that most
practical signals can be represented concisely in a transform
domain.
F. Teixeira, S. Bergen, A. Antoniou (UVic)A Signal Recovery Method for CSIEEE ISCAS 20112 / 16

Page 4

Compressive SensingSparse-Signal RecoveryProposed MethodNumerical SimulationsConclusions
Main idea
Introduction
Compressive sensing (CS) is a process of representing a large signal by
a small number of measurements.
The price that must be paid for compact signal representation is a
nontrivial signal recovery process.
The recovery process can be formulated as an undetermined
least-squares problem where the solution is known to be sparse.
The solution sparsity assumption is based on the fact that most
practical signals can be represented concisely in a transform
domain.
F. Teixeira, S. Bergen, A. Antoniou (UVic) A Signal Recovery Method for CSIEEE ISCAS 20112 / 16

Page 5

Compressive SensingSparse-Signal RecoveryProposed MethodNumerical Simulations Conclusions
Main idea
Introduction
Compressive sensing (CS) is a process of representing a large signal by
a small number of measurements.
The price that must be paid for compact signal representation is a
nontrivial signal recovery process.
The recovery process can be formulated as an undetermined
least-squares problem where the solution is known to be sparse.
The solution sparsity assumption is based on the fact that most
practical signals can be represented concisely in a transform
domain.
F. Teixeira, S. Bergen, A. Antoniou (UVic)A Signal Recovery Method for CSIEEE ISCAS 20112 / 16