A simple GPU-based approach for 3D Voronoi diagram construction and visualization

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Abstract

In this paper we propose a simple GPU-based approach for discrete incremental approximation of 3D Voronoi diagram. By constructing region maps via GPU. Nearest sites, space clustering, and shortest distance query can be quickly answered by looking up the region map. In addition, we propose another representation of the 3D Voronoi diagram for visualization.

Introduction

The Voronoi diagram is a natural and fundamental concept related to geometric proximity that has important applications in many fields of science. Given a set of primitives called Voronoi sites, the Voronoi diagram partitions the space into convex regions so that all of the subspaces belonging to a region have the same closest site under a given distance function. There are many modified Voronoi diagrams by taking sites as different shapes or nature, applying different distance functions, associating weights, or changing metrics. Voronoi diagram is also used in many fundamental problems such as nearest neighbor query, clustering, triangulation, and minimum spanning tree.

Constructing a Voronoi diagram in 2D or 3D is a fundamental problem and there exist efficient and optimal algorithms running on CPU to calculate Voronoi vertices which are the intersections of the boundaries of domains. Voronoi diagram is structured by connected segments of edges or curves in 2D, or connected polygons or paraboloids in 3D. However, it needs additional computation to handle queries for nearest sites or shortest distance to sites. Therefore, we propose a discrete incremental approximation of 3D Voronoi diagram by constructing region maps by GPU. Nearest sites and shortest distance query can be quickly answered by looking up region maps. In addition, we can visualize the Voronoi diagram in a different representation.

In this paper, we introduce a simple but flexible approximate construction of the 3D Voronoi diagram using GPU for visualization. It could generate a map for visualization or generate a region map for query of nearest neighbor, space clustering, and shortest distance. The Voronoi diagram construction in GPU is simple. At first, we subdivide the space into subspaces by GPU. Then for each Voronoi site, distance between a subspace, called a voxel, and the site is calculated and compared with the shortest distance stored in the voxel. Each voxel keeps the shortest distance and the region ID it belongs. In current implementation, we use color as region ID and z-buffer for storing shortest distance of each voxel.

The rest of paper is organized as follows. Some related works are surveyed in Section 2. In Section 3, we present the rendering framework. The experimental results are illustrated in Section 4. Finally, we conclude the proposed approach and point out some future works.

Section snippets

Related works

For computing Voronoi diagrams of points in 2D, 3D, and higher dimensions, Shamos and Hoey [1] proposed a divide-and-conquer algorithm; Fortune [2] proposed a sweepline algorithm. Gold et al. [3] proposed a simple method to construct Voronoi diagram of line segments sites by incrementally expanding line segments and using kinematic Voronoi diagram methods to maintain the Voronoi diagram. Besides, numerically robust algorithms for constructing topologically consistent Voronoi diagrams have been

Basic definition

Let S = {s1, s2 ,  , sn} denotes a set of point sites in 3D. For two distinct sites sp, sq  S, the space v can be partitioned into two half spaces or domains. Domain of sp over sq can be formed byDom(sp,sq)={vR3|σ(v,sp)σ(v,sq)},where σ is a distance function. Here we use squared Euclidean distance function for lower computation cost. Then the Voronoi region of sp can be formed byR(sp)=sqS-{sp}Dom(sp,sq).Therefore each region R(sp) will cluster the subspace nearest to the site sp by the given

Experimental results

We have implemented our fragment program using HLSL on a Pentium 4 3.0 MHz PC with 1G RAM and a nVidia Geforce FX5800 graphics card running on Windows XP with DirectX 9.0c. We use Vertex Shader 1.1 and Pixel Shader 2.0 to implement fragment program in scattering pairs of voxel positions and Voronoi site data and in distance calculation and Voronoi diagram visualization.

3D Voronoi diagrams in the experiment are calculated on a slice under voxel resolution of 5122. All Voronoi sites are randomly

Conclusion

In this paper, we introduce a GPU-based approach for 3D Voronoi diagram construction and visualization. We calculate the minimum distance between pairs of sampled voxels and Voronoi sites in different shapes for guarantee of Hausdorff distance. With programmable hardware vertex/pixel processors, a Voronoi diagram can be incrementally constructed by rendering simple quads with associated data of Voronoi sites and visualized with different representation.

However, in current implementation,

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