Orientador: Jayme Vaz Jr. / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T04:12:56Z (GMT). No. of bitstreams: 1 Souza_JoseVicenteCiprianode_M.pdf: 1091003 bytes, checksum: 9c23a3d5fde13a39b607a4ada7ade738 (MD5) Previous issue date: 2010 / Resumo: Nessa dissertação apresentamos a Álgebra Geométrica do Espaço Euclidiano e estudamos algumas de suas propriedades. Para exemplificar suas aplicações, estudamos a Transformação Kustaanheimo-Stiefel em termos de Álgebra Geométrica. Para isso apresentamos inicialmente a Transformação KS, que regulariza o movimento de Kepler em três dimensões removendo uma singularidade na origem, da forma como foi originalmente formulada, baseando-se em álgebra de matrizes. Feito isso, a Transformação KS é apresentada com Álgebra Geométrica, o que torna o seu entendimento geométrico mais claro e seu desenvolvimento mais simplificado. Para tal o uso do conceito de spinors é de grande importância / Abstract: In this dissertation we presented the Geometric Algebra of Euclidean Space and studied some of its properties. To exemplify its applications, we studied the Kustaanheimo-Stiefel Transformation in terms of Geometric Algebra. This purpose we presented initially the KS Transformation which regularizes the Kepler motion in three dimensions by removing a singularity at the origin, as it was originally formulated, based on matrix algebra. Done, the KS transformation is presented with Geometric Algebra, making clearer its geometric understanding and its development more simplified. With this goal the spinors concept use is of great importance / Mestrado / Fisica-Matematica / Mestre em Matemática Aplicada

Orientador: Carlile Campos Lavor / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T07:27:34Z (GMT). No. of bitstreams: 1 Alves_RafaelSantosdeOliveira_M.pdf: 2108746 bytes, checksum: 26f9217f1127ef34f9a7ae1692c995b8 (MD5) Previous issue date: 2008 / Resumo: O Algoritmo de Grover é um algoritmo quântico de busca em um conjunto desordenado. Com o uso de propriedades da mecânica quântica, ele apresenta um ganho quadrático em relação a um algoritmo clássico. Neste trabalho, apresentamos uma outra visão deste algoritmo, através da Álgebra Geométrica, motivados pela interpretação geométrica dos operadores, e verificamos que é possível escrevê-lo com uma nova linguagem, e ainda apresentar uma expressão mais simples para o operador de Grover (G) além de expressões gerais para estados resultantes de aplicações sucessivas deste operador / Abstract: Grover¿s algorithm is a quantum algorithm for searching in unstructured databases. Due to the properties of quantum mechanics, it provides a quadratic speedup over their classical counterparts. Using the Geometric Algebra, we present a new way to understand and simplify the operators of Grover¿s algorithm / Mestrado / Computação Quantica / Mestre em Matemática Aplicada

This thesis deals with conformal geometric algebra CGA for colour image processing, particularly with colour segmentation. For this reason it is not sufficient to work in RGB colour space. It is more convenient to use a colour space called CIELAB. CIELAB is endowed by Euclidean metric corresponding with human perception of colours. Afterwards an algorithm for an object detection via CGA based on colour differences is included. The final part of the thesis deals with least squares fitting of sphere to points using CGA. The sphere fitting is then used to adjust colour differences in an image to improve the algorithm for an object detection.

Tato diplomová práce má za cíl seznámit čtenáře se stále ještě relativně novou a neznámou oblastí matematiky, s geometrickou algebrou. Nejdříve jsou uvedeny základní definice a poté jsou studovány vlastnosti obecné geometrické algebry. Další velká část textu se věnuje Konformní geometrické algebře, která je v současnosti jedna z nejvíce zkoumaných a aplikovaných geometrických algeber. Jsou popsány její algebraické a geometrické vlastnosti, konkrétně schopnost reprezentovat určité geometrické objekty jako vektory. Taktéž umožňuje počítat jejich průniky a konformní transformace. Další část textu je zaměřena na aplikace Konformní geometrické algebry, nejdříve k popisu kinematiky robotické ruky a poté v binokulárním viděni.

Tato diplomová práce se zabývá využitím geometrické algebry pro kuželosečky (GAC) v autonomní navigaci, prezentované na pohybu robota v trubici. Nejprve jsou zavedeny teoretické pojmy z geometrických algeber. Následně jsou prezentovány kuželosečky v GAC. Dále je provedena implementace enginu, který je schopný provádět základní operace v GAC, včetně zobrazování kuželoseček zadaných v kontextu GAC. Nakonec je ukázán algoritmus, který odhadne osu trubice pomocí bodů, které umístí do prostoru pomocí středů elips, umístěných v obrazu, získaných obrazovým filtrem a fitovacím algoritmem.

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-19T18:29:28Z No. of bitstreams: 1 Dissertação-Valéria F de Oliveira.pdf: 243211 bytes, checksum: d5fc6e0d051a43fa2bf976c6162d42ae (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:42:42Z (GMT) No. of bitstreams: 1 Dissertação-Valéria F de Oliveira.pdf: 243211 bytes, checksum: d5fc6e0d051a43fa2bf976c6162d42ae (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:47:41Z (GMT) No. of bitstreams: 1 Dissertação-Valéria F de Oliveira.pdf: 243211 bytes, checksum: d5fc6e0d051a43fa2bf976c6162d42ae (MD5) / Made available in DSpace on 2015-07-06T18:47:42Z (GMT). No. of bitstreams: 1 Dissertação-Valéria F de Oliveira.pdf: 243211 bytes, checksum: d5fc6e0d051a43fa2bf976c6162d42ae (MD5) Previous issue date: 2015-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents an introduction to Geometric Algebra building it on the plane from axioms. This approach allows for easy generalization to space and other dimensions. We believe that it can be introduced in high school and be successfully applied as a methodological alternative to teaching basic math concepts. This is a modern approach that reflects the current mathematical knowledge, unifying concepts involving geometry, trigonometry, systems of equations, complex numbers, and others which are usually presented in separation. / Este trabalho apresenta uma introdução à Álgebra Geométrica construindo-a no plano a partir de axiomas, permitindo assim que possa ser generalizada para o espaço e outras dimensões. Acreditamos que ela pode ser introduzida no Ensino Médio e aplicada com sucesso como alternativa metodológica ao ensino de conceitos básicos de matemática. Trata-se de uma abordagem moderna que reflete o conhecimento matemático atual, unificando o ensino de conceitos que envolvem geometria, trigonometria, sistemas de equações, números complexos, entre outros e que, usualmente, são apresentados de forma separada.

La arquitectura semántica diseñada ha sido probada dentro del entorno de la gestión de los derechos de la propiedad intelectual. Esta permite asimilar nuevos módulos y cambios en la estructura. Se ha diseñado específicamente una propia ontología: IPROnto, que es totalmente interoperable con el resto de estándares, y propone una nueva forma de gestionar los derechos de la propiedad intelectual.Se realizó un análisis estadístico de la Web Semántica así como de los elementos que estábamos utilizando, IPROnto. Este estudio reveló que la Web Semántica se comportaba como un sistema complejo, que poseía propiedades por las cuales se podía caracterizar un comportamiento microscópico. Algunas ontologías son discutidas como paradigma de conocimiento común como DOLCE o FrameNet, además de procesos para poder interiorizarlas cercanos a la ciencia cognitiva utilizando 'image-schemas' para conectarlos a la web semántica, además de utilizar álgebra geométrica para conectar el simbolismo (álgebra) y semántica (geometría entendida como significado geométrico) como último paso. / The main issue of this work is to discover and face new challenges in negotiation over the WWW, concretely over the Semantic Web (SW) because it provides a new paradigm not only in language expression but also in its manipulation.As a result, a heterogeneous architecture is provided (with Multi-agent Systems) and IPR knowledge is formalized in an IPR ontology.Nowadays, agents have to know about other agents and their environments. Ontologies are been used to model agents' knowledge. In order to provide a model of the SW as real as possible, a deep statistical analysis of it has been made. It reveals that the SW behaves as a complex system and shares some properties with them. Concepts classification could not be as objective as we expect. An effort is done by SW community to establish a shared basis of knowledge for common understanding. The contribution is a way to connect it to the physical domain.

Orientador: Carlile Campos Lavor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:47:27Z (GMT). No. of bitstreams: 1 Melo_Nolmar_D.pdf: 2834013 bytes, checksum: 5639deabb953aa019e4e1c9c905e856d (MD5) Previous issue date: 2011 / Resumo: Este trabalho apresenta uma linguagem algébrica para dois elementos básicos da teoria da informação quântica (os bits quânticos e os operadores densidade), baseada nas propriedades de uma álgebra de Clifford de assinatura (n,3n). Demonstramos que a nova descrição desses elementos preserva as mesmas propriedades matemáticas obtidas com a descrição clássica. Com isso, estendemos alguns resultados apresentados na literatura que relaciona Álgebra de Clifford e Informação Quântica. / Abstract: This work presents an algebraic language for two basic elements of quantum information theory (the quantum bits and density operators), based in the properties of a Clifford algebra of signature (n,3n). We prove that the new description of these elements preserves the same mathematical properties obtained with the classical description. We also extend some results presented in the literature that relate Clifford algebra and quantum information. / Doutorado / Matematica Aplicada / Doutor em Matemática

Endometriosis is an oestrogen-dependent gynaecological condition defined as the presence of endometrial tissue outside the uterus cavity. The condition is predominantly found in women in their reproductive years, and associated with significant pelvic and abdominal chronic pain and infertility. The disease is believed to affect approximately 33% of women by a recent study. Currently, surgical intervention, often laparoscopic surgery, is the gold standard for diagnosing the disease and it remains an effective and common treatment method for all stages of endometriosis. Magnetic resonance imaging (MRI) of the patient is performed before surgery in order to locate any endometriosis lesions and to determine whether a multidisciplinary surgical team meeting is required. In this dissertation, our goal is to use image processing techniques to aid surgical planning. Specifically, we aim to improve quality of the existing images, and to automatically detect bladder endometriosis lesion in MR images as a form of bladder wall thickening. One of the main problems posed by abdominal MRI is the sparse anisotropic frequency sampling process. As a consequence, the resulting images consist of thick slices and have gaps between those slices. We have devised a method to fuse multi-view MRI consisting of axial/transverse, sagittal and coronal scans, in an attempt to restore an isotropic densely sampled frequency plane of the fused image. In addition, the proposed fusion method is steerable and is able to fuse component images in any orientation. To achieve this, we apply the Riesz transform for image decomposition and reconstruction in the frequency domain, and we propose an adaptive fusion rule to fuse multiple Riesz-components of images in different orientations. The adaptive fusion is parameterised and switches between combining frequency components via the mean and maximum rule, which is effectively a trade-off between smoothing the intrinsically noisy images while retaining the sharp delineation of features. We first validate the method using simulated images, and compare it with another fusion scheme using the discrete wavelet transform. The results show that the proposed method is better in both accuracy and computational time. Improvements of fused clinical images against unfused raw images are also illustrated. For the segmentation of the bladder wall, we investigate the level set approach. While the traditional gradient based feature detection is prone to intensity non-uniformity, we present a novel way to compute phase congruency as a reliable feature representation. In order to avoid the phase wrapping problem with inverse trigonometric functions, we devise a mathematically elegant and efficient way to combine multi-scale image features via geometric algebra. As opposed to the original phase congruency, the proposed method is more robust against noise and hence more suitable for clinical data. To address the practical issues in segmenting the bladder wall, we suggest two coupled level set frameworks to utilise information in two different MRI sequences of the same patients - the T2- and T1-weighted image. The results demonstrate a dramatic decrease in the number of failed segmentations done using a single kind of image. The resulting automated segmentations are finally validated by comparing to manual segmentations done in 2D.

This document introduces a new class of adaptive filters, namely Geometric- Algebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics, among others, a general leastsquares cost function is posed, from which two types of GAAFs are designed. The first one, called standard, provides a generalization of regular adaptive filters for any subalgebra of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing almost perfect agreement for different levels of measurement noise. The second type, called pose estimation, is designed to estimate rigid transformations { rotation and translation - in n-dimensional spaces. The GA-LMS performance is assessed in a 3-dimensional registration problem, in which it is able to estimate the rigid transformation that aligns two point clouds that share common parts. / Este documento introduz uma nova classe de filtros adaptativos, entitulados Geometric-Algebra Adaptive Filters (GAAFs). Eles s~ao projetados via formulação do problema de minimização (uma função custo de mínimos quadrados) do ponto de vista de álgebra geométrica (GA), uma abrangente linguagem matemática apropriada para a descrição de transformações geométricas. Adicionalmente, diferente do que ocorre na formulação com álgebra linear, cálculo geométrico (a extensão de álgebra geométrica que possibilita o uso de cálculo diferencial) permite aplicar as mesmas técnicas de derivação independentemente do tipo de dados (subálgebra), isto é, números reais, números complexos, quaternions etc. Usando essas e outras características, uma função custo geral de mínimos quadrados é proposta, da qual dois tipos de GAAFs são gerados. O primeiro, chamado standard, generaliza filtros adaptativos da literatura concebidos sob a perspectiva de subálgebras de GA. As seguintes variantes do filtro least-mean squares (LMS) s~ao obtidas como casos particulares: LMS real, LMS complexo e LMS quaternions. Uma análise mean-square é desenvolvida e corroborada por simulações para diferentes níveis de ruído de medição em um cenário de identificação de sistemas. O segundo tipo, chamado pose estimation, é projetado para estimar transformações rígidas - rotação e translação { em espaços n-dimensionais. A performance do filtro GA-LMS é avaliada em uma aplicação de alinhamento tridimensional na qual ele estima a tranformação rígida que alinha duas nuvens de pontos com partes em comum.