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

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dc.contributor.advisorZhang, Yin
dc.contributor.committeeMemberTapia, Richard
dc.contributor.committeeMemberHand, Paul
dc.contributor.committeeMemberKelly, Kevin
dc.creatorCastanon, Jorge Castanon Alberto
dc.date.accessioned2016-01-06T20:52:57Z
dc.date.available2016-01-06T20:52:57Z
dc.date.created2015-05
dc.date.issued2015-04-20
dc.date.submittedMay 2015
dc.date.updated2016-01-06T20:52:57Z
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.
dc.format.mimetypeapplication/pdf
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>.
dc.identifier.urihttps://hdl.handle.net/1911/87728
dc.language.isoeng
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.
dc.subjectnumerical optimization
dc.subjectregularization
dc.subjectlinear inverse problems: machine learning
dc.subjectimage recovery
dc.subjectcompressive sensing
dc.titleA Spectrum-based Regularization Approach to Linear Inverse Problems: Models, Learned Parameters and Algorithms
dc.typeThesis
dc.type.materialText
thesis.degree.departmentComputational and Applied Mathematics
thesis.degree.disciplineEngineering
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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