Smoothing and Decomposition for Analysis Sparse Recovery

• Published in: IEEE transactions on signal processing Vol. 62; no. 7; pp. 1762 - 1774
• Main Authors: Tan, Zhao, Eldar, Yonina C., Beck, Amir, Nehorai, Arye
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Analysis model; Applied sciences; Biological and medical sciences; Computerized, statistical medical data processing and models in biomedicine; Convergence analysis; Decomposition; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Exact solutions; fast iterative shrinkage-thresholding algorithm; Frames; Image processing; Information, signal and communications theory; Iterative methods; Mathematical models; Medical management aid. Diagnosis aid; Medical sciences; Reconstruction; Recovery; restricted isometry property; Signal and communications theory; Signal processing; Signal, noise; Smoothing; smoothing and decomposition; sparse recovery; Telecommunications and information theory; Performance evaluation; Isometry; Image processing; Threshold detection; fast iterative shrinkage-thresholding algorithm; Iterative method; sparse recovery; Closed form equation; Algorithm; Nuclear magnetic resonance imaging; Implementation; Optimization; Image reconstruction; Convergence speed; convergence analysis; restricted isometry property; Analysis model; Convergence rate; Signal processing; Medical imagery; Numerical simulation; Signal reconstruction; smoothing and decomposition
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2014.2304932

We consider algorithms and recovery guarantees for the analysis sparse model in which the signal is sparse with respect to a highly coherent frame. We consider the use of a monotone version of the fast iterative shrinkage-thresholding algorithm (MFISTA) to solve the analysis sparse recovery problem. Since the proximal operator in MFISTA does not have a closed-form solution for the analysis model, it cannot be applied directly. Instead, we examine two alternatives based on smoothing and decomposition transformations that relax the original sparse recovery problem, and then implement MFISTA on the relaxed formulation. We refer to these two methods as smoothing-based and decomposition-based MFISTA. We analyze the convergence of both algorithms and establish that smoothing-based MFISTA converges more rapidly when applied to general nonsmooth optimization problems. We then derive a performance bound on the reconstruction error using these techniques. The bound proves that our methods can recover a signal sparse in a redundant tight frame when the measurement matrix satisfies a properly adapted restricted isometry property. Numerical examples demonstrate the performance of our methods and show that smoothing-based MFISTA converges faster than the decomposition-based alternative in real applications, such as MRI image reconstruction.