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This book presents a survey of the state-of-the art in the exciting and timely topic of compressed sensing for distributed systems. It has to be noted that, while compressed sensing has been studied for some time now, its distributed applications are relatively new. Remarkably, such applications are ideally suited to exploit all the benefits that compressed sensing can provide. The objective of this book is to provide the reader with a comprehensive survey of this topic, from the basic concepts to different classes of centralized and distributed reconstruction algorithms, as well as a comparison of these techniques. This book collects different contributions on these aspects. It presents the underlying theory in a complete and unified way for the first time, presenting various signal models and their use cases. It contains a theoretical part collecting latest results in rate-distortion analysis of distributed compressed sensing, as well as practical implementations of algorithms obtaining performance close to the theoretical bounds. It presents and discusses various distributed reconstruction algorithms, summarizing the theoretical reconstruction guarantees and providing a comparative analysis of their performance and complexity. In summary, this book will allow the reader to get started in the field of distributed compressed sensing from theory to practice. We believe that this book can find a broad audience among researchers, scientists, or engineers with very diverse backgrounds, having interests in mathematical optimization, network systems, graph theoretical methods, linear systems, stochastic systems, and randomized algorithms. To help the reader become familiar with the theory and algorithms presented, accompanying software is made available on the authors’ web site, implementing several of the algorithms described in the book. The only background required of the reader is a good knowledge of advanced calculus and linear algebra.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This chapter motivates the problem of distributed compressed sensing. It recalls the basic properties of compressed sensing and describes why it can be an appealing technique for distributed systems. It highlights existing and possible future applications in this area.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli

Chapter 2. Distributed Compressed Sensing

Abstract
This chapter first introduces CS in the conventional setting where one device acquires one signal and sends it to a receiver, and then extends it to the distributed framework in which multiple devices acquire multiple signals. In particular, we focus on two key problems related to the distributed setting. The former is the definition of sparsity models for an ensemble of signals, as opposed to just one signal. The second is the structure of the corresponding recovery algorithm, which can be centralized or distributed; each solution entails specific advantages and drawbacks that are preliminarily discussed in this chapter, whereas a detailed description of the corresponding recovery algorithms is given in Chaps. 4 and 5.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli

Chapter 3. Rate-Distortion Theory of Distributed Compressed Sensing

Abstract
In this chapter, correlated and distributed sources without cooperation at the encoder are considered. For these sources, the best achievable performance in the rate-distortion sense of any distributed compressed sensing scheme is derived, under the constraint of high-rate quantization. Moreover, under this model we derive a closed-form expression of the rate gain achieved by taking into account the correlation of the sources at the receiver and a closed-form expression of the average performance of the oracle receiver for independent and joint reconstruction. Finally, we show experimentally that the exploitation of the correlation between the sources performs close to optimal and that the only penalty is due to the missing knowledge of the sparsity support as in (non-distributed) compressed sensing. Even if the derivation is performed in the large system regime, where signal and system parameters tend to infinity, numerical results show that the equations match simulations for parameter values of practical interest.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli

Chapter 4. Centralized Joint Recovery

Abstract
This chapter surveys the basic concepts and algorithms for joint reconstruction from compressive measurements in a network of nodes. The nodes acquire measurements of a set of signals obeying a specific joint sparsity model, while a centralized fusion center collects the measurements of the entire network and jointly processes them to reconstruct the acquired signals.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli

Chapter 5. Distributed Recovery

Abstract
This chapter surveys a few basic algorithms for distributed reconstruction from compressive measurements in a network of nodes, which may be sensors or nodes that collect measurements from different sensors. This estimation problem can be recast into an optimization problem where a convex and separable loss function should be minimized subject to sparsity constraints. The goal of the network is to handle distributed sparse estimation. Clearly, to achieve such a goal, the nodes must share, at least partially, their estimation. A single node typically has limited memory and processing capability; therefore, cooperation is the key to compensate for this lack and achieve satisfactory performance. Cooperation, however, raises the problem of communication among nodes, which is known to be the largest consumer of the limited energy of a node, compared to other functions such as sensing and computation. Particular attention is devoted to energy efficiency, in terms of transmissions and memory requirements.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli

Chapter 6. Conclusions

Abstract
This chapter wraps up the book, drawing some conclusions, and outlining open problems and research directions.
Giulio Coluccia, Chiara Ravazzi, Enrico Magli
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