Tuesday, June 24, 2014

From Denoising to Compressed Sensing

 Rich just sent me the following:

hi igor -

we have been working on integrating advanced denoising algorithms into compressive sensing recovery algorithms and have found that approximate message passing (AMP) provides a flexible platform that can support a variety of denoisers since the Onsager correction Gaussianizes the error at each iteration. for image recovery, coupling AMP with the BM3D denoiser and the correct Onsager correction offers state-of-the-art performance. our preprint has a range of comparisons with other approaches and also an in depth analysis of the approach. i thought it might be interesting to your audience. thanks!

Richard G. Baraniuk
Victor E. Cameron Professor of Electrical and Computer Engineering
Founder and Director, Connexions and OpenStax College
Rice University 

Sure Rich.

A denoising algorithm seeks to remove perturbations or errors from a signal. The last three decades have seen extensive research devoted to this arena, and as a result, today's denoisers are highly optimized algorithms that effectively remove large amounts of additive white Gaussian noise. A compressive 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, in this paper, we develop a denoising-based approximate message passing (D-AMP) algorithm that is capable of high-performance reconstruction. We demonstrate that, for an appropriate choice of denoiser, D-AMP offers state-of-the-art CS recovery performance for natural images. We explain the exceptional performance of D-AMP by analyzing some of its theoretical features. A critical insight in our approach is the use of an appropriate Onsager correction term in the D-AMP 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.
( Update: Implementation is here )
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