# Voronoi diagram

(redirected from Voronoi decomposition)

## Voronoi diagram

(mathematics, graphics)
(After G. Voronoi) For a set S of points in the Euclidean plane, the partition Vor(S) of the plane into the voronoi polygons associated with the members of S. Vor(S) is the dual of the Delaunay triangulation of S.
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To study the statistical properties of the distance (denoted as R throughout this manuscript) from a fixed vertex of a triangle to a random point that is uniformly distributed on the interior of this triangle is important because many distance-optimization problems depend on Voronoi decomposition, in which the entire plane is divided into polygons called Voronoi cells, which can be further decomposed into triangles.
(3) How different types of Voronoi decomposition can affect the aesthetic qualities of the rendered images?
The cloud model-based rendering consists of four main steps, including random reference points-based or image reference-based Voronoi decomposition, uncertain line representation using the extended cloud model, uncertain Voronoi polygon approximation filling, and then generating an aesthetic image with the Voronoi art style.
Image-Guided Voronoi Decomposition. For image-guided Voronoi art, the generation of Voronoi diagram is completely dependent on the image content.
Then the Voronoi decomposition is generated with an outer-boundary constraint, and we use the VoronoiLimit method proposed by Jakob [26].
Given the width w, the height h, and the density factor p, the reference points are randomly generated, and the Voronoi decomposition result is varied for each different running.
After the Voronoi polygons are determined by Voronoi decomposition, the rendering can be done in a natural way.
Given the f-sided polygon of a Voronoi decomposition, the vertex set {[P.sub.1], [P.sub.2], ..., [P.sub.t]} (t [greater than or equal to] 3), and its centroid [P.sub.r] = (p[x.sub.r], p[y.sub.r]) (also as the reference point), the sides are determined by [P.sub.1] [P.sub.2], [P.sub.2] [P.sub.3], ..., [P.sub.t-1] [P.sub.t], [P.sub.t] [P.sub.1], and each side is represented by n cloud drops according to the previous section; then the polygon corresponds to nt sample points.
Using the provided GUI, the users control the type of Voronoi decomposition and fix the parameters.
Caption: Figure 3: Voronoi decomposition: (a) the dithered IE logo image, (b) Voronoi decomposition for (a), and (c) an unsupervised decomposition.

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