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Sampling of Procedural Shaders Using Affine Arithmetic
, 1996
"... 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 efficient. While procedural shaders provide an interface for evalua ..."
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Cited by 36 (4 self)
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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 efficient. 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 affine 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 affine arithmetic from within shaders implemented in a ...
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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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
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 32 (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 32 (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 26 (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.
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 23 (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.
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 21 (4 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
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.
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 17 (9 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
Fast Ray Tracing of Arbitrary Implicit Surfaces with Interval and Affine Arithmetic
"... Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but tr ..."
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Cited by 16 (4 self)
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Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but traditionally slow. Nonetheless, implemented efficiently using a stackdriven iterative algorithm and SIMD vector instructions, these methods can achieve interactive performance for common algebraic surfaces on the CPU. A similar algorithm can also be implemented stacklessly, allowing for efficient ray tracing on the GPU. This paper presents these algorithms, as well as an inclusionpreserving reduced affine arithmetic (RAA) for faster raysurface intersection. Shader metaprogramming allows for immediate and automatic generation of symbolic expressions and their interval or affine extensions. Moreover, we are able to render even complex forms robustly, in realtime at high resolution.