Compressed sensing recovery with Bayesian approximate message passing using empirical least squares estimation without an explicit prior

Abstract

Compressed Sensing (CS) is a signal processing technique that allows for high-quality reconstruction of a source signal vector of dimension N from a number M N of linear measurements (undersampling!), and Approximate Message Passing (AMP) is a recovery technique that works particularly well at very low complexity. It has been shown that CS recovery in the AMP framework can be seen as recovery of the N independent and identically distributed (iid) components of the source signal in a decoupled measurement model, with a Gaussian noise of a variance that is estimated adaptively during the AMP- iterations; this variance contains the measurement noise as well as extra noise due to undersampling (i.e. M/N1). Hence, CS recovery in the AMP framework boils down to estimation of a signal in Gaussian noise of known variance. The Bayesian version of AMP (BAMP), which has the best performance, seems to require knowledge of the probability density function (pdf) of the signal prior; this may be seen as a major drawback in practice. It is known, however, that observations of a signal corrupted by Gaussian noise can be de-noised without knowing the signal prior explicitly, and the performance can be very close to that of the optimal Bayesian estimator knowing the pdf of the signal components. The contribution of this thesis is the use of a kernel density estimator as an Empirical Bayes Least Squares Estimator in the BAMP recovery framework and to compare the performance with other, "semi-blind" approaches that exploit partial knowledge of the signal prior, e.g. schemes that assume a shape of the prior pdf and optimize for its parameters from the observed data.