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

Distributed Functional Scalar Quantization Simplified

Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to sources with bounded distributions. We rigorously show that a much simpler decoder has equivalent asymptotic performance as the conditional expectation estimator previously explored, thus reducing decoder design complexity. The simpler decoder has the feature of decoupled communication and computation blocks. Moreover, we extend the DFSQ framework with the simpler decoder to acquire sources with infinite-support distributions such as Gaussian or exponential distributions. Finally, through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis, and we give new insight on the rate of convergence

The Pros and Cons of Compressive Sensing for Wideband Signal Acquisition: Noise Folding vs. Dynamic Range

Compressive sensing (CS) exploits the sparsity present in many signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors and any associated communication links. This paper examines the use of CS in the design of a wideband radio receiver in a noisy environment. We formulate the problem statement for such a receiver and establish a reasonable set of requirements that a receiver should meet to be practically useful. We then evaluate the performance of a CS-based receiver in two ways: via a theoretical analysis of its expected performance, with a particular emphasis on noise and dynamic range, and via simulations that compare the CS receiver against the performance expected from a conventional implementation. On the one hand, we show that CS-based systems that aim to reduce the number of acquired measurements are somewhat sensitive to signal noise, exhibiting a 3dB SNR loss per octave of subsampling, which parallels the classic noise-folding phenomenon. On the other hand, we demonstrate that since they sample at a lower rate, CS-based systems can potentially attain a significantly larger dynamic range. Hence, we conclude that while a CS-based system has inherent limitations that do impose some restrictions on its potential applications, it also has attributes that make it highly desirable in a number of important practical settings

The CCSDS 123.0-B-2 Low-Complexity Lossless and Near-Lossless Multispectral and Hyperspectral Image Compression Standard: A comprehensive review

The Consultative Committee for Space Data Systems (CCSDS) published the CCSDS 123.0-B-2, “Low- Complexity Lossless and Near-Lossless Multispectral and Hyperspectral Image Compression” standard. This standard extends the previous issue, CCSDS 123.0-B-1, which supported only lossless compression, while maintaining backward compatibility. The main novelty of the new issue is support for near-lossless compression, i.e., lossy compression with user-defined absolute and/or relative error limits in the reconstructed images. This new feature is achieved via closed-loop quantization of prediction errors. Two further additions arise from the new near lossless support: first, the calculation of predicted sample values using sample representatives that may not be equal to the reconstructed sample values, and, second, a new hybrid entropy coder designed to provide enhanced compression performance for low-entropy data, prevalent when non lossless compression is used. These new features enable significantly smaller compressed data volumes than those achievable with CCSDS 123.0-B-1 while controlling the quality of the decompressed images. As a result, larger amounts of valuable information can be retrieved given a set of bandwidth and energy consumption constraints