Applications of Surface Networks to Sampling Problems in Computer Graphics

This thesis develops the theory, algorithms and data structures for adaptive sampling of parametric functions, which can represent the shapes and motions of physical objects. For the first time, ensured methods are derived for determining collisions and other interactions for a broad class of parametric functions. A new data structure, called a surface network, is developed for the collision algorithm and for other sampling problems in computer graphics. A surface network organizes a set of parametric samples into a hierarchy. Surface networks are shown to be good for rendering images, for approximating surfaces, and for modeling physical environments. The basic notion of a surface network is generalized to higher-dimensional problems such as collision detection. We may think of a two-dimensional network covering a three-dimensional solid, or an n-dimensional network embedded in a higher-dimensional space. Surface networks are applied to the problems of adaptive sampling of static parametric surfaces, to adaptive sampling of time-dependent parametric surfaces, and to a variety of applications in computer graphics, robotics, and aviation. First we develop the theory for adaptive sampling of static surfaces. We explore bounding volumes that enclose static surfaces, subdivision mechanisms that adjust the sampling density, and subdivision criteria that determine where samples should be placed. A new method is developed for creating bounding ellipsoids of parametric surfaces using a Lipschitz condition to place bounds on the derivatives of parametric functions. The bounding volumes are arranged in a hierarchy based on the hierarchy of the surface network. The method ensures that the bounding volume hierarchy contains the parametric surface completely. The bounding volumes are useful for computing surface intersections. They are potentially useful for ray tracing of parametric surfaces. We develop and examine a variety of subdivision mechanisms to control the sampling process for parametric functions. Some of the methods are shown to improve the robustness of adaptive sampling. Algorithms for one mechanism, using bintrees of right parametric triangles, are particularly simple and robust. A set of empirical subdivision criteria determine where to sample a surface, when we have no additional information about the surface. Parametric samples are concentrated in regions of high curvature, and along intersection boundaries. Once the foundations of adaptive sampling for static surfaces are described, we examine time-dependent surfaces. Based on results with the empirical subdivision criteria for static surfaces, we derive ensured criteria for collision determination. We develop a new set of rectangular bounding volumes, apply a standard L-dimensional subdivision mechanism called k-d trees, and develop criteria for ensuring that we detect collisions between parametric surfaces. We produce rectangular bounding boxes using a "Jacobian''-style matrix of Lipschitz conditions on the parametric function. The rectangular method produces even tighter bounds on the surface than the ellipsoidal method, and is effective for computing collisions between parametric surfaces. A new collision determination technique is developed that can detect collisions of parametric functions, based on surface network hierarchies. The technique guarantees that the first collision is found, to within the temporal accuracy of the computation, for surfaces with bounded parametric derivatives. Alternatively, it is possible to guarantee that no collisions occur for the same class of surfaces. When a collision is found, the technique reports the location and parameters of the collision as well as the time of first collision. Finally, we examine several applications of the sampling methods. Surface networks are applied to the problem of converting a two-dimensional image, or texture map, into a set of triangles that tile the plane. Many polygon-rendering systems do not provide the capability of rendering surfaces with textures. The technique converts textures to triangles that can be rendered directly by a polygon system. In addition, potential applications of the collision determination techniques are discussed, including robotics and air-traffic control problems

SIGNAL PROCESSING AT 250 MHZ USING HIGH-PERFORMANCE FPGA's

This paper describes an application in high-performance signal processing using reconfigurable computing engines: a 250 MHz cross-correlator for radio astronomy. Experimental results indicate that CMOS FPGA's can perform useful computation at 250 MHz. The notion of an "event horizon" for FPGA's leads to clear design constraints for high-speed application developers, and can be applied to a variety of real-time signal processing algorithms. Recent estimates indicate that higher-performance FPGA's available early in 1998 can attain speeds of over 300 MHz using 20% fewer logic elements than current designs. The results of this design work provide important clues on how to improve FPGA architectures for signal processing at hundreds of MHz. Direct routing channels between logic elements can significantly increase performance. Routing architectures with four-way symmetry allow for rotations and reflections of subcircuits needed for optimal packing density. Experimental results indicate that..

Accurate triangulations of deformed, intersecting surfaces

A quadtree algorithm is developed to triangulate deformed, intersecting parametric surfaces. The biggest problem with adaptive sampling is to guarantee that the triangulation is accurate within a given tolerance. A new method guarantees the accuracy of the triangulation, given a "Lipschitz" condition on the surface definition. The method constructs a hierarchical set of bounding volumes for the surface, useful for ray tracing and solid modeling operations. The task of adaptively sampling a surface is broken into two parts: a subdivision mechanism for recursively subdividing a surface, and a set of subdivision criteria for controlling the subdivision process. An adaptive sampling technique is said to be robust if it accurately represents the surface being sampled. A new type of quadtree, called a restricted quadtree, is more robust than the traditional unrestricted quadtree at adaptive sampling of parametric surfaces. Each sub-region in the quadtree is half the width of the previous region. The restricted quadtree requires that adjacent regions be the same width within a factor of two, while the traditional quadtree makes no restriction on neighbor width. Restricted surface quadtrees are effective at recursively sampling a parametric surface. Quadtree samples are concentrated in regions of high curvature, and along intersection boundaries, using several subdivision criteria. Silhouette subdivision improves the accuracy of the silhouette boundary when a viewing transformation is available at sampling time. The adaptive sampling method is more robust than uniform sampling, and can be more efficient at rendering deformed, intersecting parametric surfaces

Real-Time Stereo Vision on the PARTS Reconfigurable Computer

This paper describes a powerful, scalable, reconfigurable computer called the PARTS engine. The PARTS engine consists of 16 Xilinx 4025 FPGAs, and 16 one-megabyte SRAMs. The FPGAs are connected in a partial torus--- each associated with two adjacent SRAMs. The SRAMs are tightly coupled to the FPGAs so that all the SRAMs can be accessed concurrently. The PARTS engine fits on a standard PCI card in a personal computer or workstation. The first application implemented on the PARTS engine is a depth from stereo vision algorithm that computes 24 stereo disparities on 320 by 240 pixel images at 42 frames per second. Running at this speed, the engine is performing approximately 2.3 billion RISC-equivalent operations per second, accessing memory at a rate of 500 million bytes per second and attaining throughput of over 70 million point disparity measurements per second. 1. INTRODUCTION This paper describes a powerful, scalable, reconfigurable computer called the PARTS engine. The acronym P..

Accurate triangulations of deformed, intersecting surfaces

A quadtree algorithm is developed to triangulate deformed, intersecting parametric surfaces. The biggest problem with adaptive sampling is to guarantee that the triangulation is accurate within a given tolerance. A new method guarantees the accuracy of the triangulation, given a "Lipschitz" condition on the surface definition. The method constructs a hierarchical set of bounding volumes for the surface, useful for ray tracing and solid modeling operations. The task of adaptively sampling a surface is broken into two parts: a subdivision mechanism for recursively subdividing a surface, and a set of subdivision criteria for controlling the subdivision process. An adaptive sampling technique is said to be robust if it accurately represents the surface being sampled. A new type of quadtree, called a restricted quadtree, is more robust than the traditional unrestricted quadtree at adaptive sampling of parametric surfaces. Each sub-region in the quadtree is half the width of the previous region. The restricted quadtree requires that adjacent regions be the same width within a factor of two, while the traditional quadtree makes no restriction on neighbor width. Restricted surface quadtrees are effective at recursively sampling a parametric surface. Quadtree samples are concentrated in regions of high curvature, and along intersection boundaries, using several subdivision criteria. Silhouette subdivision improves the accuracy of the silhouette boundary when a viewing transformation is available at sampling time. The adaptive sampling method is more robust than uniform sampling, and can be more efficient at rendering deformed, intersecting parametric surfaces

Accurate triangulations of deformed, intersecting surfaces

A quadtree algorithm is developed to triangulate deformed, intersecting parametric surfaces. The biggest problem with adaptive sampling is to guarantee that the triangulation is accurate within a given tolerance. A new method guarantees the accuracy of the triangulation, given a "Lipschitz" condition on the surface definition. The method constructs a hierarchical set of bounding volumes for the surface, useful for ray tracing and solid modeling operations. The task of adaptively sampling a surface is broken into two parts: a subdivision mechanism for recursively subdividing a surface, and a set of subdivision criteria for controlling the subdivision process. An adaptive sampling technique is said to be robust if it accurately represents the surface being sampled. A new type of quadtree, called a restricted quadtree, is more robust than the traditional unrestricted quadtree at adaptive sampling of parametric surfaces. Each sub-region in the quadtree is half the width of the previous region. The restricted quadtree requires that adjacent regions be the same width within a factor of two, while the traditional quadtree makes no restriction on neighbor width. Restricted surface quadtrees are effective at recursively sampling a parametric surface. Quadtree samples are concentrated in regions of high curvature, and along intersection boundaries, using several subdivision criteria. Silhouette subdivision improves the accuracy of the silhouette boundary when a viewing transformation is available at sampling time. The adaptive sampling method is more robust than uniform sampling, and can be more efficient at rendering deformed, intersecting parametric surfaces

Ray Tracing Volume Densities

This paper presents new algorithms to trace objects represented by densities within a volume grid, e.g. clouds, fog, flames, dust, particle systems. We develop the light scattering equations, discuss previous methods of solu-tion, and present a new approximate solution to the full three-dimensional radiative scattering problem suitable for use in computer graphics. Additionally we review dynamical models for clouds used to make an animated movie