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Interval Analysis For Computer Graphics
 Computer Graphics
, 1992
"... 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 ..."
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Cited by 151 (2 self)
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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...
Guaranteeing the Topology of an Implicit Surface Polygonization for Interactive Modeling
, 1997
"... Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficie ..."
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Cited by 111 (9 self)
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Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficiently during parameter changes. Changes in the function value at these critical points cause changes in the topology. Techniques for modifying the polygonization to accommodate such changes in topology are given. These techniques are robust enough to guarantee the topology of an implicit surface polygonization, and are efficient enough to maintain this guarantee during interactive modeling. The impact of this work is a topologicallyguaranteed polygonization technique, and the ability to directly and accurately manipulate polygonized implicit surfaces in real time.
Interactive ray tracing of arbitrary implicits with simd interval arithmetic
 In Proceedings of the 2nd IEEE/EG Symposium on Interactive Ray Tracing
, 2007
"... We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any progra ..."
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Cited by 29 (7 self)
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We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions.
Polygonizing Implicit Surfaces With Guaranteed Topology
, 1997
"... An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuses on blended ellipsoids, but could be extended to include any smooth, bounded implicit ..."
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Cited by 7 (1 self)
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An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuses on blended ellipsoids, but could be extended to include any smooth, bounded implicit surface. A polygonization algorithm and an incremental repolygonization algorithm are provided. Treating an implicit surface as a gradient system allows theorems from Morse theory to describe implicit surface topology. An implicit surface changes topology only when a critical value of its defining function changes sign. These critical points may be found using interval analysis. Techniques for modifying the polygonization to accommodate such changes in topology are given.
Robust Adaptive Polygonal Approximation of Implicit Curves
"... We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmet ..."
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Cited by 1 (0 self)
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We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmetic and automatic differentiation.
Interval methods for ray casting implicit surfaces with affine arithmetic
"... Abstract. We study the performance of affine arithmetic as a replacement for interval arithmetic in interval methods for ray casting implicit surfaces. Affine arithmetic is a variant of interval arithmetic designed to handle the dependency problem, and which has improved several interval algorithms ..."
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Abstract. We study the performance of affine arithmetic as a replacement for interval arithmetic in interval methods for ray casting implicit surfaces. Affine arithmetic is a variant of interval arithmetic designed to handle the dependency problem, and which has improved several interval algorithms in computer graphics.
ContourBased Polygonization of Regular Grid Terrain Data
"... Terrain visualization has been an interesting issue for graphics people for long, and there have been a lot of attempts in various ways for more e#cient visualization. In most of the works the terrain data was considered as a triangular mesh as it is, or as an implicit surface in which the original ..."
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Terrain visualization has been an interesting issue for graphics people for long, and there have been a lot of attempts in various ways for more e#cient visualization. In most of the works the terrain data was considered as a triangular mesh as it is, or as an implicit surface in which the original shape#topology# of the terrain can be slightly changed.
National Centre for Computer Animation
"... In this paper we present a method for interactive rendering general procedurally defined functionally represented (FRep) objects using the acceleration with graphics hardware, namely Graphics Processing Units (GPU). We obtain interactive rates by using GPU acceleration for all computations in render ..."
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In this paper we present a method for interactive rendering general procedurally defined functionally represented (FRep) objects using the acceleration with graphics hardware, namely Graphics Processing Units (GPU). We obtain interactive rates by using GPU acceleration for all computations in rendering algorithm, such as raysurface intersection, function evaluation and normal computations. We compute primary rays as well as secondary rays for shadows, reflection and refraction for obtaining high quality of the output visualization and further extension to raytracing of FRep objects. The algorithm is wellsuited for modern GPUs and provides acceptable interactive rates with good quality of the results. A wide range of objects can be rendered including traditional skeletal implicit surfaces, constructive solids, and purely procedural objects such as 3D fractals.
Accurate SamplingBased Algorithms for Surface Extraction and Motion Planning
, 2005
"... Boolean operations, Minkowski sum evaluation, configuration space computation, and motion planning are fundamental problems in solid modeling and robotics. Their applications include computeraided design, numericallycontrolled machining, tolerance verification, packing, assembly planning, and dyna ..."
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Boolean operations, Minkowski sum evaluation, configuration space computation, and motion planning are fundamental problems in solid modeling and robotics. Their applications include computeraided design, numericallycontrolled machining, tolerance verification, packing, assembly planning, and dynamic simulation. Prior algorithms for solving these problems can be classified into exact and approximate approaches. The exact approaches are difficult to implement and are prone to robustness problems. Current approximate approaches may not solve these problems accurately. Our work aims to bridge this gap between exact and approximate approaches. We present a samplingbased approach to solve these geometric problems. Our approach