Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions

buir.advisorÇetin, A. Enis
dc.contributor.authorTofighi, Mohammad
dc.departmentDepartment of Electrical and Electronics Engineeringen_US
dc.descriptionCataloged from PDF version of article.en_US
dc.description.abstractThis thesis focuses on image restoration and reconstruction problems. These inverse problems are solved using a convex optimization algorithm based on orthogonal Projections onto the Epigraph Set of a Convex Cost functions (PESC). In order to solve the convex minimization problem, the dimension of the problem is lifted by one and then using the epigraph concept the feasibility sets corresponding to the cost function are defined. Since the cost function is a convex function in R N , the corresponding epigraph set is also a convex set in R N+1. The convex optimization algorithm starts with an arbitrary initial estimate in R N+1 and at each step of the iterative algorithm, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The PESC algorithm provides globally optimal solutions for different functions such as total variation, `1-norm, `2-norm, and entropic cost functions. Denoising, deconvolution and compressive sensing are among the applications of PESC algorithm. The Projection onto Epigraph Set of Total Variation function (PES-TV) is used in 2-D applications and for 1-D applications Projection onto Epigraph Set of `1-norm cost function (PES-`1) is utilized. In PES-`1 algorithm, first the observation signal is decomposed using wavelet or pyramidal decomposition. Both wavelet denoising and denoising methods using the concept of sparsity are based on soft-thresholding. In sparsity-based denoising methods, it is assumed that the original signal is sparse in some transform domain such as Fourier, DCT, and/or wavelet domain and transform domain coefficients of the noisy signal are soft-thresholded to reduce noise. Here, the relationship between the standard soft-thresholding based denoising methods and sparsity-based wavelet denoising methods is described. A deterministic soft-threshold estimation method using the epigraph set of `1-norm cost function is presented. It is demonstrated that the size of the `1-ball can be determined using linear algebra. The size of the `1-ball in turn determines the soft-threshold. The PESC, PES-TV and PES-`1 algorithms, are described in detail in this thesis. Extensive simulation results are presented. PESC based inverse restoration and reconstruction algorithm is compared to the state of the art methods in the literature.en_US
dc.description.provenanceMade available in DSpace on 2016-07-01T11:11:03Z (GMT). No. of bitstreams: 1 0006961.pdf: 5582649 bytes, checksum: da8195fb3f7f8c3620f5b8251104d7b3 (MD5) Previous issue date: 2015en
dc.description.statementofresponsibilityTofighi, Mohammaden_US
dc.format.extentxxi, 102 leaves, Chartsen_US
dc.publisherBilkent Universityen_US
dc.subjectConvex optimizationen_US
dc.subjectepigraph of a convex cost functionsen_US
dc.subjectprojection onto convex setsen_US
dc.subjecttotal variation functionen_US
dc.subject`1-norm functionen_US
dc.subjectcompressive sensingen_US
dc.titleImage restoration and reconstruction using projections onto epigraph set of convex cost fuchtionsen_US
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