Shearlet-based Analysis of Image Inpainting and Convolutional Framelets

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dc.contributor.advisorLabate, Demetrio
dc.contributor.committeeMemberPapadakis, Emanuel I.
dc.contributor.committeeMemberBodmann, Bernhard G.
dc.contributor.committeeMemberHebert, Thomas J.
dc.creatorRodriguez Ayllon, Jose Pedro
dc.date.accessioned2021-06-09T17:28:01Z
dc.date.createdAugust 2020
dc.date.issued2020-08
dc.date.submittedAugust 2020
dc.date.updated2021-06-09T17:28:03Z
dc.description.abstractThe main part of my dissertation deals with image inpainting - a classical problem in image analysis that I analyze from the point of view of microlocal analysis and the theory of sparse approximation. My most important result provides a new set of theoretical performance guarantees for the exact recovery of missing data in images where the information is dominated by curvilinear singularities. In fact, my study shows that a shearlet-based approach for the recovery of missing curvilinear edges in images is provably superior to methods based on conventional wavelets in a precises sense and gives a quantitative assessment on the size of the region that can reliably recovered. As a consequence, this result offers the theoretical underpinning for algorithms based on directional multiscale methods such as shearlets in applications to image inpainting. The arguments in my proofs rely on a new application of the microlocal properties of shearlets and techniques from oscillatory integrals that are inspired in part by a seminal paper by Donoho and Kutyniok, who first introduced methods from microlocal analysis in combination with ideas from sparse representations for problems of image analysis. The second part of my dissertation is a new study of convolutional framelets - a method recently introduced to provide a mathematical framework for the patch-based analysis of images - using tools from tensor analysis. This method gives an alternative approach to analyze framelets and a deeper insight into the mathematical properties of convolutional framelets. The first part of the dissertation follows rather closely some material published by the author in his first journal paper.
dc.description.departmentMathematics, Department of
dc.format.digitalOriginborn digital
dc.format.mimetypeapplication/pdf
dc.identifier.citationPortions of this document appear in: Guo, K., Labate, D., & Rodriguez Ayllon, J. (2020). Image inpainting using sparse multiscale representations: Image recovery performance guarantees. Applied and Computational Harmonic Analysis, 49(2), 343-380. doi:10.1016/j.acha.2020.05.001
dc.identifier.urihttps://hdl.handle.net/10657/7758
dc.language.isoeng
dc.rightsThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).
dc.subjectShearlet, Inpainting
dc.titleShearlet-based Analysis of Image Inpainting and Convolutional Framelets
dc.type.dcmiText
dc.type.genreThesis
local.embargo.lift2022-08-01
local.embargo.terms2022-08-01
thesis.degree.collegeCollege of Natural Sciences and Mathematics
thesis.degree.departmentMathematics, Department of
thesis.degree.disciplineMathematics
thesis.degree.grantorUniversity of Houston
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

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