Lorentzian Iterative Hard Thresholding: Robust Compressed Sensing With Prior Information

• Published in: IEEE transactions on signal processing Vol. 61; no. 19; pp. 4822 - 4833
• Main Authors: Carrillo, Rafael E., Barner, Kenneth E.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Compressed; Compressed sensing; Cost function; Detection, estimation, filtering, equalization, prediction; Economic models; Exact sciences and technology; impulse noise; Information, signal and communications theory; Iterative methods; Noise; nonlinear estimation; Reconstruction; Recovery; sampling methods; Sampling, quantization; Signal and communications theory; Signal processing; signal reconstruction; Signal, noise; Telecommunications and information theory; Performance evaluation; Threshold detection; Gaussian distribution; Iterative method; L2 approximation; nonlinear estimation; Outlier; Least squares method; impulse noise; Cost function; Signal reconstruction; Robustness; Prior information; Non linear estimation; sampling methods; Algorithm; Computational complexity; Pulse noise; Sufficient condition; Signal restoration; Residual impurity; Signal processing; Numerical simulation; Metric; Compressed sensing
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2013.2274275

Commonly employed reconstruction algorithms in compressed sensing (CS) use the L2 norm as the metric for the residual error. However, it is well-known that least squares (LS) based estimators are highly sensitive to outliers present in the measurement vector leading to a poor performance when the noise no longer follows the Gaussian assumption but, instead, is better characterized by heavier-than-Gaussian tailed distributions. In this paper, we propose a robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse signals in the presence of impulsive noise. To address this problem, we use a Lorentzian cost function instead of the L 2 cost function employed by the traditional IHT algorithm. We also modify the algorithm to incorporate prior signal information in the recovery process. Specifically, we study the case of CS with partially known support. The proposed algorithm is a fast method with computational load comparable to the LS based IHT, whilst having the advantage of robustness against heavy-tailed impulsive noise. Sufficient conditions for stability are studied and a reconstruction error bound is derived. We also derive sufficient conditions for stable sparse signal recovery with partially known support. Theoretical analysis shows that including prior support information relaxes the conditions for successful reconstruction. Simulation results demonstrate that the Lorentzian-based IHT algorithm significantly outperform commonly employed sparse reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments. Numerical results also demonstrate that the partially known support inclusion improves the performance of the proposed algorithm, thereby requiring fewer samples to yield an approximate reconstruction.