Fast Reliable Ray-tracing of Procedurally Defined Implicit Surfaces Using Revised Affine Arithmetic

Fast and reliable rendering of implicit surfaces is an important area in the field of implicit modelling. Direct rendering, namely ray-tracing, is shown to be a suitable technique for obtaining good-quality visualisations of implicit surfaces. We present a technique for reliable ray-tracing of arbitrary procedurally defined implicit surfaces by using a modification of Affine Arithmetic called Revised Affine Arithmetic. A wide range of procedurally defined implicit objects can be rendered using this technique including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others. We compare our technique with other reliable techniques based on Interval and Affine Arithmetic to show that our technique provides the fastest, while still reliable, ray-surface intersections and ray-tracing. We also suggest possible modifications for the GPU implementation of this technique for real-time rendering of relatively simple implicit models and for near real-time for complex implicit models

Fast reliable interrogation of procedurally defined implicit surfaces using extended revised affine arithmetic.

Techniques based on interval and previous termaffine arithmetic next term and their modifications are shown to provide previous term reliable next term function range evaluation for the purposes of previous termsurface interrogation.next term In this paper we present a technique for the previous termreliable interrogation of implicit surfacesnext term using a modification of previous termaffine arithmeticnext term called previous term revised affine arithmetic.next term We extend the range of functions presented in previous termrevised affine arithmeticnext term by introducing previous termaffinenext term operations for arbitrary functions such as set-theoretic operations with R-functions, blending and conditional operators. The obtained previous termaffinenext term forms of arbitrary functions provide previous termfasternext term and tighter function range evaluation. Several case studies for operations using previous termaffinenext term forms are presented. The proposed techniques for previous termsurface interrogationnext term are tested using ray-previous termsurfacenext term intersection for ray-tracing and spatial cell enumeration for polygonisation. These applications with our extensions provide previous termfast and reliablenext term rendering of a wide range of arbitrary previous termprocedurally defined implicit surfacesnext term (including polynomial previous termsurfaces,next term constructive solids, pseudo-random objects, previous termprocedurally definednext term microstructures, and others). We compare the function range evaluation technique based on previous termextended revised affine arithmeticnext term with other previous termreliablenext term techniques based on interval and previous termaffine arithmeticnext term to show that our technique provides the previous termfastestnext term and tightest function range evaluation for previous termfast and reliable interrogation of procedurally defined implicit surfaces.next term Research Highlights The main contributions of this paper are as follows. ► The widening of the scope of previous termreliablenext term ray-tracing and spatial enumeration algorithms for previous termsurfacesnext term ranging from algebraic previous termsurfaces (definednext term by polynomials) to general previous termimplicit surfaces (definednext term by function evaluation procedures involving both previous termaffinenext term and non-previous termaffinenext term operations based on previous termrevised affine arithmetic)next term. ► The introduction of a technique for representing procedural models using special previous termaffinenext term forms (illustrated by case studies of previous termaffinenext term forms for set-theoretic operations in the form of R-functions, blending operations and conditional operations). ► The detailed derivation of special previous termaffinenext term forms for arbitrary operators

Interactive ray tracing of arbitrary implicits with SIMD interval arithmetic

Journal ArticleWe 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

Robustness and Randomness

Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust

Multi-scale space-variant FRep cellular structures

Existing mesh and voxel based modeling methods encounter difficulties when dealing with objects containing cellular structures on several scale levels and varying their parameters in space. We describe an alternative approach based on using real functions evaluated procedurally at any given point. This allows for modeling fully parameterized, nested and multi-scale cellular structures with dynamic variations in geometric and cellular properties. The geometry of a base unit cell is defined using Function Representation (FRep) based primitives and operations. The unit cell is then replicated in space using periodic space mappings such as sawtooth and triangle waves. While being replicated, the unit cell can vary its geometry and topology due to the use of dynamic parameterization. We illustrate this approach by several examples of microstructure generation within a given volume or along a given surface. We also outline some methods for direct rendering and fabrication not involving auxiliary mesh and voxel representations

A Note on Some Applications of Interval Arithmetic in Hierarchical Solid Modeling

Techniques of reliable computing like interval arithmetic can be used to guarantee a reliable solution even in the presence of numerical round-off errors. The need to trace bounds for the error function separately can be eliminated using these techniques. In this talk, we focus on some demonstrations how the techniques and algorithms of reliable computing can be applied to the construction and further processing of hierarchical solid representations using the octree model as an example. An octree is a common hierarchical data structure to represent 3D geometrical objects in solid modeling systems or to reconstruct a real scene. The solid representation is based on recursive cell decompositions of the space. Unfortunately, the data structure may require a large amount of memory when it uses a set of very small cubic nodes to approximate a solid. In this talk, we present a novel generalization of the octree model created from a CSG object that uses interval arithmetic and allows us to extend the tests for classifying points in space as inside, on the boundary or outside the object to handle whole sections of the space at once. Tree nodes with additional information about relevant parts of the CSG object are introduced in order to reduce the depth of the required subdivision. Furthermore, this talk is concerned with interval-based algorithms for reliable proximity queries between the extended octrees and with further processing of the structure. We conclude the talk with some examples of implementations

Applied computational geometry: Towards robust solutions of basic problems

AbstractGeometric computations, like all numerical procedures, are extremely prone to roundoff error. However, virtually none of the numerical analysis literature directly applies to geometric calculations. Even for line intersection, the most basic geometric operation, there is no robust and efficient algorithm. Compounding the difficulties, many geometric algorithms perform iterations of calculations reusing previously computed data. In this paper, we explore some of the main issues in geometric computations and the methods that have been proposed to handle roundoff errors. In particular, we focus on one method and apply it to a general iterative intersection problem. Our initial results seem promising and will hopefully lead to robust solutions for more complex problems of applied computational geometry