From Denoising to Compressed Sensing

Abstract

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 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.1 Note that for notational simplicity in our current derivations and algorithms we restrict A to be in R mˆn . However, an extension to C mˆn is also possible.2 A QQplot is a visual inspection tool for checking the Gaussianity of the data. In a QQplot, deviation from a straight line is an evidence of non-Gaussianity.

Compressed quantitative acoustic microscopy

“…Given x = 0 and z = y as an initial condition, the algorithm iterates sequentially (2) and (3) until satisfying a stopping criterion or reaching a pre-set iteration number. The last term of the right hand side in (3) is referred to as the Onsager reaction term and serves at balancing the sparsity-undersampling tradeoff [9], [10]. An extended wavelet-based AMP system can be generated by integrating a wavelet transform (denoted by W ) into (2) and (3) using the following transformation.…”

“…P v|w (v|w) ∼ N (0, σ 2 ), and using Bayes' rule, the MAP estimator is given in (10). Note that (10) is obtained using the log-posterior and ignoring the evidence P v (v) which is constant for all inputs.…”

Statistical estimation: from denoising to sparse regression and hidden cliques

“…Indeed similar ideas were developed and applied to a large number of problems, see [49,49,50,7,51,52,53,54] for a a very incomplete list of examples.…”

Complex domain nonlocal block-matching denoising based on high-order singular value decomposition (HOSVD)

“…Even more, a denoiser as a component of the algorithm appears as a solution in many variational setups (e.g. [9], [10]). …”