Optimal Quantization for Sparse Reconstruction with Relaxed Belief Propagation

Compressive sensing theory has demonstrated that sparse signals can be recovered from a small number of random linear measurements. However, for practical purposes, like storage, transmission, or processing with modern digital equipment, continuous-valued compressive sensing measurements need to be quantized. In this thesis we examine the topic of optimal quantization of compressive sensing measurements under reconstruction with messagepassing algorithms by following the work on generalization of relaxed belief propagation (BP) for arbitrary measurement channels. Relaxed BP is an iterative reconstruction algorithm proposed for the task of estimation from random linear measurements. It was inspired by the traditional belief propagation algorithm widely used in decoding of low-density parity-check (LDPC) codes. One of the aspects that makes relaxed belief propagation so appealing is the state evolution framework, which predicts asymptotic error behavior of the algorithm. We utilize the predictive capability of the framework to design mean-square optimal scalar quantizers under relaxed BP signal reconstruction. We demonstrate that error performance of the reconstruction can be significantly improved by using state evolution optimized quantizers, compared to quantizers obtained via traditional design schemes. We finally propose relaxed BP as a practical algorithm for reconstruction from measurements digitized with binned quantizers, which further improve error performance of the reconstruction.