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Sparse Approximation and Its Applications

Title: Sparse Approximation and Its Applications.
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Name(s): Li, Qin, 1982-, author
Erlebacher, Gordon, professor directing thesis
Wang, Xiaoming, professor co-directing thesis
Hart, Robert, university representative
Peterson, Janet, committee member
Sussman, Mark, committee member
Gallivan, Kyle A., committee member
Department of Mathematics, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2011
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: In this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of a new Iterative Shrinkage algorithm based on Surrogate Function, ISSF-K, for finding the best K-term approximation to an image. In this problem, we seek to represent an image with K elements from an overcomplete dictionary. We present a proof that this algorithm converges to a local minimum of the NP hard sparsity constrained optimization problem. In addition, we choose curvelets as the dictionary. The approximation obtained by our approach achieves higher PSNR than that of the best K-term wavelet (Cohen-Daubechies-Fauraue 9-7) approximation. We extends ISSF to the application of Morphological Component Analysis, which leads to the second contribution, a new algorithm MCA-ISSF with an adaptive thresholding strategy. The adaptive MCA-ISSF algorithm approximates the problem from the synthesis approach, and it is the only algorithm that incorporate an adaptive strategy to update its algorithmic parameter. Compared to the existent MCA algorithms, our method is more efficient and is parameter free in the thresdholding update. The third contribution concerns the non-convex optimization problems in Compressive Sensing (CS), which is an important extension of sparse approximation. We propose two new iterative reweighted algorithms based on Alternating Direction Method of Multiplier, IR1-ADM and IR2-ADM, to solve the ell-p,0.
Identifier: FSU_migr_etd-1399 (IID)
Submitted Note: A Thesis submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester, 2011.
Date of Defense: May 23, 2011.
Keywords: underdetermined linear system of equations, compressive sensing, sparse approximation, nonconvex optimization
Bibliography Note: Includes bibliographical references.
Advisory Committee: Gordon Erlebacher, Professor Directing Thesis; Xiaoming Wang, Professor Co-Directing Thesis; Robert Hart, University Representative; Janet Peterson, Committee Member; Mark Sussman, Committee Member; Kyle A. Gallivan, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-1399
Owner Institution: FSU

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Li, Q. (2011). Sparse Approximation and Its Applications. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-1399