Computing Three-dimensional Constrained Delaunay Refinement Using the GPU

We propose the first GPU algorithm for the 3D triangulation refinement problem. For an input of a piecewise linear complex $\mathcal{G}$ and a constant $B$, it produces, by adding Steiner points, a constrained Delaunay triangulation conforming to $\mathcal{G}$ and containing tetrahedra mostly of radius-edge ratios smaller than $B$. Our implementation of the algorithm shows that it can be an order of magnitude faster than the best CPU algorithm while using a similar amount of Steiner points to produce triangulations of comparable quality

Optimal Two-Dimensional Triangulations

A(geometric) triangulation in the plane is a maximal connected plane graph with straight edges. It is thus a plane graph whose bounded faces are triangles. For a xed set of vertices, there are, in general, exponentially many ways to form a triangulation. Various criteria related to the geometry of triangles are used to de ne what one could mean by a triangulation that is optimal over all possibilities. The general problem studied in this thesis is the following: given a nite set S of vertices, possibly with some prescribed edges, how canwe choose the rest of the edges to obtain an optimal triangulation? Just to mention an example, we areinterested in computing a min-max angle triangulation of S, that is, a triangulation whose maximum angle over all its triangles is the smallest among all triangulations of S. This thesis presents a number of new algorithms to construct optimal triangulations useful in engineering and scienti c computations, such as nite element analysis and surface interpolation. All algorithms are the rst and, currently, the only ones that construct the de ned optimal triangulations in time polynomial in the input size. These main results are described in three parts

INTEGRATED CIRCUIT DETAILED ROUTING

Master'sMASTER OF SCIENC

Optimal bound for conforming quality triangulations

Proceedings of the Annual Symposium on Computational Geometry240-24915

Optimal Triangulation Problems

This paper surveys some recent solutions to triangulation problems in 2D plane and surface. In particular, it focuses on three efficient and practical schemes in computing optimal triangulations useful in engineering and scientific computations, such as nite element analysis and surface interpolation. The edge-insertion paradigm can compute for a set of n vertices, with or without constraining edges, a minmax angle and a max-min height triangulation in O(n 2 log n) time and O(n) storage, and a min-max slope anda min-max eccentricity triangulation in O(n³) time and O(n²) storage. The subgraph scheme can compute a min-max length triangulation for a set of n vertices in O(n²) time and storage. Length refers to edge length and is measured by some normed metric such as the Euclidean or any other ` p metric. Additionally, the scheme provides some insight to the minimum weight triangulation problem. The wall scheme can compute for a given set of n vertices and m constraining edges, a conforming Delaunay triangulation of O(m² n) vertices. Additionally, an extension of the wall scheme can refine a triangulation of size O(n) to a quality triangulation of size O(n²) that has no angle measuring more than 11 15

Optimal Two-Dimensional Triangulations

122 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993.A triangulation in the plane is a maximal connected plane graph with straight edges. It is thus a plane graph whose bounded faces are triangles. For a fixed set of vertices, there are, in general, exponentially many ways to form a triangulation. Various criteria related to the geometry of triangles are used to define what one could mean by a triangulation that is optimal over all possibilities. The general problem studied in this thesis is the following: given a finite set S of vertices, possibly with some prescribed edges, how can we choose the rest of the edges to obtain an optimal triangulation? For example, we want to compute a min-max angle triangulation of S, i.e., a triangulation whose maximum angle over all its triangles is the smallest among all triangulations of S.This thesis presents a number of new algorithms to construct optimal triangulations useful in engineering the scientific computations, such as finite element and surface interpolation. All algorithms are currently the only ones that construct the defined optimal triangulations in time polynomial in the input size. These results are described in three parts.First, we develop a new algorithmic technique called the edge-insertion paradigm. It computes for a set of n vertices an optimal triangulation defined by some generic criterion. We then deduce that a min-max angle and a max-min height triangulation can be computed in O(n\sp2\ \log n) time, and a min-max slope and a min-max eccentricity triangulation in cubic time.Second, we show that a min-max length triangulation for a set of n vertices can be computed in quadratic time. Length refers to edge length and is measured by some normed metric such as any l\sb{p} metric.Third, for a given plane graph of n vertices and m non-crossing edges, we prove that there is a set of O(m\sp2n) points so that, for each adjacent pair of points on an edge, there exists a circle passing through the two points that encloses no other points. This implies an efficient way to construct a Delaunay triangulation that subdivides the plane graph.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

TEXTURE SIMPLIFICATION

A texture simplification problem has emerged from our need to present a city model effectively. One way of doing this is by rendering it in non-photorealistic (NPR) manner. We are exploring this area and use the idea of modifying and simplifying texture to deliver NPR rendering. We map the texture from spatial domain to feature space domain and consider the local maximum as the important features. Each other points in the feature space are associated with one of the local maximum. This is done by applying mean shift algorithm. Texture simplification is carried out by manipulating the mean shift process. Our results show the effectiveness of our texture simplification algorithm. We also demonstrate the application of our algorithm in non-photorealistic city model rendering.