From Denoising to Compressed Sensing

TL;DR

This paper introduces D-AMP, a novel algorithm that integrates advanced denoisers into compressed sensing reconstruction, achieving state-of-the-art results with high speed and theoretical support.

Contribution

It extends the AMP framework to incorporate generic denoisers, enabling improved CS recovery performance and efficiency.

Findings

D-AMP achieves superior reconstruction quality on natural images.

D-AMP operates tens of times faster than competing methods.

Theoretical analysis explains the effectiveness of the Onsager correction in D-AMP.

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, today's 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.