SIGNAL RECONSTRUCTION FROM COMPRESSIVE SAMPLES

Abstract

The emerging fi eld of Compressive Sensing (CS), is a novel sampling paradigm that exploits the sparsity/compressibility of signals to reconstruct them from far fewer samples, than what is required by the traditional Shannon-Nyquist sampling theorem. Unlike conventional methods, which use linear sinc interpolation to recover signals/images from the acquired samples, CS relies on non-linear optimization-based methods to fi nd the sparsest signal among the set of all feasible solutions. In the current literature of compressive sampling, sparsest signal corresponds to the one with the minimum value of L0 norm (e.g. number of nonzero elements). However, it is acknowledged that solving the equivalent optimization problem is computationally unwieldy in view of its NP-hard nature. Therefore, in the majority of CS literature, the reconstruction is done using convex L1-based optimization. This thesis presents some efficient and practical methodologies for reconstruction of high dimensional signals from compressive measurements that overcome the current limitations of state-of-the-art CS recovery methods. The key contributions include: Developing a stochastic-based method for achieving as close an approximation to L0-norm as is computationally feasible in signal reconstruction from compressive samples; Exploring properties of the Gini index (GI) as an sparsity measure in the problem of signal/image reconstruction; Demonstrating the robustness and reliability of GI as an alternative to the currently popular Lp norm-based (for 0 < p 1) sparsity measures, through extensive experiments. In the case of time-varying signals, a novel approach for recursively reconstructing sequences of sparse signals is proposed, where sparsity changes smoothly with time. In this approach reconstructed signal of the previous time instant is used to extract a probability model. This priori-knowledge is then incorporated into the reconstruction of the next time instant signal, to signi cantly reduce the number of needed samples, compared to other state-of-art CS methods. Lastly, the application of the developed method in low power ECG wireless-enabled monitoring devices and medical imaging modalities, is tested.