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dc.rights.licenseRestricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.
dc.contributorLai, Rongjie
dc.contributorKovacic, Gregor
dc.contributorMitchell, John E.
dc.contributorYazici, Birsen
dc.contributor.authorSchonsheck, Stefan C.
dc.date.accessioned2021-11-03T09:22:55Z
dc.date.available2021-11-03T09:22:55Z
dc.date.created2021-02-22T15:34:48Z
dc.date.issued2020-08
dc.identifier.urihttps://hdl.handle.net/20.500.13015/2640
dc.descriptionAugust 2020
dc.descriptionSchool of Science
dc.description.abstractLeo Tolstoy opened his monumental novel Anna Karenina with the now famous words:
dc.description.abstractThe genesis of the so-called 'big data era' and the proliferation of social and scientific databases of increasing size has led to a need for algorithms that can efficiently process, analyze and, even generate high dimensional data. However, the \emph{curse of dimensionality} leads to the fact that many classical approaches do not scale well with respect to the size of these problems. One technique to avoid some of these ill-effects is to exploit the geometric structure of coherent data. In this thesis, we will explore geometric methods for shape processing and data analysis.
dc.description.abstractMore specifically, we will study techniques for representing manifolds and signals supported on them through a variety of mathematical tools including, but not limited to, computational differential geometry, variational PDE modeling and deep learning. First, we will explore non-isometric shape matching through variational modeling. Next, we will use ideas from parallel transport on manifolds to generalize convolution and convolutional neural networks to deformable manifolds. Finally, we conclude by proposing a novel auto-regressive model for capturing the intrinsic geometry and topology of data. Throughout this work, we will use the idea of computing correspondences as a though-line to both motivate our work and analyze our results
dc.description.abstractOne of the advantages of working in this manner is that questions which arise from very specific problems will have far reaching consequences. There are many deep connections between concise models, harmonic analysis, geometry and learning that have only started to emerge in the past few years, and the consequences will continue to shape these fields for many years to come. Our goal in this work is to explore these connections and develop some useful tools for shape analysis, signal processing and representation learning.
dc.description.abstract"Happy families are all alike; every unhappy family is unhappy in its own way"
dc.description.abstractA similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study.
dc.language.isoENG
dc.publisherRensselaer Polytechnic Institute, Troy, NY
dc.relation.ispartofRensselaer Theses and Dissertations Online Collection
dc.subjectMathematics
dc.titleComputational analysis of deformable manifolds : from geometric modeling to deep learning
dc.typeElectronic thesis
dc.typeThesis
dc.digitool.pid180415
dc.digitool.pid180416
dc.digitool.pid180417
dc.rights.holderThis electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.
dc.description.degreePhD
dc.relation.departmentDept. of Mathematical Sciences


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