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Applications of Mesh Laplacian in Topology.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Applications of Mesh Laplacian in Topology./
Author:	Abdelrahman, Hayam.
Description:	1 online resource (87 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-07, Section: B.
Contained By:	Dissertations Abstracts International85-07B.
Subject:	Computer science. -
Online resource:	click for full text (PQDT)
ISBN:	9798381381207

Applications of Mesh Laplacian in Topology.
Abdelrahman, Hayam.

Applications of Mesh Laplacian in Topology. - 1 online resource (87 pages)

Source: Dissertations Abstracts International, Volume: 85-07, Section: B.

Thesis (Ph.D.)--Michigan State University, 2024.

Includes bibliographical references

Locating neck-like features, or locally narrow parts, of a surface is crucial in various applications such as segmentation, shape analysis, path planning, and robotics. Topological methods are often utilized to find the set of shortest loops around handles and tunnels. However, there are abundant neck-like features on genus-0 shapes without any handles. While 3D geometry-aware topological approaches exist to find such loops, their construction can be cumbersome and may even lead to unintuitive loops. Here we present two methods for efficiently computing a complete set of surface loops that are not limited to the topologically nontrivial independent loops.Our first approach is an efficient "topology-aware geometric" method to compute the tightest loops around neck features on surfaces, including genus-0 surfaces. We use the critical points of a processed distance function (such as Morse function) to find both the location and evaluate the significance of possible neck-like features. Critical points of a Morse function defined on a volume provide rich topological and geometric information about the structure of the shape. Our algorithm starts with a volumetric representation of an input surface and then calculates the distance function of mesh points to the boundary surface as a Morse function. We directly create a cutting plane through each neck feature. Each resulting loop can then be tightened to form a closed geodesic representation of the neck feature. Moreover, we offer criteria to measure the significance of a neck feature through the evolution of critical points during the smoothing of the distance function. Furthermore, we speed up the detection process through mesh simplification without compromising the quality of the output loops.It is known that reducing the dimension of a problem typically boosts efficiency drastically. Hence, we propose our second approach, which is a novel, efficient method that uses the skeleton of the shape to compute surface loops. Given a closed surface mesh, our algorithm produces a practically complete set of loops around narrow regions of the volume enclosed by or outside the surface. Moreover, as our approach accepts a 1D representation of the shape as input, it significantly simplifies and accelerates computations. In particular, the handle-type loops are found by examining a subset of the skeleton points as candidate loop centers and tunnel-type loops are found by examining only high-valence skeleton points. 1D Laplacian and 1D persistent homology are the two alternative methods we utilize to find the best skeleton points. Similar to our first approach, we use cutting planes for constructing surface loops and then optimize them to their shortest geodesic form.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798381381207Subjects--Topical Terms:

573171
Computer science.
Subjects--Index Terms:

GeometryIndex Terms--Genre/Form:

554714
Electronic books.

Applications of Mesh Laplacian in Topology.
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520 $a Locating neck-like features, or locally narrow parts, of a surface is crucial in various applications such as segmentation, shape analysis, path planning, and robotics. Topological methods are often utilized to find the set of shortest loops around handles and tunnels. However, there are abundant neck-like features on genus-0 shapes without any handles. While 3D geometry-aware topological approaches exist to find such loops, their construction can be cumbersome and may even lead to unintuitive loops. Here we present two methods for efficiently computing a complete set of surface loops that are not limited to the topologically nontrivial independent loops.Our first approach is an efficient "topology-aware geometric" method to compute the tightest loops around neck features on surfaces, including genus-0 surfaces. We use the critical points of a processed distance function (such as Morse function) to find both the location and evaluate the significance of possible neck-like features. Critical points of a Morse function defined on a volume provide rich topological and geometric information about the structure of the shape. Our algorithm starts with a volumetric representation of an input surface and then calculates the distance function of mesh points to the boundary surface as a Morse function. We directly create a cutting plane through each neck feature. Each resulting loop can then be tightened to form a closed geodesic representation of the neck feature. Moreover, we offer criteria to measure the significance of a neck feature through the evolution of critical points during the smoothing of the distance function. Furthermore, we speed up the detection process through mesh simplification without compromising the quality of the output loops.It is known that reducing the dimension of a problem typically boosts efficiency drastically. Hence, we propose our second approach, which is a novel, efficient method that uses the skeleton of the shape to compute surface loops. Given a closed surface mesh, our algorithm produces a practically complete set of loops around narrow regions of the volume enclosed by or outside the surface. Moreover, as our approach accepts a 1D representation of the shape as input, it significantly simplifies and accelerates computations. In particular, the handle-type loops are found by examining a subset of the skeleton points as candidate loop centers and tunnel-type loops are found by examining only high-valence skeleton points. 1D Laplacian and 1D persistent homology are the two alternative methods we utilize to find the best skeleton points. Similar to our first approach, we use cutting planes for constructing surface loops and then optimize them to their shortest geodesic form.
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650 4 $a Mathematics. $3 527692
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653 $a Topology
653 $a Computer graphics
653 $a Mesh Laplacian
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710 2 $a Michigan State University. $b Computer Science - Doctor of Philosophy. $3 1188138
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