Optimality of the Delaunay triangulation in ℝ d

In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ ^d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ ^d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.     

Density based fuzzy c-means clustering of non-convex patterns

We propose a new technique to perform unsupervised data classification (clustering) based on density induced metric and non-smooth optimization. Our goal is to automatically recognize multidimensional clusters of non-convex shape. We present a modification of the fuzzy c-means algorithm, which uses the data induced metric, defined with the help of Delaunay triangulation. We detail computation of the distances in such a metric using graph algorithms. To find optimal positions of cluster prototypes we employ the discrete gradient method of non-smooth optimization. The new clustering method is capable to identify non-convex overlapped d-dimensional clusters.     

Efficiently updating constrained Delaunay triangulations

TheConstrained Delaunay Triangulation of a set of obstacle line segments in the plane is the Delaunay triangulation of the endpoint set of these obstacles with the restriction that the edge set of the triangulation contains all these obstacles. In this paper we present an optimal (logn +k) algorithm for inserting an obstacle line segment or deleting an obstacle edge in the constrained Delaunay triangulation of a set ofn obstacle line segments in the plane. Herek is the number of Delaunay edges deleted and added in the triangulation during the updates.This work is supported by NSERC grant OPG0041629.     

Optimal Delaunay Triangulations

The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in L^P-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function ‖x‖^2 among all the triangulations with a given set of vertices. For a more general function, a functiondependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-depend entoptimal Delaunay triangulation is proved to exist for any given convex continuous function.On an optimal Delaunay triangulation associated with f, it is proved that △↓f at the interior vertices can be exactly recovered by the function values on its neighboring vertices.Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.     

A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in R³ may be quadratic in the worst case, we show in this paper that it is only linear when the points are distributed on a fixed set of well-sampled facets of R³ (e.g. the planar polygons in a polyhedron). Our bound is deterministic and the constants are explicitly given.     

Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

It is well known that the complexity of the Delaunay triangulation of $n$ points in $\RR ^d$, i.e., the number of its simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In particular, in $\RR ^3$, the number of tetrahedra can be quadratic. Put another way, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the three-dimensional Delaunay triangulation of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^{1.8})$ for general polyhedral surfaces and $O(n\sqrt{n})$ for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points.     

Generalized delaunay triangulation for planar graphs

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graphG=(V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs inO(|V|²) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-and-conquer algorithm based on the polygon cutting theorem of Chazelle is given that runs inO(|V| log |V|) time.Supported in part by the National Science Foundation under Grants DCR 8420814 and ECS 8340031.     

Quality mesh generation for molecular skin surfaces using restricted union of balls

Quality surface meshes for molecular models are desirable in the studies of protein shapes and functionalities. However, there is still no robust software that is capable to generate such meshes with good quality. In this paper, we present a Delaunay-based surface triangulation algorithm generating quality surface meshes for the molecular skin model. We expand the restricted union of balls along the surface and generate an ε-sampling of the skin surface incrementally. At the same time, a quality surface mesh is extracted from the Delaunay triangulation of the sample points. The algorithm supports robust and efficient implementation and guarantees the mesh quality and topology as well. Our results facilitate molecular visualization and have made a contribution towards generating quality volumetric tetrahedral meshes for the macromolecules.     

The greedy triangulation can be computed from the Delaunay triangulation in linear time

The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that does not cross those already in the plane. The Delaunay triangulation, which is the straight-line dual of the Voronoi diagram, can be produced in O(nlogn) worst-case time, and often even faster, by several practical algorithms. In this paper we show that for any planar point set S, if the Delaunay triangulation of S is given, then the greedy triangulation of S can be computed in linear worst-case time (and linear space).     

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

An O (n2) heuristic algorithm for the directed Steiner minimal tree problem

This paper describes the design of an O(N²) heuristic algorithm for the Directed Steiner Minimal Tree (DSMT) problem in the plane. The algorithm employs the Delaunay triangulation, a minimum spanning arborescence algorithm, and a triangulation of the arborescence to develop solutions to the DSMT problem. Example applications are also included.     

Aspects of the Facet-Edge Data Structure for the Construction of Voronoi Cells

The construction of Voronoi diagrams has two main aspects: The construction algorithm and the data structure. For the construction of planar Voronoi diagrams e.g. the well known plane sweep algorithm can be applied. Another efficient method is the incremental construction of the Delaunay triangulation by using the quad-edge data structure. Within this data structure the Delaunay triangulation is treated as a planar graph. Incremental construction implies that manipulations of the triangulation are allowed. Its Voronoi diagram is obtained simply by accessing the triangulation's dual graph. We extend this method to tree dimensions, by implementing a 3D Delaunay triangulation on the facet-edge data structure. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Meshing skin surfaces with certified topology

Skin surfaces are used for the visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used for approximation purposes.

We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal.

We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.     

On the stretch factor of Delaunay triangulations of points in convex position

Let S be a finite set of points in the Euclidean plane. Let G be a geometric graph in the plane whose point set is S. The stretch factor of G is the maximum ratio, among all points p and q in S, of the length of the shortest path from p to q in G over the Euclidean distance |pq|. Keil and Gutwin in 1989 [11] proved that the stretch factor of the Delaunay triangulation of a set of points S in the plane is at most 2π/(3cos(π/6))≈2.42. Improving on this upper bound remains an intriguing open problem in computational geometry.In this paper we consider the special case when the points in S are in convex position. We prove that in this case the stretch factor of the Delaunay triangulation of S is at most ρ=2.33.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

Reconstruction Using Witness Complexes

We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch

A local C ¹ positivity preserving scheme is developed using Bernstein-Bézier rational cubic function. The domain is triangulated by Delaunay triangulation method. Simple sufficient conditions are derived on the inner and boundary Bézier ordinates to preserve the shape of positive data. These inner and boundary Bézier ordinates involve weights in their definition. In any triangular patch if the Bézier ordinates do not satisfy the derived conditions of positivity, then these are modified by the weights (free parameters) involved in the construction of Bernstein-Bézier rational cubic function to preserve the shape of positive scattered data.     

基于Delaunay三角剖分的地表平均温度模拟

利用Delaunay三角网对目标区域进行剖分,在对地表温度进行高度插值后,运用二重积分的思想建立了基于Delaunayr三角剖分的地表平均温度测量模型.同时以南极地表平均温度的测量为例,将67个自动气象地表台站、46个气象地表台站以及56个高空气象观测站的加权平均温度与地表平均温度的数据进行分析,得到南极2015全年地表平均温度均在-8℃以下,最低温约为-20℃,符合南极大陆地表温度的实际情况.