Improvements in the ray tracing of implicit surfaces based on interval arithmetic

Las superfícies implícitas son útiles en muchas áreasde los gráficos por ordenador. Una de sus principales ventajas es que pueden ser fácilmente usadas como primitivas para modelado. Aun asi, no son muy usadas porque su visualización toma bastante tiempo. Cuando se necesita una visualización precisa, la mejor opción es usar trazado de rayos. Sin embargo, pequeñas partes de las superficies desaparecen durante la visualización. Esto ocurre por la truncación que se presenta en la representación en punto flotante de los ordenadores; algunos bits se puerden durante las operaciones matemáticas en los algoritmos de intersección. En este tesis se presentan algoritmos para solucionar esos problemas. La investigación se basa en el uso del Análisis Intervalar Modal el cual incluye herramientas para resolver problemas con incertidumbe cuantificada. En esta tesis se proporcionan los fundamentos matemáticos necesarios para el desarrollo de estos algoritmos.

A Mathematical Basis for an Interval Arithmetic Standard

Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied in section 7. The desirability of variable length interval arithmetic is also discussed in the paper. The requirement to adapt the digital computer to the needs of interval arithmetic is as old as interval arithmetic. An obvious, simple possible solution is shown in section 8.

Function interval arithmetic

We propose an arithmetic of function intervals as a basis for convenient rigorous numerical computation. Function intervals can be used as mathematical objects in their own right or as enclosures of functions over the reals. We present two areas of application of function interval arithmetic and associated software that implements the arithmetic: (1) Validated ordinary differential equation solving using the AERN library and within the Acumen hybrid system modeling tool. (2) Numerical theorem proving using the PolyPaver prover. © 2014 Springer-Verlag.

A New Approach to Fuzzy Arithmetic

This work shows an application of a generalized approach for constructing dilation-erosion adjunctions on fuzzy sets. More precisely, operations on fuzzy quantities and fuzzy numbers are considered. By the generalized approach an analogy with the well known interval computations could be drawn and thus we can define outer and inner operations on fuzzy objects. These operations are found to be useful in the control of bioprocesses, ecology and other domains where data uncertainties exist.

Grid-based histogram arithmetic for the probabilistic analysis of functions

The selection of predefined analytic grids (partitions of the numeric ranges) to represent input and output functions as histograms has been proposed as a mechanism of approximation in order to control the tradeoff between accuracy and computation times in several áreas ranging from simulation to constraint solving. In particular, the application of interval methods for probabilistic function characterization has been shown to have advantages over other methods based on the simulation of random samples. However, standard interval arithmetic has always been used for the computation steps. In this paper, we introduce an alternative approximate arithmetic aimed at controlling the cost of the interval operations. Its distinctive feature is that grids are taken into account by the operators. We apply the technique in the context of probability density functions in order to improve the accuracy of the probability estimates. Results show that this approach has advantages over existing approaches in some particular situations, although computation times tend to increase significantly when analyzing large functions.

On the Arithmetic of Errors

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.

Reliable Solid Modelling Using Subdivision Surfaces

Les surfaces de subdivision fournissent une méthode alternative prometteuse dans la modélisation géométrique, et ont des avantages sur la représentation classique de trimmed-NURBS, en particulier dans la modélisation de surfaces lisses par morceaux. Dans ce mémoire, nous considérons le problème des opérations géométriques sur les surfaces de subdivision, avec l'exigence stricte de forme topologique correcte. Puisque ce problème peut être mal conditionné, nous proposons une approche pour la gestion de l'incertitude qui existe dans le calcul géométrique. Nous exigeons l'exactitude des informations topologiques lorsque l'on considère la nature de robustesse du problème des opérations géométriques sur les modèles de solides, et il devient clair que le problème peut être mal conditionné en présence de l'incertitude qui est omniprésente dans les données. Nous proposons donc une approche interactive de gestion de l'incertitude des opérations géométriques, dans le cadre d'un calcul basé sur la norme IEEE arithmétique et la modélisation en surfaces de subdivision. Un algorithme pour le problème planar-cut est alors présenté qui a comme but de satisfaire à l'exigence topologique mentionnée ci-dessus.

Die aeroelastische Stabilitätsanalyse – ein praxisnaher Ansatz zur intervalltheoretischen Betrachtung von Modellierungsunsicherheiten am Flugzeug

Die Untersuchung des dynamischen aeroelastischen Stabilitätsverhaltens von Flugzeugen erfordert sehr komplexe Rechenmodelle, welche die wesentlichen elastomechanischen und instationären aerodynamischen Eigenschaften der Konstruktion wiedergeben sollen. Bei der Modellbildung müssen einerseits Vereinfachungen und Idealisierungen im Rahmen der Anwendung der Finite Elemente Methode und der aerodynamischen Theorie vorgenommen werden, deren Auswirkungen auf das Simulationsergebnis zu bewerten sind. Andererseits können die strukturdynamischen Kenngrößen durch den Standschwingungsversuch identifiziert werden, wobei die Ergebnisse Messungenauigkeiten enthalten. Für eine robuste Flatteruntersuchung müssen die identifizierten Unwägbarkeiten in allen Prozessschritten über die Festlegung von unteren und oberen Schranken konservativ ermittelt werden, um für alle Flugzustände eine ausreichende Flatterstabilität sicherzustellen. Zu diesem Zweck wird in der vorliegenden Arbeit ein Rechenverfahren entwickelt, welches die klassische Flatteranalyse mit den Methoden der Fuzzy- und Intervallarithmetik verbindet. Dabei werden die Flatterbewegungsgleichungen als parameterabhängiges nichtlineares Eigenwertproblem formuliert. Die Änderung der komplexen Eigenlösung infolge eines veränderlichen Einflussparameters wird mit der Methode der numerischen Fortsetzung ausgehend von der nominalen Startlösung verfolgt. Ein modifizierter Newton-Iterations-Algorithmus kommt zur Anwendung. Als Ergebnis liegen die berechneten aeroelastischen Dämpfungs- und Frequenzverläufe in Abhängigkeit von der Fluggeschwindigkeit mit Unschärfebändern vor.

Uma fundamentação para sinais e sistemas intervalares

In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given

Constrained intervals and interval spaces

Constrained intervals, intervals as a mapping from [0, 1] to polynomials of degree one (linear functions) with non-negative slopes, and arithmetic on constrained intervals generate a space that turns out to be a cancellative abelian monoid albeit with a richer set of properties than the usual (standard) space of interval arithmetic. This means that not only do we have the classical embedding as developed by H. Radström, S. Markov, and the extension of E. Kaucher but the properties of these polynomials. We study the geometry of the embedding of intervals into a quasilinear space and some of the properties of the mapping of constrained intervals into a space of polynomials. It is assumed that the reader is familiar with the basic notions of interval arithmetic and interval analysis. © 2013 Springer-Verlag Berlin Heidelberg.

Síndrome do X Frágil com variante de Dandy-Walker: estudo clínico das manifestações comunicativas orais e escritas

A síndrome do X Frágil é a causa mais frequente de deficiência intelectual hereditária. A variante de Dandy-Walker trata-se de uma constelação específica de achados neurorradiológicos. Este estudo relata achados da comunicação oral e escrita de um menino de 15 anos com diagnóstico clínico e molecular da síndrome do X-Frágil e achados de neuroimagem do encéfalo compatíveis com variante de Dandy-Walker. A avaliação fonoaudiológica foi realizada por meio da Observação do Comportamento Comunicativo, aplicação do ABFW - Teste de Linguagem Infantil - Fonologia, Perfil de Habilidades Fonológicas, Teste de Desempenho Escolar, Teste Illinois de Habilidades Psicolinguísticas, avaliação do sistema estomatognático e avaliação audiológica. Observou-se: alteração de linguagem oral quanto às habilidades fonológicas, semânticas, pragmáticas e morfossintáticas; déficits nas habilidades psicolinguísticas (recepção auditiva, expressão verbal, combinação de sons, memória sequencial auditiva e visual, closura auditiva, associação auditiva e visual); e alterações morfológicas e funcionais do sistema estomatognático. Na leitura verificou-se dificuldades na decodificação dos símbolos gráficos e na escrita havia omissões, aglutinações e representações múltiplas com o uso predominante de vogais e dificuldades na organização viso-espacial. Em matemática, apesar do reconhecimento numérico, não realizou operações aritméticas. Não foram observadas alterações na avaliação audiológica periférica. A constelação de sintomas comportamentais, cognitivos, linguísticos e perceptivos, previstos na síndrome do X-Frágil, somada às alterações estruturais do sistema nervoso central, pertencentes à variante de Dandy-Walker, trouxeram interferências marcantes no desenvolvimento das habilidades comunicativas, no aprendizado da leitura e escrita e na integração social do indivíduo.

A Compact and Scalable RNS Architecture

Conferência: IEEE 24th International Conference on Application-Specific Systems, Architectures and Processors (ASAP)- Jun 05-07, 2013

An efficient scalable RNS architecture for large dynamic ranges

This paper proposes an efficient scalable Residue Number System (RNS) architecture supporting moduli sets with an arbitrary number of channels, allowing to achieve larger dynamic range and a higher level of parallelism. The proposed architecture allows the forward and reverse RNS conversion, by reusing the arithmetic channel units. The arithmetic operations supported at the channel level include addition, subtraction, and multiplication with accumulation capability. For the reverse conversion two algorithms are considered, one based on the Chinese Remainder Theorem and the other one on Mixed-Radix-Conversion, leading to implementations optimized for delay and required circuit area. With the proposed architecture a complete and compact RNS platform is achieved. Experimental results suggest gains of 17 % in the delay in the arithmetic operations, with an area reduction of 23 % regarding the RNS state of the art. When compared with a binary system the proposed architecture allows to perform the same computation 20 times faster alongside with only 10 % of the circuit area resources.

ROM-less RNS-to-binary converter moduli {22N − 1, 22N + 1, 2N − 3, 2N + 3}

In this paper, a novel ROM-less RNS-to-binary converter is proposed, using a new balanced moduli set {22n-1, 22n + 1, 2n-3, 2n + 3} for n even. The proposed converter is implemented with a two stage ROM-less approach, which computes the value of X based only in arithmetic operations, without using lookup tables. Experimental results for 24 to 120 bits of Dynamic Range, show that the proposed converter structure allows a balanced system with 20% faster arithmetic channels regarding the related state of the art, while requiring similar area resources. This improvement in the channel's performance is enough to offset the higher conversion costs of the proposed converter. Furthermore, up to 20% better Power-Delay-Product efficiency metric can be achieved for the full RNS architecture using the proposed moduli set. © 2014 IEEE.

Estudi GPGPU dedicat al processament de neuroimatges

La tecnologia GPGPU permet paral∙lelitzar càlculs executant operacions aritmètiques en els múltiples processadors de que disposen els xips gràfics. S'ha fet servir l'entorn de desenvolupament CUDA de la companyia NVIDIA, que actualment és la solució GPGPU més avançada del mercat. L'algorisme de neuroimatge implementat pertany a un estudi VBM desenvolupat amb l'eina SPM. Es tracta concretament del procés de segmentació d'imatges de ressonància magnètica cerebrals, en els diferents teixits dels quals es composa el cervell: matèria blanca, matèria grisa i líquid cefaloraquidi. S'han implementat models en els llenguatges Matlab, C i CUDA, i s'ha fet un estudi comparatiu per plataformes hardware diferents.