Model-based compressive sensing for signal ensembles

Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Instead of taking N periodic samples, we measure M ¿ N inner products with random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. A new framework for CS based on unions of subspaces can improve signal recovery by including dependencies between values and locations of the signal's significant coefficients. In this paper, we extend this framework to the acquisition of signal ensembles under a common sparse supports model. The new framework provides recovery algorithms with theoretical performance guarantees. Additionally, the framework scales naturally to large sensor networks: the number of measurements needed for each signal does not increase as the network becomes larger. Furthermore, the complexity of the recovery algorithm is only linear in the size of the network. We provide experimental results using synthetic and real-world signals that confirm these benefits.

I. Introduction

Compressive sensing (CS) is a new approach to simultaneous sensing and compression that enables a potentially large reduction in the sampling and computation costs at a sensor for signals having a sparse representation in some basis. CS builds on the work of Candes, Romberg, and Tao [1] and Donoho [2], who showed that a signal having a sparse representation in one basis can be recovered from a small set of projections onto a second measurement basis that is incoherent with the first. ¹

Roughly speaking, incoherence means that no element of one basis has a sparse representation in terms of the other basis.

Random projections play a central role as a universal measurement basis in the sense that they are incoherent with any fixed basis with high probability. The CS measurement process is nonadaptive, and the recovery process is nonlinear, for which a variety of algorithms have been proposed [1]–[6].