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14
Comparison of Interval Methods for Plotting Algebraic Curves
 Comput. Aided Geom. Des
, 2002
"... This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. ..."
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Cited by 24 (2 self)
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This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.
Statistical Cue Integration in DAG Deformable Models
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... Deformable models are a useful modeling paradigm in computer vision. A deformable model is a curve, a surface, or a volume, whose shape, position, and orientation are controlled through a set of parameters. They can represent manufactured objects, human faces and skeletons, and even bodies of flu ..."
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Cited by 21 (6 self)
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Deformable models are a useful modeling paradigm in computer vision. A deformable model is a curve, a surface, or a volume, whose shape, position, and orientation are controlled through a set of parameters. They can represent manufactured objects, human faces and skeletons, and even bodies of fluid.
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 10 (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.
Modified Affine Arithmetic Is More Accurate than Centered Interval Arithmetic or Affine Arithmetic
 Martin (Eds.), Lecture Notes in Computer Science 2768, Mathematics of Surfaces, SpringerVerlag
, 2003
"... In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a boxshaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than i ..."
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Cited by 9 (1 self)
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In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a boxshaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified a#ne arithmetic is not only more accurate but also much faster than standard a#ne arithmetic. We thus suggest that modified a#ne arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.
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 8 (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.
Approximating Parametric Curves with Strip Trees using Affine Arithmetic
"... We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance ..."
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Cited by 7 (2 self)
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We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance fields for parametric curves.
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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Cited by 6 (0 self)
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Intersecting and Trimming Parametric Meshes on FiniteElement Shells
 International Journal for Numerical Methods in Engineering
, 2000
"... We present an algorithm for intersecting finiteelement meshes defined on parametric surface patches. The intersection curves are modeled precisely and both meshes are adjusted to the newly formed borders, without unwanted reparametrizations. The algorithm is part of an interactive shell modeling pr ..."
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Cited by 5 (0 self)
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We present an algorithm for intersecting finiteelement meshes defined on parametric surface patches. The intersection curves are modeled precisely and both meshes are adjusted to the newly formed borders, without unwanted reparametrizations. The algorithm is part of an interactive shell modeling program that has been used in the design of large offshore oil structures. To achieve good interactive response, we represent meshes with a topological data structure that stores its entities in spatial indexing trees instead of linear lists. These trees speed up the intersection computations required to determine points of the trimming curves; moreover, when combined with the topological information, they allow remeshing using only local queries. keywords: surface intersection; finiteelement mesh generation; parametric representation; geometric modeling; topological data structures; constrained triangulation. 1 Introduction Surface modeling and mesh generation on surfaces are important pro...
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 1 (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.
Robust Approximation of Offsets, Bisectors, and Medial Axes of Plane Curves
, 2002
"... Most methods for computing offsets, bisectors, and medial axes of parametric curves in the plane are based on a local formulation of the distance to a curve. As a consequence, the computed objects may contain spurious parts and components, and have to be trimmed. We approach these problems as global ..."
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Cited by 1 (0 self)
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Most methods for computing offsets, bisectors, and medial axes of parametric curves in the plane are based on a local formulation of the distance to a curve. As a consequence, the computed objects may contain spurious parts and components, and have to be trimmed. We approach these problems as global optimization problems, and solve them using interval arithmetic, thus generating robust approximations that need not be trimmed.