Discrete representation of a surface, especially the triangle mesh, is ubiquitous in numerical
simulation and computer graphics. Compared with isotropic triangle meshes,
anisotropic triangle meshes provide more accurate results in numerical simulation by
capturing anisotropic features more faithfully. Furthermore, emerging applications in
computer graphics and geometric modeling require reliable differential geometry information
estimated on these anisotropic meshes. The first part of this thesis proposes
a special type of anisotropic meshes, called shape-preserving meshes, provides guaranteed
convergence of discrete differential operators on these meshes and devises an
algorithm for generating shape-preserving meshes on free-form surfaces based on the
mesh optimization framework with centroidal Voronoi tessellation (CVT). To improve
the numerical stability in simulation, we discuss how to reduce the number of obtuse
triangles in the mesh. The second part of the thesis discusses the non-uniqueness
of anisotropic meshes to represent the same anisotropy defined on a domain, shows
that of all anisotropic meshes, there exists one instance minimizing the number of
obtuse triangles, and proposes a variational approach to suppressing obtuse triangles
in anisotropic meshes by introducing a Minkowski metric in the CVT framework.
On a complex shape, its topological information is also highly useful to guide the
mesh generation. To extract topology properties, the Morse-Smale complex (MSC) is
a classical tool and widely used in computer graphics. However, on a manifold with
boundary, its MSC is not well defined. The final part of this thesis generalizes the MSC
to manifolds with boundaries. Based on this generalized MSC (GMSC), an operator to
merge n GMSCs of manifolds partitioning a large manifold is proposed. The merging
operator is used in a divide-and-conquer approach on a massive data set, providing
the potential to employ the computational power in a parallel manner. / published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy
| Identifer | oai:union.ndltd.org:HKU/oai:hub.hku.hk:10722/161523 |
| Date | January 2011 |
| Creators | Sun, Feng, 孙峰 |
| Contributors | Wang, WP |
| Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
| Source Sets | Hong Kong University Theses |
| Language | English |
| Detected Language | English |
| Type | PG_Thesis |
| Source | http://hub.hku.hk/bib/B4786963X |
| Rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works., Creative Commons: Attribution 3.0 Hong Kong License |
| Relation | HKU Theses Online (HKUTO) |
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