Doctoral theses
Current Record: 2005 of 2435
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| Identifier |
000355148 |
| Title |
Bayesian compressed sensing using alpha-stable distributions |
| Alternative Title |
Μπεϋζιανή συμπιεστική δειγματοληψία με χρήση άλφα-ευσταθών κατανομών |
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Author
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Τζαγκαράκης, Γεώργιος Ιωάννη
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Thesis advisor
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Τσακαλίδης, Παναγιώτης
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| Abstract |
Nyquist-Shannon’s sampling theorem is one of the fundamental principles in signal processing. However, the high-resolution capabilities of many modern digital devices result in a huge amount of data when sampling at Nyquist’s rate and thus increased processing power and storage are required. A mathematical theory that emerged recently presents the background for developing novel sensing/sampling paradigms that go against the common tenet in data acquisition. Compressed sensing (CS) is a technique for acquiring and reconstructing a signal at sub-Nyquist sampling rates by exploiting its sparsity or compressibility in a transform domain. Most of the proposed CS methods for representing and reconstructing sparse signals do not take into account the true underlying statistics of the given sparse signal. Only recently CS has been set in a probabilistic (Bayesian) framework. However, the vast majority of the previous algorithms is based on a Gaussian assumption for the statistical characterization of the sparse signal and/or the noise.
This thesis introduces the family of Alpha-Stable distributions as a suitable modeling tool for designing efficient CS algorithms for the sparse representation and reconstruction of a given signal exploiting its sparsity in an appropriate transform domain. The proposed Bayesian CS algorithms are based on modeling the prior distribution of a sparse signal using members of the Alpha-Stable family, which are heavy-tailed and thus enforce its sparsity. In the first method, a Gaussian Scale Mixture (GSM) is employed to model the sparse structure. Then, finding a sparse representation reduces to estimating the parameters of the GSM model. Furthermore, we extend this method in the case of multiple observations characterized by a common sparsity structure with high probability (e.g., in a sensor network), yielding an efficient CS method amenable to a distributed implementation.
There are also real-world environments where the signal and/or the noise are highly impulsive and thus resulting in even sparser transform-domain representations. In these cases, the Gaussian assumption is inadequate for approximating the true signal statistics. The second of our proposed methods models the statistics of such sparse signals using a Cauchy distribution, with its parameters being estimated via a tree structure, which is a common approach in several Bayesian learning tasks. Finally, we generalize by developing a CS algorithm which employs an arbitrary symmetric Alpha-Stable distribution. The method proceeds by solving a constrained optimization problem iteratively using duality theory and sub-gradients. For this purpose, we introduce a novel Lagrangian function based on Fractional Lower-Order Moments. In addition, it is shown that the objective function and the constraints are separable and thus this algorithm is amenable to a distributed implementation from the nodes of a sensor network.
The performance of the proposed CS algorithms is illustrated and compared with recently introduced state-of-the-art CS techniques by carrying out a series of experiments for reconstructing simulated signals, for solving the problem of Direction-of-Arrival (DOA) estimation, as well as for recovering real image data from their transform coefficients.
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| Language |
English |
| Issue date |
2009-11-06 |
| Collection
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School/Department--School of Sciences and Engineering--Department of Computer Science--Doctoral theses
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Type of Work--Doctoral theses
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| Permanent Link |
https://elocus.lib.uoc.gr//dlib/b/2/a/metadata-dlib-b403b98150537167a1ec59e545c7e493_1275636144.tkl
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| Views |
680 |
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