Hamming Compressed Sensing

Compressed sensing (CS) and 1-bit CS cannot directly recover quantized signals and require time consuming recovery. In this paper, we introduce \textit{Hamming compressed sensing} (HCS) that directly recovers a k-bit quantized signal of dimensional $n$ from its 1-bit measurements via invoking $n$ times of Kullback-Leibler divergence based nearest neighbor search. Compared with CS and 1-bit CS, HCS allows the signal to be dense, takes considerably less (linear) recovery time and requires substantially less measurements ($\mathcal O(\log n)$). Moreover, HCS recovery can accelerate the subsequent 1-bit CS dequantizer. We study a quantized recovery error bound of HCS for general signals and "HCS+dequantizer" recovery error bound for sparse signals. Extensive numerical simulations verify the appealing accuracy, robustness, efficiency and consistency of HCS.Comment: 33 pages, 8 figure

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($\Sigma\Delta$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing and Its Applications", 201

Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.Comment: Keywords: Quantized compressed sensing, quantization, consistency, error decay, low-rank, sparsity. 10 pages, 3 figures. Note abbout this version: title change, typo corrections, clarification of the context, adding a comparison with BPD

Variational Bayesian algorithm for quantized compressed sensing

Compressed sensing (CS) is on recovery of high dimensional signals from their low dimensional linear measurements under a sparsity prior and digital quantization of the measurement data is inevitable in practical implementation of CS algorithms. In the existing literature, the quantization error is modeled typically as additive noise and the multi-bit and 1-bit quantized CS problems are dealt with separately using different treatments and procedures. In this paper, a novel variational Bayesian inference based CS algorithm is presented, which unifies the multi- and 1-bit CS processing and is applicable to various cases of noiseless/noisy environment and unsaturated/saturated quantizer. By decoupling the quantization error from the measurement noise, the quantization error is modeled as a random variable and estimated jointly with the signal being recovered. Such a novel characterization of the quantization error results in superior performance of the algorithm which is demonstrated by extensive simulations in comparison with state-of-the-art methods for both multi-bit and 1-bit CS problems.Comment: Accepted by IEEE Trans. Signal Processing. 10 pages, 6 figure