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An Introduction to Affine Arithmetic
, 2003
"... Affine arithmetic (AA) is a model for selfvalidated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard I ..."
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Cited by 14 (0 self)
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Affine arithmetic (AA) is a model for selfvalidated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard IA, the quantity representations used by AA are firstorder approximations, whose error is generally quadratic in the width of input intervals. In many practical applications, the higher asymptotic accuracy of AA more than compensates for the increased cost of its operations.
Affine Arithmetic: Concepts and Applications
, 2003
"... Affine arithmetic is a model for selfvalidated numerical computation that affine arithmetic keeps track of firstorder correlations between computed and input quantities. We explain the main concepts in affine arithmetic and it handles the dependency problem in standard interval arithmetic. We also ..."
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Cited by 13 (1 self)
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Affine arithmetic is a model for selfvalidated numerical computation that affine arithmetic keeps track of firstorder correlations between computed and input quantities. We explain the main concepts in affine arithmetic and it handles the dependency problem in standard interval arithmetic. We also describe some of its applications.
Ray casting implicit procedural noises with reduced affine arithmetic
 Dept. of Comp. Science, The University of Sheffield
, 2005
"... A method for ray casting implicit surfaces, defined with procedural noise models, is presented. The method is robust in that it is able to guarantee correct intersections at all image pixels and for all types of implicit surfaces. This robustness comes from the use of an affine arithmetic representa ..."
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A method for ray casting implicit surfaces, defined with procedural noise models, is presented. The method is robust in that it is able to guarantee correct intersections at all image pixels and for all types of implicit surfaces. This robustness comes from the use of an affine arithmetic representation for the quantity that expresses the variation of the implicit function along a ray. Affine arithmetic provides a bounding interval estimate which is tighter than the interval estimates returned by conventional interval arithmetic. Our ray casting method is also efficient due to a proposed modification in the data structure used to hold affine arithmetic quantities. This modified data structure ultimately leads to a reduced affine arithmetic model. We show that such a reduced affine arithmetic model is able to retain all the tight estimation capabilities of standard affine arithmetic, in the context of ray casting implicit procedural noises, while being faster to compute and more efficient to store. We also show that, without this reduced model, affine arithmetic would not have any advantage over the more conventional interval arithmetic for ray casting the class of implicit procedural surfaces that we are interested in visualizing.
A Progressive Refinement Approach for the Visualisation of Implicit Surfaces
"... Visualising implicit surfaces with the ray casting method is a slow procedure. The design cycle of a new implicit surface is, therefore, fraught with long latency times as a user must wait for the surface to be rendered before being able to decide what changes should be introduced in the next iterat ..."
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Cited by 2 (1 self)
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Visualising implicit surfaces with the ray casting method is a slow procedure. The design cycle of a new implicit surface is, therefore, fraught with long latency times as a user must wait for the surface to be rendered before being able to decide what changes should be introduced in the next iteration. In this paper, we present an attempt at reducing the design cycle of an implicit surface modeler by introducing a progressive refinement rendering approach to the visualisation of implicit surfaces. This progressive refinement renderer provides a quick previewing facility. It first displays a low quality estimate of what the final rendering is going to be and, as the computation progresses, increases the quality of this estimate at a steady rate. The progressive refinement algorithm is based on the adaptive subdivision of the viewing frustrum into progressively smaller cells. An estimate for the average value and the variance of the implicit function inside each cell is obtained with an affine arithmetic range estimation technique. Overall, we show that our progressive refinement approach not only provides the user with visual feedback as the rendering advances but is also capable of completing the image faster than a conventional implicit surface rendering algorithm based on ray casting.
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"... Abstract: Ambient isotopic approximations are fundamental for correct representation of the embedding of geometric objects in R 3, with a detailed geometric construction given here. Using that geometry, an algorithm is presented for efficient update of these isotopic approximations for dynamic visua ..."
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Abstract: Ambient isotopic approximations are fundamental for correct representation of the embedding of geometric objects in R 3, with a detailed geometric construction given here. Using that geometry, an algorithm is presented for efficient update of these isotopic approximations for dynamic visualization with a molecular simulation. 1 Approximation and Topology for Visualization Figure 1(a) depicts a knot 5 and Figure 1(b) shows a visually similar protein model 6. prompting two criteria for efficient algorithms for visualization:
Manuscript Adaptive Curve Approximation by Bending Energy
"... Let c denote a C 2 parametric curve c: [0, 1] → R 3.Itisshownthatc has an adaptive piecewise linear (PL) approximation that can be chosen optimally in terms ofthe bending energy. Furthermore, the approximation technique provides bounds upon the Hausdorff distance and the tangency differences betwee ..."
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Let c denote a C 2 parametric curve c: [0, 1] → R 3.Itisshownthatc has an adaptive piecewise linear (PL) approximation that can be chosen optimally in terms ofthe bending energy. Furthermore, the approximation technique provides bounds upon the Hausdorff distance and the tangency differences between the approximant and c. The method leads to efficient algorithms for a broad class of curves that includes splines, as well as many other representations used in computer graphics and animation. Some examples will be given. A novel contribution is the proofthat this method produces an asymptotically optimal number ofapproximating segments as the error bound goes to zero. This approach was motivated by robustness concerns for dynamic visualization in high performance computing environments. Key words: parametric curves, adaptive approximation, compactness, computational topology ∗ Corrresponding author.