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dc.contributor.advisorSchönlieb, Carola
dc.contributor.authorCalatroni, Luca
dc.date.accessioned2016-06-02T13:27:21Z
dc.date.available2016-06-02T13:27:21Z
dc.date.issued2016-03-01
dc.identifier.otherPhD.39488
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/256139
dc.description.abstractVariational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed.en
dc.language.isoenen
dc.subjectResearch Subject Categories::MATHEMATICS::Algebra, geometry and mathematical analysisen
dc.subjecttotal variationen
dc.subjecthigher-order PDEsen
dc.subjectdirectional splittingen
dc.subjectquasi-Newton methodsen
dc.subjectimage denoisingen
dc.subjectimage inpaintingen
dc.subjectmixed noise distributionen
dc.subjectbilevel optimisationen
dc.subjectparameter learningen
dc.subjectSemiSmooth Newton methodsen
dc.subjectimage segmentationen
dc.subjectgraph clusteringen
dc.subjectmatrix completionen
dc.subjectHough transformen
dc.titleNew PDE models for imaging problems and applicationsen
dc.typeThesisen
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridgeen
dc.publisher.departmentDepartment of Applied Mathematics and Theoretical Physicsen
dc.publisher.departmentCambridge Centre for Analysisen
dc.publisher.departmentFaculty of Mathematicsen
dc.publisher.departmentHughes Hallen
dc.identifier.doi10.17863/CAM.79


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