A Spectrum-based Regularization Approach to Linear Inverse Problems: Models, Learned Parameters and Algorithms

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dc.contributor.advisorZhang, Yinen_US
dc.contributor.committeeMemberTapia, Richarden_US
dc.contributor.committeeMemberHand, Paulen_US
dc.contributor.committeeMemberKelly, Kevinen_US
dc.creatorCastanon, Jorge Castanon Albertoen_US
dc.date.accessioned2016-01-06T20:52:57Zen_US
dc.date.available2016-01-06T20:52:57Zen_US
dc.date.created2015-05en_US
dc.date.issued2015-04-20en_US
dc.date.submittedMay 2015en_US
dc.date.updated2016-01-06T20:52:57Zen_US
dc.description.abstractIn this thesis, we study the problem of recovering signals, in particular images, that approximately satisfy severely ill-conditioned or underdetermined linear systems. For example, such a linear system may represent a set of under-sampled and noisy linear measurements. It is well-known that the quality of the recovery critically depends on the choice of an appropriate regularization model that incorporates prior information about the target solution. Two of the most successful regularization models are the Tikhonov and Total Variation (TV) models, each of which is used in a wide range of applications. We design and investigate a class of spectrum-based models that generalize and improve upon both the Tikhonov and the TV methods, as well as their combinations or so-called hybrids. The proposed models contain "spectrum parameters" that are learned from training data sets through solving optimization problems. This parameter-learning feature gives these models the flexibility to adapt to desired target solutions. We devise efficient algorithms for all the proposed models and conduct comprehensive numerical experiments to evaluate their performance as compared to established models. Numerical results show a generally superior quality in recovered images by our approach from under-sampled linear measurements. Using the proposed algorithms, one can often obtain much improved quality at a moderate increase in computational time.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationCastanon, Jorge Castanon Alberto. "A Spectrum-based Regularization Approach to Linear Inverse Problems: Models, Learned Parameters and Algorithms." (2015) Diss., Rice University. <a href="https://hdl.handle.net/1911/87728">https://hdl.handle.net/1911/87728</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/87728en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectnumerical optimizationen_US
dc.subjectregularizationen_US
dc.subjectlinear inverse problems: machine learningen_US
dc.subjectimage recoveryen_US
dc.subjectcompressive sensingen_US
dc.titleA Spectrum-based Regularization Approach to Linear Inverse Problems: Models, Learned Parameters and Algorithmsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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