This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

Robust and time-efficient skeletonization of a (planar) shape, which is connectivity preserving and based on Euclidean metrics, can be achieved by first regularizing the Voronoi diagram (VD) of a shape's boundary points, i.e., by removal of noise-sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents. Each component of the VD is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise. The second processing step, a hierarchic clustering of skeleton branches, leads to a multiresolution representation of the skeleton, termed skeleton pyramid.

This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n ).

Vortex methods for inviscid incompressible two-dimensional fluid flow are usually based on blob approximations. This paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a Delaunay triangulation of the vortices. An efficient reconstruction of the Delaunay triangulation at each step makes the method accurate for long times. The vertices of the triangulation move with the fluid velocity, which is reconstructed from the vorticity via a simplified fast multipole method for the Biot-Savart law with a continuous source distribution. The initial distribution of vortices is constructed from the initial vorticity field by an adaptive approximation method which produces good accuracy even for discontinuous initial data. Numerical results show that the method is highly accurate over long time intervals. Experiments with single and multiple circular and elliptical rotating patches of both piecewise constant and smooth vorticity indicate that the me...

Centroidal Voronoi tessellation (CVT) is a fundamental geometric structure that finds many applications

Let S be a set of n sites chosen independently from a uniform distribution in a cube in 3 dimensional Euclidean space. In this paper, work by Bentley, Weide and Yao is extended to show that the Voronoi diagram for S has an expected O(n) number of faces. A consequence of the proof of this result is that the Voronoi diagram for S can be constructed in expected O(n) time. 1. INTRODUCTION Consider a set S = fp 1 ; : : : ; p n g of n sites in d dimensional Euclidean space E d . The Voronoi diagram for S is a sequence V (p 1 ), : : : , V (p n ) of convex polyhedra covering E d , where for each i, i = 1; : : : ; n, V (p i ) is the Voronoi polyhedron of p i relative to S, i. e. the set of all points x in the space such that p i is as close to x as is any other site in S. The Voronoi diagram is an important geometrical concept that is used for solving a large number of problems in many areas. Accordingly, several algorithms have been devised and implemented for constructing it in two and...

The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image. The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms. In contrast to the generic case, it can be shown that the topology of the grid permits the algorithm to run faster on complex scenes. Consequently, in Computer Vision applications, the computation of the Voronoi tessellation represents an attractive alternative to rasterbased techniques in terms of both computational complexity and quality of data structures. Index terms ---Tessellation, Computational Geometry, Delaunay triangulation, Voronoi diagram. 1. Introduction The concept of the Voronoi diagram (also termed Voronoi or Dirichlet tessellation) refers to one of the basic closest point probl...

Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd’s method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd’s method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.

In geographical databases for navigation, users raise various types of queries concerning route guidance. The most fundamental query is a shortest-route query, but, as dynamical traffic information newly becomes available and the static geographical database of roads itself has grown up further, more flexible queries are required to realize a user-friendly interface meeting the current settings. One important query among them is a detour query which provides information about detours, say listing several candidates for useful detours. We have proposed efficient algorithms for enumerating meaningful detours [6, 14]. In this paper, we first review our algorithms for the static case, and discuss their extensions to incorporate dynamical information in an efficient manner. Also, in connection with the user interface part, animation of the proposed algorithm is performed, and its prototype version is made public via WWW. In a more general setting, we discuss data mining of this rapidly grow...

A mathematical description and representation of 2D and 3D shape, capable of hierarchically decomposing complex objects, is at the focus of this paper. The development is based on a hierarchic extension to the Medial Axis Transformation (MAT). Our implementation of the hierarchic MAT (HMAT) combines full Voronoi tessellation generated by the set of border points with regularization procedures to obtain a hierarchy of geometrically and topologically correct medial manifolds. This hierarchy defines a skeleton pyramid allowing shape-driven decomposition in 2D and 3D. The proposed methodology is illustrated in 2D on a planar section through a 3D MRI data set of the human brain. The hierarchical decomposition indicates the process history of brain development. For further analysis, it is converted into a boundary representation. The 3D extension of the HMAT concept is tested and illustrated on synthetic objects. It is applied to the full 3D MRI data set to obtain a description of the sulci ...