Fast computation of generalized Voronoi diagrams using graphics hardware

Abstract

We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects.

Creating Watercolor Style Images Taking Into Account Painting Techniques

“…The vector field is generated using an algorithm similar to the method of Hausner [9]. After the edges are detected, a distance image whose pixels contain the distance to the nearest edge is created using the hardwareaccelerated method of Hoff [12] that creates the Voronoi diagrams by drawing cones with their apexes at the edge pixels on the screen. The vector field of stroke directions that follow the orientation of the nearby edges is created by calculating the direction perpendicular to the gradient of the distance image.…”

Medical image processing on the GPU – Past, present and future

“…A GPU can accelerate this algorithm significantly, by running many seeds in parallel. Hoff et al (1999) were one of the first to use graphics hardware to accelerate the computation of Voronoi diagrams. Sigg et al (2003) implemented the signed distance transform, using OpenGL's ARB fragment program.…”

Accelerated Discovery of Discrete M-Clusters/Outliers on the Raster Plane Using Graphical Processing Units

“…Given the nature of GPUs and the displays upon which solutions are ultimately rendered, it is natural to work in the domain of pixels, using integer coordinates. For example, Hoff et al [7] describe an algorithm to calculate generalized Voronoi diagrams. Fan et al [8] based their algorithm for natural neighbor interpolation on the same discretization of the plane.…”

“…Consider a raster surface of size 12 × 12 with 4 seeds located at the coordinates: (4,4), (5,10), (6,2) and (7,9). Assume that we have specified that we need m = 3 seeds per cluster and that the radius of every m-cluster should be r = 4.2 or smaller.…”