In this paper, we introduce the fuzzy Voronoi diagram as an extension of the Voronoi diagram. We assume Voronoi sites to be fuzzy points and then define the Voronoi diagram for this kind of sites, then we provide an algorithm for computing this diagram based on Fortune’s algorithm which costs O(n log n) time. Also we introduce the fuzzy Voronoi diagram for a set of fuzzy circles, rather than fu...

We study a probabilistic algorithm for the computation of the centroidal Voronoi tessellation which is a Voronoi tessellation of a given set such that the associated generating points are centroids (centers of mass) of the corresponding Voronoi regions. We discuss various issues related to the implementation of the algorithm and provide numerical results. Some measures to improve the performanc...

We present linear-time algorithms to construct tree-like Voronoi diagrams with disconnected regions after the sequence of their faces along an enclosing boundary (or at infinity) is known. We focus on the farthest-segment Voronoi diagram, however, our techniques are also applicable to constructing the order-(k+1) subdivision within an order-k Voronoi region of segments and updating a nearest-ne...

This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvature-adapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d or larger). In this high dimensional space, the mesh is optimized by computing a Centroidal Voron...

We present a novel algorithm to compute a homotopy preserving bounded-error approximate Voronoi diagram of a 3D polyhedron. Our approach uses spatial subdivision to generate an adaptive volumetric grid and computes an approximate Voronoi diagram within each grid cell. Moreover, we ensure each grid cell satisfies a homotopy preserving criterion by computing an arrangement of 2D conics within a p...

Vector quantization is a classical signal-processing technique with significant applications in data compression, pattern recognition, clustering, and data stream mining. It is well known that for critical points of the quantization energy, the tessellation of the domain is a centroidal Voronoi tessellation. However, for dimensions greater than one, rigorously verifying a given centroidal Voron...

We introduce a new approach to approximate generalized 3D Voronoi diagrams for different site shapes (points, spheres, segments, lines, polyhedra, etc) and different distance functions (Euclidean metrics, convex distance functions, etc). The approach is based on an octree data structure denoted Voronoi-Octree (VO) that encodes the information required to generate a polyhedral approximation of t...

We present structural properties of the farthest line-segment Voronoi diagram in the piecewise linear L∞ and L1 metrics, which are computationally simpler than the standard Euclidean distance and very well suited for VLSI applications. We introduce the farthest line-segment hull, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram, and is related...

Motivated by the deterministic single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem for the Euclidean norm, we study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show that for strictly convex and smooth norms the geometry of Voronoi cells of lattices in any dimensio...

Given a set of points in IR , called sites, we consider the problem of approximating the Voronoi cell of a site by a convex polyhedron with a small number of facets or, equivalently, of finding a small set of approximate Voronoi neighbors of . More precisely, we define an -approximate Voronoi neighborhood of , denoted , to be a subset of satisfying the following property: is an -approximate nea...