John Keyser

Fast computation of generalized Voronoi diagrams using graphics hardware

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. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation. Additional

Simultaneous shape decomposition and skeletonization

Shape decomposition and skeletonization share many common properties and applications. However, they are generally treated as independent computations. In this paper, we propose an iterative approach that simultaneously generates a hierarchical shape decomposition and a corresponding set of multi-resolution skeletons. In our method, a skeleton of a model is extracted from the components of its decomposition --- that is, both processes and the qualities of their results are interdependent. In particular, if the quality of the extracted skeleton does not meet some user specified criteria, then the model is decomposed into finer components and a new skeleton is extracted from these components. The process of simultaneous shape decomposition and skeletonization iterates until the quality of the skeleton becomes satisfactory. We provide evidence that the proposed framework is efficient and robust under perturbation and. deformation. We also demonstrate that our results can readily be used in problems including skeletal deformations and virtual reality navigation.

Interactive motion planning using hardware-accelerated computation of generalized Voronoi diagrams

We present techniques for fast motion planning by using discrete approximations of generalized Voronoi 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 3D distance mesh for each Voronoi site. The sites can be points, line segments, polygons, polyhedra, curves and surfaces. The computation of the generalized Voronoi diagram provides fast proximity query toolkits for motion planning. The tools provide the distance to the nearest obstacle stored in the Z-buffer, as well as the Voronoi boundaries, Voronoi vertices and weighted Voronoi graphs extracted from the frame buffer using continuation methods. We have implemented these algorithms and demonstrated their performance for path planning in a complex dynamic environment composed of more than 140,000 polygons.

Accurate computation of the medial axis of a polyhedron

We present an accurate and efficient algorithm to compute the internal Voronoi region and medial axis of a 3-D polyhedron. It uses exact arithmetic and representations for accurate computation of the medial axis. The sheets, seams, and junctions of the medial axis are represented as trimmed quadric surfaces, algebraic space curves, and points with algebraic coordinates, respectively. The algorithm works by recursively finding neighboring junctions along the seam curves. It uses spatial decomposition and linear programming to speed up the search step. We also present a new algorithm for analysis of the topology of an algebraic plane curve, which is the core of our medial axis algorithm. To speed up the computation, we have designed specialized algorithms for fast computation on implicit geometric structures. These include lazy evaluation based on multivariate Stiirm sequences, fast resultant computation, curve topology analysis, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

Wave particles

We present a new method for the real-time simulation of fluid surface waves and their interactions with floating objects. The method is based on the new concept of wave particles, which offers a simple, fast, and unconditionally stable approach to wave simulation. We show how graphics hardware can be used to convert wave particles to a height field surface, which is warped horizontally to account for local wave-induced flow. The method is appropriate for most fluid simulation situations that do not involve significant global flow. It is demonstrated to work well in constrained areas, including wave reflections off of boundaries, and in unconstrained areas, such as an ocean surface. Interactions with floating objects are easily integrated by including wave forces on the objects and wave generation due to object motion. Theoretical foundations and implementation details are provided, and experiments demonstrate that we achieve plausible realism. Timing studies show that the method is scalable to allow simulation of wave interaction with several hundreds of objects at real-time rates.

Exact computation of the medial axis of a polyhedron

We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding neighboring junctions along the seam curves. To speed up the computation, we have designed specialized algorithms for fast computation with algebraic curves and surfaces. These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

Mesh colors

The coloring of 3D models using 2D or 3D texture mapping has well-known intrinsic problems, such as mapping discontinuities and limitations to model editing after coloring. Workarounds for these problems often require adopting very complex approaches. Here we propose a new technique, called mesh colors, for associating color data directly with a polygonal mesh. The approach eliminates problems deriving from using a map from texture space to model space. Mesh colors is an extension of vertex colors where, in addition to keeping color values on each vertex, they are also kept on edges and faces. Like texture mapping, the approach allows higher texture resolution than model resolution, but at the same time it guarantees one-to-one correspondence between the model surface and the color data, and eliminates discontinuities. We show that mesh colors integrate well with the current graphics pipeline and can be used to generate very high-quality textures.

Dual scattering approximation for fast multiple scattering in hair

When rendering light colored hair, multiple fiber scattering is essential for the right perception of the overall hair color. In this context, we present a novel technique to efficiently approximate multiple fiber scattering for a full head of human hair or a similar fiber based geometry. In contrast to previous ad-hoc approaches, our method relies on the physically accurate concept of the Bidirectional Scattering Distribution Functions and gives physically plausible results with no need for parameter tweaking. We show that complex scattering effects can be approximated very well by using aggressive simplifications based on this theoretical model. When compared to unbiased Monte-Carlo path tracing, our approximations preserve photo-realism in most settings but with rendering times at least two-orders of magnitude lower. Time and space complexity are much lower compared to photon mapping-based techniques and we can even achieve realistic results in real-time on a standard PC with consumer graphics hardware.

Deep Opacity Maps

We present a new method for rapidly computing shadows from semi‐transparent objects like hair. Our deep opacity maps method extends the concept of opacity shadow maps by using a depth map to obtain a per pixel distribution of opacity layers. This approach eliminates the layering artifacts of opacity shadow maps and requires far fewer layers to achieve high quality shadow computation. Furthermore, it is faster than the density clustering technique, and produces less noise with comparable shadow quality. We provide qualitative comparisons to these previous methods and give performance results. Our algorithm is easy to implement, faster, and more memory efficient, enabling us to generate high quality hair shadows in real‐time using graphics hardware on a standard PC.

ESOLID¿a system for exact boundary evaluation

Abstract We present a system, ESOLID, that performs exact boundary evaluation of low-degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating-point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating-point filters, arbitrary floating-point arithmetic with error bounds, and lower-dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating-point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating-point boundary evaluation system on most cases.