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Adaptive Enumeration of Implicit Surfaces with Affine Arithmetic
 Computer Graphics Forum
, 1996
"... . We discuss adaptive enumeration and rendering methods for implicit surfaces, using octrees computed with affine arithmetic, a new tool for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally ..."
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Cited by 30 (15 self)
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. We discuss adaptive enumeration and rendering methods for implicit surfaces, using octrees computed with affine arithmetic, a new tool for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally providing much tighter bounds for the computed quantities. The resulting octrees are accordingly much smaller, and the rendering faster. We also describe applications of affine arithmetic to intersection and ray tracing of implicit surfaces. keywords: cellular models, interval analysis, rendering, implicit surfaces. 1 Introduction Implicit surfaces have recently become popular in computer graphics and solid modeling. In order to exploit existing hardware and algorithms, it is often necessary to approximate such surfaces by models with simpler geometry, such as polygonal meshes or voxel arrays. Let S be a surface defined implicitly by the equation h(x; y; z) = 0. A simple and general techn...
Raytracing Procedural Displacement Shaders
 In Graphics Interface
, 1998
"... 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 Reyesarchitecture, i ..."
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Cited by 28 (1 self)
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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 Reyesarchitecture, it is much harder to efficiently integrate displacementmapped surfaces in raytracers. Many commercial raytracers 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 raytracing 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 raytracing of B'ezier surfaces and implicit surfaces using B'ezier clipping and interval methods, respectively. Keyw...
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.
Towards an industrial use of Fluctuat on safetycritical avionics software
 In FMICS
, 2009
"... Abstract. Most modern safetycritical control programs, such as those embedded in flybywire control systems, perform a lot of floatingpoint computations. The wellknown pitfalls of IEEE 754 arithmetic make stability and accuracy analyses a requirement for this type of software. This need is tradi ..."
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Cited by 21 (2 self)
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Abstract. Most modern safetycritical control programs, such as those embedded in flybywire control systems, perform a lot of floatingpoint computations. The wellknown pitfalls of IEEE 754 arithmetic make stability and accuracy analyses a requirement for this type of software. This need is traditionally addressed through a combination of testing and sophisticated intellectual analyses, but such a process is both costly and errorprone. FLUCTUAT is a static analyzer developed by CEALIST for studying the propagation of rounding errors in C programs. After a long time research collaboration with CEALIST on this tool, Airbus is now willing to use FLUCTUAT industrially, in order to automate part of the accuracy analyses of some control programs. In this paper, we present the IEEE 754 standard, the FLUCTUAT tool, the types of codes to be analyzed and the analysis methodology, together with code examples and analysis results. 1
Geometric and Arithmetic Culling Methods for Entire Ray Packets
, 2006
"... Recent interactive ray tracing performance has been mainly derived from the use of ray packets. Larger ray packets allow for significant amortization of both computations and memory accesses; however, the majority of primitives are still intersected by each ray in a packet. This paper discusses seve ..."
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Cited by 20 (9 self)
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Recent interactive ray tracing performance has been mainly derived from the use of ray packets. Larger ray packets allow for significant amortization of both computations and memory accesses; however, the majority of primitives are still intersected by each ray in a packet. This paper discusses several methods to cull entire ray packets against common primitives (box, triangle, and sphere) that allows an arbitrary number of rays to be tested by a single test. This provides cheap “all miss ” or “all hit ” tests and may substantially improve the performance of an interactive ray tracer. The paper surveys current methods, provides details on three particular approaches using interval arithmetic, bounding planes, and corner rays, describes how the respective bounding primitives can be easily and efficiently constructed, and points out the relation among the different fundamental concepts.
Surface Intersection Using Affine Arithmetic
 In Graphics Interface
, 1996
"... We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Aff ..."
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Cited by 18 (7 self)
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We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally providing much tighter bounds for the computed quantities. As a consequence, the quadtree domain decompositions are much smaller and the intersection algorithm runs faster. keywords: surface intersection, trimming surfaces, range analysis, interval analysis, CAGD.
S.: Static analysis of finite precision computations
 In: VMCAI’11. LNCS
, 2011
"... Abstract. We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantic ..."
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Cited by 14 (3 self)
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Abstract. We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program. We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine sets, especially well suited for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments. 1
Extentions of Affine Arithmetic: Application to Unconstrained Global Optimization
 Journal of Universal Computer Science
"... Abstract: Global optimization methods in connection with interval arithmetic permit to determine an accurate enclosure of the global optimum, and of all the corresponding optimizers. One of the main features of these algorithms consists in the construction of an interval function which produces an e ..."
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Cited by 12 (0 self)
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Abstract: Global optimization methods in connection with interval arithmetic permit to determine an accurate enclosure of the global optimum, and of all the corresponding optimizers. One of the main features of these algorithms consists in the construction of an interval function which produces an enclosure of the range of the studied function over a box (right parallelepiped). We use here affine arithmetic in global optimization algorithms, in order to elaborate new inclusion functions. These techniques are implemented and then discussed. Three new affine and quadratic forms are introduced. On some polynomial examples, we show that these new tools often yield more efficient lower bounds (and upper bounds) compared to several wellknown classical inclusion functions. The three new methods, presented in this paper, are integrated into various Branch and Bound algorithms. This leads to improve the convergence of these algorithms by attenuating some negative effects due to the use of interval analysis and standard affine arithmetic.
Reliable TwoDimensional Graphing Methods for Mathematical Formulae with Two Free Variables
, 2001
"... present s a series of new algorit hms for reliably graphingt wodimensional implicit equat ions and inequalit ies. A clear st andard for int erpret ingt he graphs generat ed byt wodimensional graphing soft ware is int roduced and used t o evaluat et he present ed algorit hms. The first approach pr ..."
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Cited by 12 (0 self)
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present s a series of new algorit hms for reliably graphingt wodimensional implicit equat ions and inequalit ies. A clear st andard for int erpret ingt he graphs generat ed byt wodimensional graphing soft ware is int roduced and used t o evaluat et he present ed algorit hms. The first approach present ed uses a st andard int erval arit hmet ic library. This approach is shownt o be fault y; an analysis oft he failure reveals a limit at ion of st andard int erval arit hmet ic. Subsequent algorit hms are developed in parallel wit h improvement s and ext#E sions t# t# e int erval ari t#met# c used byt he graphing algorit hms. Graphs exhibit ing a variet y of mat hemat ical and art ist ic phenomena are shownt o be graphed correct ly byt he present ed algorit hms. A brief comparison of t he final algorit hm present edt o ot her graphing algorit hms is included.
A logical product approach to zonotope intersection
 In CAV’10, LNCS 6174
, 2010
"... Abstract. We define and study a new abstract domain which is a finegrained combination of zonotopes with (sub)polyhedric domains such as the interval, octagon, linear template or polyhedron domains. While abstract transfer functions are still rather inexpensive and accurate even for interpreting n ..."
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Cited by 11 (8 self)
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Abstract. We define and study a new abstract domain which is a finegrained combination of zonotopes with (sub)polyhedric domains such as the interval, octagon, linear template or polyhedron domains. While abstract transfer functions are still rather inexpensive and accurate even for interpreting nonlinear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid systems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic overapproximation of the intersection. We describe some examples and an implementation of our method in the APRON library, and discuss some further interesting combinations of zonotopes with nonlinear or nonconvex domains such as quadratic templates and maxplus polyhedra. 1