Sensing and reconstruction of sparse phenomena bounds and algorithms

In this thesis we will consider the problem of sensing and reconstruction of sparse phenomena. In the first part of the thesis we will focus on the sensing aspects and consider the problem of finding fundamental performance limits of Sensor Networks (SNETs) for sensing sparse phenomena. We characterize the performance in terms of sensing capacity which is asymptotically defined as the maximum number of signal dimensions reliably identified per sensor. The notion of sensing capacity is directly related to the notion of compression rate which is useful to characterize the compressibility of sparse signals and has implications for Compressed Sensing (CS). In this work bounds to sensing capacity and compression rate are derived in an information theoretic framework. In particular we use information theoretic analysis to first derive novel upper and lower bounds to probability of error for (a) exact support recovery of sparse signals and (b) approximate signal recovery with an end to end distortion criteria and use these bounds to subsequently evaluate bounds to sensing capacity. We point out that the main difference between the CS and SNETs cases is in the way the SNR is accounted for and we reveal sharp contrasts between sensing capacities for the two cases under a fixed SNR, and linear observation model. We then consider the effect of sensing architecture on sensing capacity. Based on the results derived for approximate recovery we isolate the effect of sensing architecture in terms of mutual information between the data and sparse signal conditioned on the sensing functions. We then quantify the performance of some interesting cases of sensor network configurations considered in the literature.;In the second part of the thesis we focus on a practical problem of dispersion extraction from borehole acoustic array data, a problem that is of considerable interest to the geophysical community. Dispersion refers to a systematic variation of propagation slowness (= wavenumber divided by frequency) of the waves in the array data. In this work we present a novel broadband approach for automatic dispersion extraction. In contrast to previous approaches that are primarily narrowband and require user input such as model order for generating dispersion curves, this approach is capable of automatic model order selection and is more general than model based broadband approaches. The key idea and contribution here is recognition of a sparsity aspect of the underlying signal features, i.e. dispersion curves in the wavenumber-frequency domain, in a suitably chosen over-complete dictionary of basis elements in the f-k domain. We first propose a sparse signal reconstruction framework for dispersion extraction from the acoustic array data. Following that we propose a tractable sparsity penalized (regularized) reconstruction algorithm. For this set-up we present a novel strategy for the selection of regularization parameter based on the distribution of the residuals which is promising for model order selection. Furthermore using the time compactness of the transient signals we propose a hybrid strategy that exploits this time compactness of the waves in the space time domain in addition to sparsity in the (f -- k) domain for robust estimation of group slowness. We show the performance of the proposed methodology on synthetic data sets and also present a small error Cramer Rao Bound (CRB) for the group and the phase slowness being estimated.