This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are required: SOLVE, which computes solutions to a system of constraints, and MINIMIZE, which computes the global minimum of a function, subject to a system of constraints. We present algorithms for SOLVE and MINIMIZE using interval analysis as the conceptual framework. Crucial to the technique is the creation of "inclusion functions" for each constraint and function to be minimized. Inclusion functions compute a bound on the range of a function, given a similar bound on its domain, allowing a branch and bound approach to constraint solution and constrained minimization. Inclusion functions also allow the MINIMIZE algorithm to compute global rather than local minima, unlike many other numerica...

We propose an approach for modeling surface details such as scales, feathers, or thorns. These types of cellular textures require a representation with more detail than texture-mapping but are inconvenient to model with hand-crafted geometry. We generate

Procedural shaders have become popular tools for describing surface reflectance functions and other material properties. In comparison to fixed resolution textures they have the advantage of being resolution independent and storage e#cient. While procedural shaders provide an interface for evaluating the shader at a single point in parameter space, it is not easily possible to obtain an average value of the shader together with accurate error bounds over a finite area. Yet the ability to compute such error bounds is crucial for several interesting applications, most notably hierarchical area sampling for global illumination computations using the finite element approach and for the generation of textures used in interactive computer graphics. Using a#ne arithmetic for evaluating the shader over a finite area yields a tight, conservative error interval for the shader function. Compilers can automatically generate code for utilizing a#ne arithmetic from within shaders implemented in a ...

This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higher-dimensional shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as "reparameterize this curve by arclength", "compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces", or "solve this constraint or ODE system". The system has been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds multidimensional parametric functions. The language is bas...

Displacement maps and procedural displacement shaders are a widely used approach of specifying geometric detail and increasing the visual complexity of a scene. While it is relatively straightforward to handle displacement shaders in pipeline based rendering systems such as the Reyes-architecture, it is much harder to efficiently integrate displacement-mapped surfaces in ray-tracers. Many commercial ray-tracers tessellate the surface into a multitude of small triangles. This introduces a series of problems such as excessive memory consumption and possibly undetected surface detail. In this paper we describe a novel way of ray-tracing procedural displacement shaders directly, that is, without introducing intermediate geometry. Affine arithmetic is used to compute bounding boxes for the shader over any range in the parameter domain. The method is comparable to the direct ray-tracing of B'ezier surfaces and implicit surfaces using B'ezier clipping and interval methods, respectively. Keyw...

Graphics cards for personal computers have recently undergone a radical transformation from fixed-function graphics pipelines to multi-processor, programmable architectures. Multi-processor architectures are clearly advantageous for graphics for the simple reason that graphics computations are naturally concurrent, mapping well to stateless stream processing. They therefore parallelize easily and need no random access to memory with its problematic latencies. This paper presents Vertigo, a purely functional, Haskell-embedded language for 3D graphics and an optimizing compiler that generates graphics processor code. The language integrates procedural surface modeling, shading, and texture generation, and the compiler exploits the unusual processor architecture. The shading sublanguage is based on a simple and precise semantic model, in contrast to previous shading languages. Geometry and textures are also defined via a very simple denotational semantics. The formal semantics yields not only programs that are easy to understand and reason about, but also very efficient implementation, thanks to a compiler based on partial evaluation and symbolic optimization, much in the style of Pan [2]. Haskell’s overloading facility is extremely useful throughout Vertigo. For instance, math operators are used not just for floating point numbers, but also expressions (for differentiation and compilation), tuples, and functions. Typically, these overloadings cascade, as in the case of surfaces, which may be combined via math operators, though they are really functions over tuples of expressions on floating point numbers. Shaders may be composed with the same notational convenience. Functional dependencies are exploited for vector spaces, cross products, and derivatives.

Displacement mapping is a technique for applying fine geometric detail to a simpler base surface. The displacement is specified as a scalar function which makes it relatively easy to increase visual complexity without the difficulties inherent in more general modeling techniques. We would like to use displacement mapping in real-time applications. Ideally, a graphics accelerator should create a polygonal tessellation of the displaced surface on the fly to avoid storage and host bandwidth overheads. We present an online

We are currently developing an Automatic Target Recognition (ATR) algorithm to locate an object using multisensor data. The ATR algorithm will determine corresponding points between a range (LADAR) image, a color (CCD) image, a thermal (FLIR) image and a BRL/CAD model of the object being located. The success of this process depends in part on which features can be automatically extracted from the model database. The BRL/CAD models we have for this process contain more detail than can be productively used by our ATR algorithm and must be reduced to a more appropriate form. This paper presents algorithms we are developing in order to reduce the BRL/CAD models a level of detail appropriate for the ATR algorithm. A secondary feature of these algorithms is to also maintain a parallel version with details sufficient to appear realistic when rendered. This rendering enables the ATR system to animate its search procedure for monitoring and debugging. Model reduction begins by converting the Co...

Meshing is the process of computing, for a given surface, a representation consisting of pieces of simple surface patches, like triangles. This survey discusses all currently known surface (and curve) meshing algorithms that come with correctness and quality guarantees.

OF THESIS OBTAINING 3D SILHOUETTES AND SAMPLED SURFACES FROM SOLID MODELS FOR USE IN COMPUTER VISION Model-based object recognition algorithms identify modeled objects in a scene by relating stored geometric models to features extracted from sensor data. This process can be combinatorially explosive as the amount of information presented to the recognition algorithm increases. This thesis presents a method for extracting only relevant features from a stored three dimensional (3D) model in an attempt to reduce the difficulty of the recognition process. The development of the methods presented here were driven by the needs of the Automatic Target Recognition (ATR) algorithm being developed concurrently at Colorado State University (CSU). The ATR algorithm locates an object using multi-sensor data by determining the correspondence between a range (LADAR) image, a color image, a thermal (FLIR) image, and a Computer Aided Design (CAD) geometric model. The success of this process depends in ...