Contributions to signal analysis and processing using compressed sensing techniques

Chapter 2 contains a short introduction to the fundamentals of compressed sensing theory, which is the larger context of this thesis. We start with introducing the key concepts of sparsity and sparse representations of signals. We discuss the central problem of compressed sensing, i.e. how to adequately recover sparse signals from a small number of measurements, as well as the multiple formulations of the reconstruction problem. A large part of the chapter is devoted to some of the most important conditions necessary and/or sufficient to guarantee accurate recovery. The aim is to introduce the reader to the basic results, without the burden of detailed proofs. In addition, we also present a few of the popular reconstruction and optimization algorithms that we use throughout the thesis. Chapter 3 presents an alternative sparsity model known as analysis sparsity, that offers similar recovery guarantees as the classical synthesis sparsity model. We explore the relation between the two models, based on the idea that the analysis model can be thought of as a synthesis model with a set of additional constraints. As such, we develop a series of theorems proving that analysis-based signal recovery can be reformulated as a augmented version of synthesis-based sparse signal recovery. The practical advantage of this reformulation is that one can use existing recovery algorithms, designed solely for synthesis-based recovery, for the analysis-based recovery as well, as we show through practical simulations. The second part of the chapter consist of an experimental comparison of the two models, based on the above reformulation. The simulations reveal a fundamental difference between the two models in terms of the signals that are adequate to one or the other, as well as different sensitivity to noise and number of measurements. Chapter 4 presents an innovative solution for finding improved acquisition matrices for signals that are sparse in non-orthogonal bases and overcomplete dictionaries. We first present three existing state-of-the-art algorithms and we propose modifications for improving each of them, based on the analysis of some unfavourable scenarios. We then propose a unified approach for finding an optimal acquisition matrix that encompasses all the three modified algorithms. We formulate the general problem as a rank-constrained nearest correlation matrix problem, i.e. finding the matrix of minimal distance from a given correlation matrix, subject to rank, semipositivity and unit-diagonal constraints. The modified versions of the three algorithms correspond to different ways of choosing a distance metric. Experimental results confirm superior performance and increased robustness, particularly for higher compression ratios and with dictionaries that are highly correlated, e.g. learned from a particular data set. Chapter 5 presents a surprising result on using correlated projection vectors, i.e. random vectors drawn from a multivariate probability distribution with a non-unit covariance matrix. We develop a series of experiments that show that the covariance matrix of the reconstruction errors obtained with a number of well-known recovery algorithms is heavily dependent on the covariance matrix of the random projection vectors: the two matrices have roughly the same eigenvectors, and we also find a rather precise relation between the eigenvalues of the two matrices, up to a constant factor. This means that one can effectively control the “shape” of the error distribution by properly designing the covariance of the random projection vectors. In this way, one can control the recovery accuracy along some directions in space at the expense of others. This opens the door for a number of interesting compressed sensing applications, scarcely analyzed in the literature up to now. We validate our assumptions with two practical scenarios: recovering signals that are sparse in non-orthogonal bases, and recovering signals with unequal atom importance. In both cases, simulations show that using correlated projection vectors leads to significant improvements over the non-correlated case. Chapter 6 evaluates the possibility of using the compressed sensing theory in the practical context of electrocardiographic (ECG) signal acquisition. We start with the preprocessing of ECG signals, based on segmentation into independent heart beats, and we investigate the performance of standard wavelet bases and custom learned bases in terms of sparse representations of the ECG segments. We evaluate the possibility of classifying the ECG segments into normal/pathological classes using solely the compressed random measurements. We also investigate recovery using dictionaries of different sizes and created with different algorithms. We propose a hybrid reconstruction method based on classifying the acquired measurements in the compressed space, followed by reconstruction with a dictionary dedicated to the particular signal class. The simulation results indicate superior reconstruction accuracy compared to the single dictionary scenario. Finally, Chapter 7 summarizes the conclusions and reviews the main contributions of the thesis. An additional appendix contains further results for ECG signal classification in chapter 6.