Compressed sensing is a novel methodology in data processing, which takes an advantage of the fact that most signals are sparse (have a small number of nonzeros), or admit a sparse representation, i.e., can be represented as a linear combination of few elements of a given frame (dictionary). Sparsity then allows one to recover the signal from considerably fewer linear measurements than what is required by traditional methods. In the first part of the thesis, we exploit sparse geometric structure of the signal. We consider signals consisting of unions of a few discrete lines as the simplest case of geometric sparsity. We investigate the frame properties of the system of discrete lines and the application of compressed sensing to such signal models, for example, for separating discrete lines and points. The second type of structured sparsity that we consider is, signals which are sparse in a dictionary of time- and frequency-shifts, i.e., a Gabor system. We are interested in Gabor systems generated by difference sets, which can be seen as lines in finite projective geometry. We further view this system as a fusion frame, show that it is optimally sparse, and moreover an equidistant tight fusion frame, i.e. it is an optimal Grassmannian packing. In the second half of this thesis, we move from compressed sensing to phase retrieval: if compressed sensing studies the recovery of signals from a set of linear, non-adaptive measurements, phase retrieval tries to recover signals from only the absolute values of those measurements. In general, the signal in the phase retrieval problem is not necessarily sparse, but there has been an increased interest in the recent years in including the sparsity assumption for this problem, and by that lowering the number of measurements needed for recovery of the signal. We investigate the case when the signal itself is not sparse, but it has a sparse representation in an arbitrary dictionary. Finally, we consider the phase retrieval problem in the case when the measurements are time- and frequency-shifts of a suitably chosen generator. We prove an injectivity condition for recovery of any signal from all time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given. We discuss which generators are suitable for sparse phase retrieval from Gabor measurements, and provide an algorithm for solving this problem.
|School:||Technische Universitaet Berlin (Germany)|
|Source:||DAI-C 81/1(E), Dissertation Abstracts International|
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