Quantization of compressed sensing measurements using Analysis-by-Synthesis with Bayesian-optimal Approximate Message Passing

Compressed sensing allows for stable reconstruction of sparse source vectors from noisy, linear measurement vectors of much lower dimension than the source vectors. In many applications, low-bit rate quantization is unavoidable or even desired in further processing of the signal, and suitable algorithms need to be developed for minimizing negative effects on the recovered source signal due to the quantization of the measurements. We present an Analysis-by-Synthesis (AbS) quantization scheme in which, as a novelty, Bayesian-optimal Approximate Message Passing (BAMP) is used as a reconstruction algorithm. The focus is on source signals that can be modeled by a linear combination of a discrete component and a zero-mean Gaussian component; for those signals suitable estimation functions are given for use in the BAMP algorithm. We investigate different setups of the AbS scheme with BAMP and compare the results with an AbS scheme known from the literature, in which Orthogonal Matching Pursuit is used as the reconstruction algorithm.