Sparsity-based signal processing algorithms and applications using convex and non-convex optimization

The first part of this work describes an exponential transient excision algorithm (ETEA). The proposed method is formulated as an unconstrained convex optimization problem, regularized by smoothed `1-norm penalty function, which can be solved by majorization-minimization (MM) method. With a slight modification of the regularizer, ETEA can also suppress more irregular piecewise smooth artifacts, especially, ocular artifacts (OA) in electroencephalography (EEG) data.;The second part of this work formulates wavelet-TV (WATV) denoising as a unified problem. To strongly induce wavelet sparsity, the proposed approach uses non-convex penalty functions. At the same time, in order to draw on the advantages of convex optimization (unique minimum, reliable algorithms, simplified regularization parameter selection), the non-convex penalties are chosen so as to ensure the convexity of the total objective function. A computationally efficient, fast converging algorithm is derived.;The third part of this work addresses the detection of periodic transients in vibration signals for detecting faults in rotating machines. For this purpose, we present a method to estimate periodic-group-sparse signals in noise. The method is based on the formulation of a convex optimization problem. An extension of the this method is also proposed, which is suitable to detect compound faults in rotating machines.