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.

Abstract:We present hierarchical occlusion maps (HOM) for visibility culling on complex models with high depth complexity. The culling algorithm uses an object space bounding volume hierarchy and a hierarchy of image space occlusion maps. Occlusion maps represent the aggregate of projections of the occluders onto the image plane. For each frame, the algorithm selects a small set of objects from the model as occluders and renders them to form an initial occlusion map, from which a hierarchy of occlusion maps is built. The occlusion maps are used to cull away a portion of the model not visible from the current viewpoint. The algorithm is applicable to all models and makes no assumptions about the size, shape, or type of occluders. It supports approximate culling in which small holes in or among occluders can be ignored. The algorithm has been implemented on current graphics systems and has been applied to large models composed of hundreds of thousands of polygons. In practice, it achieves significant speedup in interactive walkthroughs of models with high depth complexity.

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Abstract:Many applications in computer graphics and virtual environments need to render datasets with large numbers of primitives and high depth complexity at interactive rates. However, standard techniques like view frustum culling and a hardware z-buffer are unable to display datasets composed of hundred of thousands of polygons at interactive frame rates on current high-end graphics systems. We add a "conservative'' visibility culling stage to the rendering pipeline, attempting to identify and avoid processing of occluded polygons. Given a moving viewpoint, the algorithm dynamically chooses a set of occluder3. Each occluder is used to compute a 3h.adow frustum, and all primitives contained within this frustum are culled. The algorithm hierarchically traverses the model, culling out parts not visible from the current viewpoint using efficient, robust, and in some cases specialized interference detection algorithms. The algorithm's performance varies with the location of the viewpoint and the depth complexity of the model. In the worst case it is linear in the input size with a small constant. In this paper, we demonstrate its performance on a city model composed of 500,000 polygons and possessing varying depth complexity. We are able to cull an average of 55% of the polygons that would not be culled by view-frustum culling and obtain a commensurate improvement in frame rate. The overall approach is ejJecti11e and •calable, is applicable to all polygonal models, and can be easily implemented on top of view-frustum culling.

We present techniques for fast motion planning by using discrete approximations of generalized V oronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a three-dimensional distance mesh for each Voronoi site. The sites can be p oints, line segments, polygons, polyhedra, curves and surfaces. The computation of the generalized V oronoi diagram provides fast proximity query toolkits for motion planning. The tools provide the distance to the nearest obstacle stored in the Z-bu er, as well as the Voronoi boundaries, Voronoi vertices and weighted V oronoi graphs extracted f r om the frame bu er using continuation methods. We have implemented these algorithms and demonstrated their performance for path planning in a complex dynamic environment composed o f m o r e than 140,000 polygons.