From Denoising to Compressed Sensing

• Published in: IEEE transactions on information theory Vol. 62; no. 9; pp. 5117 - 5144
• Main Authors: Metzler, Christopher A., Maleki, Arian, Baraniuk, Richard G.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-09-2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; approximate message passing; Approximation; Approximation algorithms; Compressed sensing; denoiser; Detection; Electronics; Error correction & detection; Gaussian; Iterative methods; Message passing; Noise; Noise control; Noise reduction; Normal distribution; Onsager correction; Perturbation methods; Reconstruction; Reconstruction algorithms; Signal processing; Signal processing algorithms
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2016.2556683

A denoising algorithm seeks to remove noise, errors, or perturbations from a signal. Extensive research has been devoted to this arena over the last several decades, and as a result, todays denoisers can effectively remove large amounts of additive white Gaussian noise. A compressed sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, we develop an extension of the approximate message passing (AMP) framework, called denoising-based AMP (D-AMP), that can integrate a wide class of denoisers within its iterations. We demonstrate that, when used with a high-performance denoiser for natural images, D-AMP offers the state-of-the-art CS recovery performance while operating tens of times faster than competing methods. We explain the exceptional performance of D-AMP by analyzing some of its theoretical features. A key element in D-AMP is the use of an appropriate Onsager correction term in its iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove.