This thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.

Turbochargers are used on many automotive internal combustion engines to increase power density. The broad operating range of the engine also requires a wide range of the turbocharger compressor. At low mass flows, however, turbo compressor operation becomes unstable and eventually enters surge. Surge is characterized by large oscillations in mass flow and pressure. Due to the associated noise, control problems, and possibility of mechanical component damage, this has to be avoided. Different indicators exist to classify compressor operation as stable or unstable on a gas stand. They are based on pressure oscillations, speed oscillations, or inlet temperature increase. In this thesis, a new stability indicator is proposed based on the Hurst exponent of the pressure signal. The Hurst exponent is a number between zero and one that describes what kind of long-term correlations are present in a time series. Data from three cold gas stand experiments are analyzed using this criterion. Results show that the Hurst exponent of the compressor outlet pressure signal has good characteristics. Stable operation is being indicated by values larger than 0.5. As compressor operation moves towards the surge line, the Hurst exponent decreases towards zero. An additional distinction between the long-term correlations of small and large amplitude fluctuations by means of higher order Hurst exponents can be used as an early warning indicator. Further tests using compressor housing accelerometers show that the Hurst exponent is not a good choice for real-time surge detection on the engine. Reasons are the long required sampling time compared to competing methods, and the fact that other periodically repeating oscillations lead to Hurst exponents close to zero independent of compressor operation. / Turboladdare används ofta på förbränningsmotorer för att öka motorns effekttäthet. Motorns breda driftområde ställer krav på ett brett driftområde för turboladdarens kompressor. Vid låga massflöden blir kompressordriften dock mindre stabil, och surge kan uppträda. Surge innebär stora oscillationer i tryck och massflöde genom kompressorn. På grund av oljud, reglerproblem och risken för mekaniska skador vill man undvika surge. Det finns indikatorer för att bedöma kompressorns stabilitet på ett gas stand. Indikatorerna är baserade på tryckoscillationer, varvtalsoscillationer, eller temperaturökning i gasen i kompressorinloppet. I denna avhandling presenteras en ny indikator baserad på Hurst-exponenten, beräknad på trycksignalen. Hurst-exponenten är ett tal mellan noll och ett som beskriver vilka typer av långtidskorrelationer det finns i signalen. Mätningar från tre gas-stand-experiment har analyserats på detta sätt. Analyserna visar att Hurst-exponenten baserad på kompressorutloppstrycket fungerar bra som som surgeindikator. Stabil drift av kompressorn indikeras av att Hurst-exponenten är större än 0.5. När kompressordriftpunkten närmar sig surgelinjen faller Hurst-exponenten mot noll. En distinktion mellan oscillationer med små och stora amplituder kan används för att få en tidig varning. Analyser av vibrationsmätningar på kompressorhuset vid motorapplikation visar att Hurst-exponenten inte är lämplig som realtidsindikator på en motor. Detta kommer sig dels av att data behöver samlas in under en längre tid än med andra tänkbara indikatorer, dels av att andra periodiska oscillationer i signalen kopplade till motorns naturliga beteende leder till Hurst-exponenter nära noll även vid stabil kompressordrift. / <p>QC 20170510</p> / CCGEx - Compressor off-Design

On sait que la matrice d'incidence associée à une substitution ne suffit pas pour déterminer complètement le système dynamique associé, même dans des cas très simples, il existe plusieurs substitutions associées à une même matrice car il existe de nombreux mots ayant le même abélianisé. Dans cette thèse, on étudie les points communs de deux lignes brisées associées à deux substitutions $sigma_1$ et $sigma_2$ irréductibles unimodulaires de type Pisot qui ont la même matrice d'incidence. On identifie les points communs de ces deux lignes brisées à partir d'un algorithme. On montre ainsi que l'intersection de ces deux lignes brisées est aussi une ligne brisée associée au point fixe d'une nouvelle substitution. On montre plus précisément que si $sigma_1$ vérifie la conjecture Pisot et $0$ est un point intérieur à son fractal de Rauzy alors ces points communs peuvent être engendrés par une substitution définie sur un alphabet appelé alphabet des paires équilibrées. Cette substitution est obtenue à partir d'un algorithme, l'algorithme des paires équilibrées. On obtient ainsi l'intersection des intérieurs des deux fractals de Rauzy. En prenant la clôture de cet ensemble on obtient un ensemble substitutif. La condition que $0$ est un point intérieur au fractal de Rauzy associé à la substitution $sigma_1$ nous permet de montrer que l'intersection des deux fractals de Rauzy est de mesure positive. Dans une deuxième partie du travail on s'intéresse à l'étude de la frontière du fractal de Rauzy. Le fractal de Rauzy est dit fractal mais c'est en fait sa frontière qui est fractale. / The matrix of a substitution is not su&#64259;cient to completely determine the dynamics associated with it, even in the simplest cases since there are many words with the same abelianization. In this paper we study the common points of the canonical broken lines associated with two di&#64256;erent irreducible Pisot unimodular substitutions &#963;1 and &#963;2 having the same incidence matrix. We prove that if &#963;1 veri&#64257;es the Pisot conjecture and 0 is an inner point to the Rauzy fractal associated with the substitution &#963;1 then these common points can be generated with a substitution on an alphabet of so-called balanced pairs, and we obtain in this way the intersection of the interior of two Rauzy fractals.

O uso de imagens é crucial na medicina, e seu uso no diagnóstico de doenças é uma das principais ferramentas clínicas da atualidade. Porém, frequentemente necessitam de pós-processamento para serem úteis. Embora ferramentas clássicas sejam utilizadas para esse fim, elas não dão tratamento específico a certas características de imagens fractais, como as provindas de sistemas biológicos. Nesse enfoque, este trabalho objetivou a criação de novas bases de interpolação utilizando a Q-Estatística para verificar se seriam estas seriam adequadas à representação de objetos com características fractais que as bases clássicas. Foram criados dois tipos de splines: uma unidimensional e outra bidimensional, que permitiram um tipo diferente de interpolação, fundamentado na q-Estatística. Os testes demonstraram a potencialidade dessas ferramentas para uso em sinais e imagens médicas, com acentuada redução do erro de interpolação no caso unidimensional (em até 351,876%) e uma redução sutil no caso bidimensional (0,3%). Como resultado adicional, foram criados filtros de imagens e avaliados seus resultados em imagens médicas, que resultaram em melhorias de até 1.340% de ganho efetivo na remoção de ruídos de natureza fractal (marrom). Os resultados sugerem que as q-bases desenvolvidas foram capazes de representar melhor imagens e sinais médicos, bem como é interessante o uso dos filtros desenvolvidos na remoção de diversos tipos de ruído do tipo 1/f^b. / The use of images is crucial in modern medicine, and diagnostic imaging is a major clinical tools used in detecting, monitoring and completion of many treatments. However, often the images need to be post-processed for display to health professionals or automated analysis, searching for signs of abnormalities. Although classical tools are used for that purpose, they do not give special treatment to certain characteristics of fractal images, such as those coming from biological systems. These characteristics are produced, in general, by complex dynamic systems as a result of internal interactions of sub-system components, giving the system a fractal character. In this context, the main objective of this work was to propose interpolation bases using the Q-statistic, creating bases of Q-interpolation, and verify if such bases would be best suited to the representation of objects with fractal characteristics than classical bases, assumed the premise that such a theory model best this kind of phenomenon than classical theory. Based on this hypothesis, we created two types of splines: one-dimensional and one-dimensional, called Q-splines, which allow a different type of interpolation and they can capture behaviors as super-additive or sub-additive among the constituents of a spline. These models have demonstrated numerically the potential use of this type of interpolation for use in signals and medical images, reducing the interpolation error by up to 351.876 % in the one-dimensional case and 0.3 % in two dimensional. As secondary results, were defined two families of image filters, called anisotropic Q-filters and isotropic Q-filters, and their results were evaluated in real medical images. In virtually all analyzes it was possible to obtain the best results from conventional approaches, sometimes with improvements of 1.340 % in some filters, in removing noise fractal nature (brown). The results were more modest for the interpolation of two-dimensional images, however, generally proved exciting and encouraging, clearly showing that these new approaches are not only viable, but also can produce better results compared to classical approaches. Based on these results, we concluded that the Q-bases developed are best able to represent not only signs but medical imaging (1D and 2D) although its use can be improved by the adoption of approaches adapted to the vector representation of information, that favor the use of splines. Similarly, the Q-filters were more suitable for the processing of medical signals when compared to conventional approaches.

Ultimamente, a análise fractal (AF) vem sendo utilizada com sucesso na área de processamento digital de voz, especialmente para fins de segmentação de palavras e fonemas, que é uma das etapas fundamentais dos sistemas de reconhecimento automático de fala (Automatic Speech Recognition - ASR ) e identificação automática de locutor (Automatic Speaker Identification - ASI). O uso prático da AF para ASR e ASI depende de dois fatores básicos: baixo custo computacional, para permitir o uso em tempo-real, e precisão nos resultados, para produzir a segmentação correta e entregar dados coerentes à etapa de classificação. Visando atender a esses objetivos, o presente trabalho propõe uma técnica de segmentação de sinais de voz baseada na dimensão do fractal, obtida com o uso da transformada wavelet discreta (DWT). Diversas famílias de wavelets são testadas e comparadas, sendo que os testes foram realizados com algumas sentenças extraídas da base de dados TIMIT do Linguistic Data Consortium (LDC). / Nowadays, fractal analysis has been successfully applied to digital speech processing, particularly for words and phonemes segmentation, which represents one of the fundamental steps in automatic speech recognition and speaker identification systems. The practical use of fractal analysis for these purposes should match two principles: low computational cost, to allow use in real-time, and accuracy in the results, to produce a correct segmentation, delivering consistent data to the classifier. Aiming at meeting these two requirements, this work proposes a technique for speech segmentation based on the fractal dimension, obtained by using the discrete wavelet transform (DWT). Many families of wavelets were tested and compared, being the experiments performed with speech data collected from TIMIT corpus provided by the Linguistic Data Consortium.

Made available in DSpace on 2015-03-04T18:57:49Z (GMT). No. of bitstreams: 1 Tese Doutorado - Marcelo Barros.pdf: 2669838 bytes, checksum: 882e9d77602451e9413c043290bd82ba (MD5) Previous issue date: 2011-06-22 / The physics associated to geometric fractal systems is investigated. Discrete and continuous models, from statics and dynamics as well as computational and physical experiments help defining and evaluating dimensions associated to the physics of the systems. It is shown the relation between the mechanical dimensions (flexibility and dynamical) and the geometric fractal dimension. Moments of order 2 are shown to be useful in identifying randomness in the generation process of geometry. Mixed fractals are defined by more than one law of formation or organization: the case of alternating laws is studied. Weierstrass-Mandelbrot systems (SWM) are defined through a properly summation of senoidal functions, each with amplitude proportional to the associated period squared. A dimension for SWM is defined. An origin for 1/f noises from SWM is proposed. A new method to determine fractal dimensions is proposed. It consists in taking successive samples from the object and relating a given property with the size of the sample, called sampling method. It is tested with Koch, mixed and Weierstrass systems. Branched systems (fractal trees) in 2D are studied under the solid mechanics approach. It is shown that Murray's law corresponds to the state of constant normal stress in solids. A mechanical efficiency (stiffness x weight) of beams with cross sections given by a Sierpinski system is studied. Defined by the proportion between geometric mechanical stiffness (moment of inertia) and the cross section area squared, the efficiency is shown to grow with the advance of orders. In this way, the more porous the more efficient is the beam. / Estuda-se a física associada a sistemas geométricos fractais. Por meio de modelos discreto e contínuo, da estática e da dinâmica e de experimentos computacionais e físicos definem-se e avaliam-se dimensões associadas à física dos sistemas. Mostra-se a relação existente entre as dimensões da mecânica (da flexibilidade e da dinâmica) e a dimensão fractal geométrica. Nota-se que momentos de ordem 2 são úteis na identificação de aleatoriedade no processo de geração da geometria. Definem-se fractais mistos como aqueles que apresentam mais de uma lei de formação ou organização. Estudou-se o caso que alterna entre duas ou mais leis. Definem-se sistemas de Weierstrass-Mandelbrot (SWM) a partir da soma apropriada de funções senoidais, cada uma com amplitude proporcional ao quadrado do período associado. Define-se uma dimensão para os SWM. Propõe-se uma origem para os ruídos do tipo 1/f a partir de SWM. Propõe-se um método para estimação de dimensões fractais a partir da relação entre amostras sucessivas do objeto, denominado método da amostragem. Testa-se numericamente o método nos sistemas de Koch, misto e Weierstrass, com êxito. Estuda-se sistemas ramificados (árvores fractais) em 2D sob a abordagem da mecânica dos sólidos. Mostra-se que a lei de Murray tem sua equivalência na mecânica dos sólidos pelo estado de tensão normal constante em todas as ordens. É estudada a eficiência mecânica (rigidez x peso) de vigas com seções transversais dadas por um sistema de Sierpinski. Mostra-se que a eficiência definida pela razão entre a rigidez mecânica geométrica (momento de inércia) e o quadrado da área da seção transversal aumenta com o avanço nas ordens. Desta forma, quanto mais porosa mais eficiente é a viga

Made available in DSpace on 2015-07-15T13:31:42Z (GMT). No. of bitstreams: 1 dissertacaoCharles.pdf: 2350650 bytes, checksum: 3a5a6710de4fe0f9f72e936ffc6d9a22 (MD5) Previous issue date: 2007-11-22 / Brazil Northeast s climate is usually described as semi-arid, characterized by hard dry seasons intermingled by hard rainfall seasons. Some areas of Northeast present annual pluviometric measure about 400 mm in mean, while in others the annual pluviometric measure is about 2000 mm. Rain events result from interplay of several physical phenomena, most of which can be individually described on the basic laws of mechanics and thermodynamics in a rather adequate way. Because of this, a huge progress has been achieved in recent years in relation to weather forecast with the use of very precise algorithms in large scale computing resources. They take into account the variables that are relevant for the atmospheric and ocean circulation and input of large amount of physical data obtained from a dense set of stations scattered around the world. In order to improve the interpretation of the accurate data resulting from the description of atmospheric phenomena and rain events, it is necessary to proceed with sophisticated analyses of recorded and simulated data, as spatial and temporal statistical correlations, scale properties, topological properties of spatial event distribution, ad so on. They indicate the extent of statistical relevance of the data, local and global effects, typical patterns, and other topological features related to the phenomena.In this work, we explore the usefulness of complex network framework for the analysis and nderstanding of rain events, based solely on recorded data from a set of stations in Northeast Brazil. The method is inspired on a proposal to characterize actual sequences of earthquake events where, like precipitation phenomena, the available data stems from complex systems with a very large number of physical variables. The potential network nodes are the meteorological stations where the rain events have been recorded, while the network edges are placed according to rules that take into account temporal and spatial correlation criteria between events occurring at different stations, for a time span as large as one month. We evaluate usual network properties based on diameter, node degrees, clustering coefficient, minimal inter-node distance along network edges. This allows for a characterization of networks based on seasonality and on spatial span of the region where the stations are distributed. The obtained results are discussed, taking into account the known precipitation patterns of the investigated region. rainfall variability, complex networks, fractals. / Climaticamente, a regi ao Nordeste do Brasil ?e marcada pela predomin?ncia de clima semi-?arido, caracterizado por per?&#305;odos de secas severas intercalados por per?odos de chuvas intensas. Eventos clim?aticos como a chuva resultam da intera??o de v?rios fen?menos f?sicos que, em sua maioria, pode ser descrita individualmente pelas leis b?sicas da mec?nica e termodin?mica de forma satisfat?ria. Por esse motivo, um imenso progresso tem sido observado, nos ?ltimos anos, com rela??o ? previs?o de tempo e clima utilizando algoritmos mais precisos em recursos computacionais de larga escala. Estes algoritmos levam em considera??o as vari?veis que s?o relevantes para a circula??o atmosf?rica e oce?nica al?m de uma grande quantidade de dados f?sicos obtidos de um conjunto denso de esta??es distribu?das ao redor do mundo. Com objetivo de prover a interpreta??o dos dados destes algoritmos, ? necess?rio proceder com an?lises sofisticadas dos dados armazenados e simulados, como correla?c oes estat?&#305;sticas temporais e espaciais, propriedades de escalas, propriedades topologicas da distribui??o espacial de eventos, etc. Os resultados falam sobre a relev?ncia estat?stica dos dados, efeitos locais e globais, padr?es t?picos e outros recursos relacionados ao fen?meno. Neste trabalho, n?s exploramos o uso da Teoria de Redes Complexas para a an?lise e interpreta??o de eventos de chuva, baseandonos somente em registros de dados de um conjunto de esta??es pluviom?tricas da regi?o Nordeste do Brasil. Este m?etodo ? inspirado em uma proposta para caracterizar sequ?ncias de eventos s?smicos, eventos em que, assim como no fen?meno das chuvas, a grande quantidade de vari?aveis f?sicas envolvidas motiva a an?lise usando m?todos da Teoria de Sistemas Complexos. Os n?s das redes geradas s?o as esta??es meteorol?gicas onde h? dados de chuva no per?odo analisado, enquanto as arestas s?o criadas de acordo com crit?rios de correla??o temporal e espacial entre eventos de chuva ocorridos em diferentes esta??es pluviom?tricas. Calculamos os ?ndices mais comuns de caracteriza??o de redes complexas, tais como: di?metro, caminho m?nimo m?dio, coeficiente de aglomera??o m?dio. As redes conectam esta??es a diferentes dist?ncias, e a fim de estudar a causalidade n?o-local desse fen?meno foram calculados ?ndices fractais de caracteriza??o. Os valores de di?metro e de caminho m?nimo m?dio s?o menores para os meses de inverno e primavera, t?picos de chuva mais localizada no litoral; enquanto que para os meses de ver?o e outono, t?picos de chuva mais distribu?&#305;da em toda a regi?o, os valores s?o maiores. A dimens?o fractal calculada para dados do Sul do Nordeste (Bahia) ? semelhante ? calculada para dados do Norte do Nordeste (demais estados da Regi?o), mas ambas s?o diferentes das dimens?es fractais de redes completas e regulares hipot?ticas, o que demonstra que a distribui??o das esta??es pluviom?tricas n?o ? homog?nea. Estes resultados sugerem o estudo mais aprofundado deste m?todo de an?lise de dados pluviom?tricos, que, atrav?s da modelagem em Sistemas Complexos.

Research into automatic retina image analysis has become increasingly important, not just in ophthalmology but also in other clinical specialities such as cardiology and neurology. In the retina, blood vessels can be directly visualised non-invasively in-vivo, and hence it serves as a "window" to cardiovascular and neurovascular complications. Biomarker research, i.e. investigating associations between the morphology of the retinal vasculature (as a means of revealing microvascular health or disease) and particular conditions affecting the body or brain could play an important role in detecting disease early enough to impact on patient treatment and care. A fundamental requirement of biomarker research is access to large datasets to achieve sufficient power and significance when ascertaining associations between retinal measures and clinical characterisation of disease. Crucially, the vascular changes that appear can affect arteries and veins differently. An essential part of automatic systems for retinal morphology quantification and biomarker extraction is, therefore, a computational method for classifying vessels into arteries and veins. Artery-vein classification enables the efficient extraction of biomarkers such as the Arteriolar to Venular Ratio, which is a well-established predictor of stroke and other cardiovascular events. While structural parameters of the retinal vasculature such as vessels calibre, branching angle, and tortuosity may individually convey some information regarding specific aspects of the health of the retinal vascular network, they do not convey a global summary of the branching pattern and its state or condition. The retinal vascular tree can be considered a fractal structure as it has a branching pattern that exhibits the property of self-similarity. Fractal analysis, therefore, provides an additional means for the quantitative study of changes to the retinal vascular network and may be of use in detecting abnormalities related to retinopathy and systemic diseases. In this thesis, new developments to fully automated retinal vessel classification and fractal analysis were explored in order to extract potential biomarkers. These novel processes were tested and validated on several datasets of retinal images acquired with fundus cameras. The major contributions of this thesis include: 1) developing a fully automated retinal blood vessel classification technique, 2) developing a fractal analysis technique that quantifies regional as well as global branching complexity, 3) validating the methods using multiple datasets, and 4) applying the proposed methods in multiple retinal vasculature analysis studies.

Ultimamente, a análise fractal (AF) vem sendo utilizada com sucesso na área de processamento digital de voz, especialmente para fins de segmentação de palavras e fonemas, que é uma das etapas fundamentais dos sistemas de reconhecimento automático de fala (Automatic Speech Recognition - ASR ) e identificação automática de locutor (Automatic Speaker Identification - ASI). O uso prático da AF para ASR e ASI depende de dois fatores básicos: baixo custo computacional, para permitir o uso em tempo-real, e precisão nos resultados, para produzir a segmentação correta e entregar dados coerentes à etapa de classificação. Visando atender a esses objetivos, o presente trabalho propõe uma técnica de segmentação de sinais de voz baseada na dimensão do fractal, obtida com o uso da transformada wavelet discreta (DWT). Diversas famílias de wavelets são testadas e comparadas, sendo que os testes foram realizados com algumas sentenças extraídas da base de dados TIMIT do Linguistic Data Consortium (LDC). / Nowadays, fractal analysis has been successfully applied to digital speech processing, particularly for words and phonemes segmentation, which represents one of the fundamental steps in automatic speech recognition and speaker identification systems. The practical use of fractal analysis for these purposes should match two principles: low computational cost, to allow use in real-time, and accuracy in the results, to produce a correct segmentation, delivering consistent data to the classifier. Aiming at meeting these two requirements, this work proposes a technique for speech segmentation based on the fractal dimension, obtained by using the discrete wavelet transform (DWT). Many families of wavelets were tested and compared, being the experiments performed with speech data collected from TIMIT corpus provided by the Linguistic Data Consortium.

Este trabalho tem como objetivo central a elaboração, implementação e reflexão sobre uma sequência didática para o estudo de matrizes a partir de tranformações geométricas. A sequência didática pretende propiciar ao aluno um estudo que justifique as definições das operações entre matrizes e suas respectivas propriedades, a partir da observação e análise de algumas transformações geométricas, de modo a se refazer o processo histórico da definição e obtenção desses conceitos. Além disso apresenta algumas atividades de aplicação de matrizes, onde a composição e iteração de transformações geométricas no software Shapari geram algumas figuras fractais. Como metodologia de trabalho adotamos a engenharia didática para a elaboração, implementação e avaliação da didática proposta. O texto também apresenta a análise das referências sobre o ensino de matrizes e tranformações geométricas. Começando pelas orientações dos documentos oficiais e passando pela apresentação de diversos estudos sobre o tema delimitamos e justificamos a nossa proposta de ensino. Além disso, apresentamos um extrato sobre o conhecimento matemático envolvido no tema, de modo que sirva de base para o docente que implementar a seuência didática em sala de aula. / This work has as its main goal the formulation, implementation and contemplation of a didactie sequence to the study of matrices from geometric transformations. The didactic sequence intends to propitiate the student a study which justifies the definitions of operations between matrices and their respective properties, from the observation and analysis of some geometrie transformations, so that they are able to redo the historical process of the definition and the acquisition of these concepts. Besides that, it shows some activities to the application of matrices, in which the composition and interation of geometric transformations in the Shapari software generate some fractals. We have adopted Didactic Engineering as a methodology to the elaboration, implementation and evaluation of the proposed didactic sequence. The text also shows an analysis of the references about the teaching of matrices and the geometric transformations. We delimited and justified our teaching proposal by staring with orientations of official documents and showing a series of studies about the theme. Besides that, we show an excerpt about the mathematical knowledge involved in the theme, so that, it can be used as basis to the teacher who decides to implement the didactic sequence in the classroom.