Author : Zhe Su
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (134 download)
Book Synopsis Elastic Shape Analysis of Curves and Surfaces by : Zhe Su
Download or read book Elastic Shape Analysis of Curves and Surfaces written by Zhe Su and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The shape analysis of curves and surfaces has caught more and more attention in recent years. For curves in Euclidean space, the SRVF framework allows us to define and efficiently compute a distance on the shape space of curves. In this dissertation we give a generalization of the SRVF to curves with values in a homogeneous space and show that, under mild conditions, there always exist optimal reparametrizations realizing the quotient distance on the space of shapes. We give concrete examples to demonstrate the efficiency of our framework. Next, we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The metric is motivated by applications in the field of shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. We calculate the geodesic equations and obtain explicit solutions for the corresponding initial value problem. Using this, we study the geodesic and metric incompleteness of the space of one-forms and exhibit some totally geodesic subspaces. We also calculate its sectional curvature and observe that, depending on the dimensions of the base manifold and the ambient Euclidean space, it either has a semidefinite sign or admits both signs. After that, we give two data-driven frameworks for analyzing shapes of immersed surfaces in $\RR^3$. In the first framework, we introduce a 4-parameter family of elastic metrics on the space of surfaces as a pullback of a family of metrics on the space of vector-valued one-forms. This 4-parameter family of elastic metrics is invariant under rigid motions and reparametrizations. It generalizes a previously studied 3-parameter family of elastic metrics called the general elastic metric, and in particular the Square Root Normal Field (SRNF) metric, which has proved to be successful in various applications. In the second framework, we define a new representation for surfaces in $\RR^3$ by combining the induced surface metric and the SRNF of each surface. Using the DeWitt metric on the space of metrics and the $L^2$ metric on the space of SRNFs, we obtain a 3-parameter family of elastic metrics on surfaces, which forms an open subset of the general elastic metric. The new representation avoids the degeneracy of the SRNF representation, while it still results in an explicit extrinsic distance function on the space of surfaces and thus makes the framework computationally efficient. For both of these two frameworks we provide numerical algorithms and show several examples to validate the frameworks. The second framework, although it does not give a computation of geodesics, provides fast and accurate registration of surfaces. Finally we conclude with a summary of this dissertation and a discussion of possible future work.
