We focus on the problem of segmenting scattered point data into multiple regions in a single segmentation pass. To solve this problem, we begin with a set of potential boundary points and use a Delaunay triangulation to complete the boundaries. We then use information from the triangulation and its dual Voronoi complex to determine for each face whether it resembles a boundary or interior face,...

We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n1/d. For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We al...

Given a set B of n black points in general position, we say that a set of white points W blocks B if in the Delaunay triangulation of [Formula: see text] there is no edge connecting two black points. We give the following bounds for the size of the smallest set W blocking B: (i) [Formula: see text] white points are always sufficient to block a set of n black points, (ii) if B is in convex posit...

In this paper, we introduce a novel definition of the anisotropic centroidal Voronoi tessellation (ACVT) corresponding to a given Riemann metric tensor. A directional distance function is used in the definition to simplify the computation. We provide algorithms to approximate the ACVT using the Lloyd iteration and the construction of anisotropic Delaunay triangulation under the given Riemannian...

Delaunay Triangulation (DT) is one of the most fundamental data structures in computational geometry. It is well known in geosciences for many years to model the real world objects. To extend its applications for simulating the real world processes, however, this data structure must support dynamic point sets (where the points are added or deleted) and kinetic point sets (where the position of ...

Various graph-theoretic definitions of neighborhood have been used in the past in computer vision and clustering applications. In this paper we study the properties of a set of four related closest-point graphs using Monte Carlo methods: (i) the Delaunay Triangulation (DT) and its dual, Voronoi tessellation, (ii) the Gabriel Graph (GG), (iii) the Relative Neighborhood Graph (RNG), and (iv) the ...

Using the domain-theoretic model for geometric computation, we define the partial Delaunay triangulation and the partial Voronoi diagram of N partial points in R2 and show that these operations are domain-theoretically computable and effectively computable with respect to Hausdorff distance and Lebesgue measure. These results are obtained by showing that the map which sends three partial points...

We interpret a Voronoi region as the shape achieved by a crystal that grows from a seed and stops growing when it reaches either the domain boundary or another crystal. Using this analogy we devise a method for generating anisotropic boundary-conforming Voronoi regions and their dual Delaunay triangulation for a set of points. The method simulates the propagation of crystals as evolving fronts ...

Delaunay triangulation is always used to construct TIN, and is also widely applied in manifold fields, for it can avoid long and skinny triangles resulting in a nice looking map. A wide variety of algorithms have been proposed to construct delaunay triangulation, such as divide-and-conquer, incremental insertion, trangulation growth, and so on. The compound algorithm is also researched to const...