Compressed sensing performance bounds under Poisson noise

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical $\ell_2$--$\ell_1$ minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity- and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the signal-dependent part of the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.Comment: 12 pages, 3 pdf figures; accepted for publication in IEEE Transactions on Signal Processin

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

Performance bounds for optimal feedback control in networks

Many important complex networks, including critical infrastructure and emerging industrial automation systems, are becoming increasingly intricate webs of interacting feedback control loops. A fundamental concern is to quantify the control properties and performance limitations of the network as a function of its dynamical structure and control architecture. We study performance bounds for networks in terms of optimal feedback control costs. We provide a set of complementary bounds as a function of the system dynamics and actuator structure. For unstable network dynamics, we characterize a tradeoff between feedback control performance and the number of control inputs, in particular showing that optimal cost can increase exponentially with the size of the network. We also derive a bound on the performance of the worst-case actuator subset for stable networks, providing insight into dynamics properties that affect the potential efficacy of actuator selection. We illustrate our results with numerical experiments that analyze performance in regular and random networks

Performance Bounds for Bi-Directional Coded Cooperation Protocols

In coded bi-directional cooperation, two nodes wish to exchange messages over a shared half-duplex channel with the help of a relay. In this paper, we derive performance bounds for this problem for each of three protocols. The first protocol is a two phase protocol were both users simultaneously transmit during the first phase and the relay alone transmits during the second. In this protocol, our bounds are tight and a multiple-access channel transmission from the two users to the relay followed by a coded broadcast-type transmission from the relay to the users achieves all points in the two-phase capacity region. The second protocol considers sequential transmissions from the two users followed by a transmission from the relay while the third protocol is a hybrid of the first two protocols and has four phases. In the latter two protocols the inner and outer bounds are not identical, and differ in a manner similar to the inner and outer bounds of Cover's relay channel. Numerical evaluation shows that at least in some cases of interest our bounds do not differ significantly. Finally, in the Gaussian case with path loss, we derive achievable rates and compare the relative merits of each protocol in various regimes. This case is of interest in cellular systems. Surprisingly, we find that in some cases, the achievable rate region of the four phase protocol sometimes contains points that are outside the outer bounds of the other protocols.Comment: 15 page