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Riemannian Approach for Computing Geodesics in Elastic Shape Space and Its Applications

Title: A Riemannian Approach for Computing Geodesics in Elastic Shape Space and Its Applications.
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Name(s): You, Yaqing, author
Gallivan, Kyle A., 1958-, professor co-directing dissertation
Absil, Pierre-Antoine, professor co-directing dissertation
Erlebacher, Gordon, 1957-, university representative
Ökten, Giray, committee member
Sussman, Mark, committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Doctoral Thesis
Issuance: monographic
Date Issued: 2018
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (107 pages)
Language(s): English
Abstract/Description: This dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limited-memory Riemannian Broyden-Fletcher-Goldfarb-Shanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors, Riemannian metrics, Riemannian gradient, as well as retraction and vector transport. The difference between this Riemannian approach to compute closed curve geodesics as well as accurate geodesic distance, the existing Path-Straightening algorithm and the existing Riemannian approach to approximate distances between closed shapes, are also discussed in this dissertation. This dissertation summarizes the implementation details and techniques for both Riemannian algorithms to achieve the most efficiency. This dissertation also contains basic experiments and applications that illustrate the value of the proposed algorithms, along with comparison tests to the existing alternative approaches. It has been demonstrated by various tests that this proposed approach is superior in terms of time and performance compared to a state-of-the-art distance computation algorithm, and has better performance in applications of shape distance when compared to the distance approximation algorithm. This dissertation applies the Riemannian geodesic computation algorithm to calculate Karcher mean of shapes. Algorithms that generate less accurate distances and geodesics are also implemented to compute shape mean. Test results demonstrate the fact that the proposed algorithm has better performance with sacrifice in time. A hybrid algorithm is then proposed, to start with the fast, less accurate algorithm and switch to the proposed accurate algorithm to get the gradient for Karcher mean problem. This dissertation also applies Karcher mean computation to unsupervised learning of shapes. Several clustering algorithms are tested with the distance computation algorithm and Karcher mean algorithm. Different versions of Karcher mean algorithm used are compared with tests. The performance of clustering algorithms are evaluated by various performance metrics.
Identifier: 2018_Su_You_fsu_0071E_14686 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester 2018.
Date of Defense: June 29, 2018.
Bibliography Note: Includes bibliographical references.
Advisory Committee: Kyle A. Gallivan, Professor Co-Directing Dissertation; Pierre-Antoine Absil, Professor Co-Directing Dissertation; Gordon Erlebacher, University Representative; Giray Okten, Committee Member; Mark Sussman, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/2018_Su_You_fsu_0071E_14686
Owner Institution: FSU

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You, Y. (2018). A Riemannian Approach for Computing Geodesics in Elastic Shape Space and Its Applications. Retrieved from http://purl.flvc.org/fsu/fd/2018_Su_You_fsu_0071E_14686