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

Lower bounds on the estimation performance in low complexity quantize-and-forward cooperative systems

Cooperative communication can effectively mitigate the effects of multipath propagation fading by using relay channels to provide spatial diversity. A relaying scheme suitable for half-duplex devices is the quantize-and-forward (QF) protocol, in which the information received from the source is quantized at the relay before being forwarded to the destination. In this contribution, the Cramer-Rao bound (CRB) is obtained for the case where all channel parameters in a QF system are estimated at the destination. The CRB is a lower bound (LB) on the mean square estimation error (MSEE) of an unbiased estimate and can thus be used to benchmark practical estimation algorithms. Additionally, the modified Cramer-Rao bound (MCRB) is also presented, which is a looser but computationally less complex bound. An importance sampling technique is developed to speed up the computation of the MCRBs, and the MSEE performance of a practical estimation algorithm is compared with the (M)CRBs. We point out that the parameters of the source-destination and relay-destination channels can be accurately estimated but that inevitably the source-relay channel estimate is poor when the instantaneous SNR on the relay-destination channel is low; however, in this case, the decoder performance is not affected by the inaccurate source-relay channel estimate

Fast Compression of Imagery with High Frequency Content

Image compression is an active area due to the many applications involving electronic media. Much research has been focused on image quality versus bit rate and/or algorithm speed. Here, we seek an effective image coder with a weighted constraint on speed. However, the compression must not taint the quality of impulsive features on the image. Moreover, the camera is operated in a mode that creates a dominant fixed pattern noise across the image array, degrading visual quality and disrupting compression performance. We propose a method that efficiently compresses such an image. We begin by characterizing and removing the fixed pattern noise from the image, thereby dramatically improving its visual quality

High-Rate Vector Quantization for the Neyman-Pearson Detection of Correlated Processes

This paper investigates the effect of quantization on the performance of the Neyman-Pearson test. It is assumed that a sensing unit observes samples of a correlated stationary ergodic multivariate process. Each sample is passed through an N-point quantizer and transmitted to a decision device which performs a binary hypothesis test. For any false alarm level, it is shown that the miss probability of the Neyman-Pearson test converges to zero exponentially as the number of samples tends to infinity, assuming that the observed process satisfies certain mixing conditions. The main contribution of this paper is to provide a compact closed-form expression of the error exponent in the high-rate regime i.e., when the number N of quantization levels tends to infinity, generalizing previous results of Gupta and Hero to the case of non-independent observations. If d represents the dimension of one sample, it is proved that the error exponent converges at rate N^{2/d} to the one obtained in the absence of quantization. As an application, relevant high-rate quantization strategies which lead to a large error exponent are determined. Numerical results indicate that the proposed quantization rule can yield better performance than existing ones in terms of detection error.Comment: 47 pages, 7 figures, 1 table. To appear in the IEEE Transactions on Information Theor

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