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Dissertation
Length space skeletonization
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2020
DOI: 10.17077/etd.005571
Abstract
Mathematically, the skeleton (also known as the medial axis or symmetric axis) is defined as the collection of centers of maximally inscribed balls in a bounded domain G. The skeleton is a set whose dimension is less than the domain G, is located centrally in the interior of the domain, and contains some geometric properties associated to the object and its boundary. In general, the skeleton can be very complicated, but under suitable conditions, it is a union of curves and surfaces. The skeleton is used in application such as a shape descriptor in image processing, computer vision, and computer graphics.
In this thesis, we study the mathematical theory of a length skeleton in a compact, Lipschitz path-connected subset G having an intrinsic metric and restricting the boundary of G to be piecewise linear with finitely many vertices and edges. Moreover, G may have finitely many "obstructions", i.e., G may be multiply-connected with finitely many holes. It is necessary for our studies to divide the boundary of G into two compact subsets, which we will call the fire and water boundary. Based on this division of the boundary of G, the length skeleton is defined as the closure of points p in G so that there exist two or more minimal geodesics from the fire boundary to p which have the same length and are distinct at p. We discuss some challenges using a maximal ball definition in metric space G to define the length skeleton, and we show the necessity of the condition distinct at p in our definition.
There are three main groups of results in this thesis. First, we identify elements of the boundary of G used to prove the nature of minimal geodesics in G from the fire boundary to interior points, which turn around the water boundary. We classify all such possibilities of minimal geodesics. Second, we prove a characterization of the level sets of the distance function in 2D to fire boundary as a C1 concatenation of parts of lines and circular arcs outside the skeleton. Lastly, we prove that the length skeleton in the interior of G is a collection of C1 curves obtained from concatenation of parts of line segments, parabolas, and hyperbolas, connected by transition points and branching points, and that the skeleton has measure zero. We show that the length skeleton has finitely many transition points, and finitely many branching points with finitely many branches at each point. Furthermore, the skeleton is a C1 curve at transition points, away from branching points and the boundary.
Details
- Title: Subtitle
- Length space skeletonization
- Creators
- Melanie King
- Contributors
- Oguz C. Durumeric (Advisor)Gary E Christensen (Advisor)Punam K Saha (Committee Member)Lihe Wang (Committee Member)Weiman Han (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Applied Mathematical and Computational Sciences
- Date degree season
- Summer 2020
- Publisher
- University of Iowa
- DOI
- 10.17077/etd.005571
- Number of pages
- xiii, 188 pages
- Copyright
- Copyright 2020 Melanie King
- Language
- English
- Description illustrations
- color illustrations
- Description bibliographic
- Includes bibliographical references (pages 183-188).
- Academic Unit
- Interdisciplinary Graduate Program in Applied Mathematical & Computational Sciences
- Record Identifier
- 9983987894602771
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