This thesis studies the problem of measuring freedom of choice. It analyzes the concept of freedom of choice, discusses conditions that a measure should satisfy, and introduces a new class of measures that uniquely satisfy ten proposed conditions. The study uses a decision-theoretical model to represent situations of choice and a metric space model to represent differences between options. The first part of the thesis analyzes the concept of freedom of choice. Different conceptions of freedom of choice are categorized into evaluative and non-evaluative, as well as preference-dependent and preference-independent kinds. The main focus is on the three conceptions of freedom of choice as cardinality of choice sets, representativeness of the universal set, and diversity of options, as well as the three conceptions of freedom of rational choice, freedom of eligible choice, and freedom of evaluated choice. The second part discusses the conceptions, together with conditions for a measure and a variety of measures proposed in the literature. The discussion mostly focuses on preference-independent conceptions of freedom of choice, in particular the diversity conception. Different conceptions of diversity are discussed, as well as properties that could affect diversity, such as the cardinality of options, the differences between the options, and the distribution of differences between the options. As a result, the diversity conception is accepted as the proper explication of the concept of freedom of choice. In addition, eight conditions for a measure are accepted. The conditions concern domain-insensitivity, strict monotonicity, no-choice situations, dominance of differences, evenness, symmetry, spread of options, and limited function growth. None of the previously proposed measures satisfy all of these conditions. The third part concerns the construction of a ratio-scale measure that satisfies the accepted conditions. Two conditions are added regarding scale-independence and function growth proportional to cardinality. Lastly, it is shown that only one class of measures satisfy all ten conditions, given an additional assumption that the measures should be analytic functions with non-zero partial derivatives with respect to some function of the differences. These measures are introduced as the Ratio root measures.

Um espaço métrico é definido por um conjunto de objetos e uma função de distância métrica, que é utilizada para avaliar o nível de similaridade entre estes objetos. Isto permite a elaboração de Métodos de Acesso Métricos (MAMs) capazes de responder consultas por similaridade nesses conjuntos em um tempo reduzido. Em geral, esses MAMs são materializados através de uma estrutura hierárquica chamada de árvore métrica. Normalmente essas árvores são mantidas balanceadas, pois isto tende a manter a altura da árvore mínima, reduzindo o número de acessos a disco necessários para responder às consultas. No entanto, é difícil manter as estruturas balanceadas sem a existência de sobreposição entre os nós que cobrem regiões de alta densidade de objetos. O efeito disto é a degradação do tempo das consultas, pois várias subárvores devem ser analisadas para compor as consultas. Em outras palavras, minimizar a sobreposição entre os nós aumenta a eficiência das árvores métricas. Um meio efetivo para isto é flexibilizar o balanceamento das árvores métricas. Este trabalho apresenta um novo MAM dinâmico, chamado de DBM-tree (Density-Based Metric tree), que permite flexibilizar o balanceamento da estrutura, minimizando o grau de sobreposição entre os nós em regiões densas e, conseqüentemente, aumentando o seu desempenho para responder às consultas. Essa flexibilização é ajustada pelo usuário e é rigidamente controlada pela estrutura. A profundidade da árvore é maior em regiões de alta densidade, procurando um equilíbrio entre o número de acessos a disco para avaliar múltiplas subárvores e para a busca em profundidade em cada subárvore. A DBM-tree possui um algoritmo de otimização chamado de DBM-Slim-Down, que melhora o desempenho das árvores através da reorganização de elementos entre os seus nós. Os experimentos feitos com dados reais e sintéticos mostram que a DBM-tree supera em desempenho os MAMs tradicionais. Ela é, em média, 50% mais rápida que os MAMs tradicionais e reduz o número de acessos a disco e cálculos de distância em até 50%. Depois de executado o algoritmo DBM-Slim-Down, o seu desempenho melhorou em até 30% para as consultas por abrangência e aos vizinhos mais próximos. Ainda, a DBM-tree é escalável considerando tempo total de processamento, número de acessos a disco e de cálculos de distância em relação ao tamanho do conjunto de dados indexado. / A metric space is defined as a set of objects and a metric distance function that is used to measure the similarity between these objects. It allows the development of Metric Access Methods (MAMs) that are able to answer similarity queries in these datasets quickly. Usually these MAMs are materialized through a hierarchical structure called metric trees. These trees are kept balanced because it tends to maintain the height of the tree small, aiming to reduce the number of disk access required to answer queries. However, it is difficult to maintain the tree balanced without overlapping nodes covering a large number of objects, leading to the degradation of query performance. In other words, reducing the overlap among nodes increases the performance of metric trees. A possible solution is to relax the need to keep metric trees balanced. This work presents a new dynamic MAM called DBM-tree (Density-Based Metric tree), which changes the rule that imposes a rigid balancing policy, allowing a small amount of unbalancing in some regions of it. This unbalancing minimizes the degree of overlapping among some high-density nodes and, consequently, increases query answering performance. The amount of relaxation is set by the user and is strongly enforced in the tree. The height of the tree is higher in high-density regions, in order to keep a balance between searching in various subtrees and searching deeply in each subtree. The DBM-tree has an optimization algorithm called DBM-Slim-Down that improves the performance in trees through reorganizing the elements among its nodes. The experiments performed over synthetic and real-world datasets showed that the DBM-tree outperforms the traditional MAMs. The DBM-tree is, in average, 50% faster than traditional MAMs and reduces the number of distance calculations and disk accesses up to 50%. After executing the DBM-Slim-Down algorithm, the performance achieves improvements up to 30% for range and k-nearest neighbor queries. Moreover, the DBM-tree is scalable regarding time, number of disk accesses and distance calculations.

Une classe de graphes est close par mineurs si, pour tout graphe dans la classe et tout mineur de ce graphe, le mineur est ́egalement dans la classe. Par un fameux th ́eor`eme de Robertson et Seymour, nous savons que car- act ́eriser une telle classe peut ˆetre fait `a l’aide d’un nombre fini de mineurs exclus minimaux. Ceux-ci sont des graphes qui n’appartiennent pas `a la classe et qui sont minimaux dans le sens des mineurs pour cette propri ́et ́e.Dans cette thèse, nous étudions trois problèmes à propos de classes de graphes closes par mineurs. Les deux premiers sont reliés à la caractérisation de certaines classes de graphes, alors que le troisième étudie une relation de “packing-covering” dans des graphes excluant un mineur.Pour le premier problème, nous étudions des plongements isométriques de graphes dont les arêtes sont pondérées dans des espaces métriques. Principalement, nous nous intêressons aux espaces ell_2 et ell_∞. E ́tant donné un graphe pondéré, un plongement isométrique associe à chaque sommet du graphe un vecteur dans l’autre espace de sorte que pour chaque arête du graphe le poids de celle-ci est égal à la distance entre les vecteurs correspondant à ses sommets. Nous disons qu’une fonction de poids sur les arêtes est une fonction de distances réalisable s’il existe un tel plongement. Le paramètre f_p(G) détermine la dimension k minimale d’un espace ell_p telle que toute fonction de distances réalisable de G peut être plongée dans ell_p^k. Ce paramètre est monotone dans le sens des mineurs. Nous caractérisons les graphes tels que f_p(G) a une grande valeur en termes de mineurs inévitables pour p = 2 et p = ∞. Une famille de graphes donne des mineurs inévitables pour un invariant monotone pour les mineurs, si ces graphes “expliquent” pourquoi l’invariant est grand.Le deuxième problème étudie les mineurs exclus minimaux pour la classe de graphes avec φ(G) borné par une constante k, où φ(G) est un paramètre lié au dominant des coupes d’un graphe G. Ce polyèdre contient tous les points qui, composante par composante, sont plus grands ou égaux à une combination convexe des vecteurs d’incidence de coupes dans G. Le paramètre φ(G) est égal au membre de droite maximum d’une description linéaire du dominant des coupes de G en forme entière minimale. Nous étudions les mineurs exclus minimaux pour la propriété φ(G) <= 4 et montrons une nouvelle borne sur φ(G) en termes du “vertex cover number”.Le dernier problème est d’un autre type. Nous étudions une relation de “packing-covering” dans les classes de graphes excluant un mineur. Étant donné un graphe G, une boule de centre v et de rayon r est l’ensemble de tous les sommets de G qui sont à distance au plus r de v. Pour un graphe G et une collection de boules donnés nous pouvons définir un hypergraphe H dont les sommets sont ceux de G et les arêtes correspondent aux boules de la collection. Il est bien connu que dans l’hypergraphe H, le “transversal number” τ(H) vaut au moins le “packing number” ν(H). Nous montrons une borne supérieure sur ν(H) qui est linéaire en τ(H), résolvant ainsi un problème ouvert de Chepoi, Estellon et Vaxès. / A class of graphs is closed under taking minors if for each graph in the class and each minor of this graph, the minor is also in the class. By a famous result of Robertson and Seymour, we know that characterizing such a class can be done by identifying a finite set of minimal excluded minors, that is, graphs which do not belong to the class and are minor-minimal for this property.In this thesis, we study three problems in minor-closed classes of graphs. The first two are related to the characterization of some graph classes, while the third one studies a packing-covering relation for graphs excluding a minor.In the first problem, we study isometric embeddings of edge-weighted graphs into metric spaces. In particular, we consider ell_2- and ell_∞-spaces. Given a weighted graph, an isometric embedding maps the vertices of this graph to vectors such that for each edge of the graph the weight of the edge equals the distance between the vectors representing its ends. We say that a weight function on the edges of the graph is a realizable distance function if such an embedding exists. The minor-monotone parameter f_p(G) determines the minimum dimension k of an ell_p-space such that any realizable distance function of G is realizable in ell_p^k. We characterize graphs with large f_p(G) value in terms of unavoidable minors for p = 2 and p = ∞. Roughly speaking, a family of graphs gives unavoidable minors for a minor-monotone parameter if these graphs “explain” why the parameter is high.The second problem studies the minimal excluded minors of the class of graphs such that φ(G) is bounded by some constant k, where φ(G) is a parameter related to the cut dominant of a graph G. This unbounded polyhedron contains all points that are componentwise larger than or equal to a convex combination of incidence vectors of cuts in G. The parameter φ(G) is equal to the maximum right-hand side of a facet-defining inequality of the cut dominant of G in minimum integer form. We study minimal excluded graphs for the property φ(G) <= 4 and provide also a new bound of φ(G) in terms of the vertex cover number.The last problem has a different flavor as it studies a packing-covering relation in classes of graphs excluding a minor. Given a graph G, a ball of center v and radius r is the set of all vertices in G that are at distance at most r from v. Given a graph and a collection of balls, we can define a hypergraph H such that its vertices are the vertices of G and its edges correspond to the balls in the collection. It is well-known that, in the hypergraph H, the transversal number τ(H) is at least the packing number ν(H). We show that we can bound τ(H) from above by a linear function of ν(H) for every graphs G and ball collections H if the graph G excludes a minor, solving an open problem by Chepoi, Estellon et Vaxès. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

During the last decade, potential theory and p-harmonic functions have been developed in the setting of doubling metric measure spaces supporting a p-Poincar´e inequality. This theory unifies, and has applications in several areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds and Carnot groups, subelliptic differential operators and potential theory on graphs. In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We show the existence and uniqueness of solutions and their continuity when the obstacles are continuous. Moreover the solution is p-harmonic in the open set where it does not touch the continuous obstacles. The boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles vary and fix the boundary values and show the convergence of the solutions. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω. / Låt oss börja med att betrakta följande situation: Vi vill förflytta oss från en plats vid ena sidan av en äng till en viss punkt på andra sidan ängen. På båda sidor om ängen finns skogsområden som vi inte får gå in i. Ängen är tyvärr inte homogen utan består av olika sorters mark som vi har noggrant beskrivet på en karta. Vi vill göra förflyttningen på smidigast sätt, men då ängen inte är homogen ska vi förmodligen inte gå rakaste vägen utan ska anpassa vägen optimalt efter terrängen. Detta är ett exempel på ett dubbelhinderproblem där hindren är skogsområdena på sidorna som vi måste hålla oss utanför. Mer abstrakt vill man minimiera energin hos funktioner som tar vissa givna randvärden (de givna start- och slutpunkterna i exemplet ovan) och som håller sig mellan ett undre och ett övre hinder. I denna avhandling studeras detta dubbelhinderproblem i väldigt allmänna situationer. För att kunna lösa hinderproblemet krävs det att vi tillåter ickekontinuerliga lösningar och då visas i avhandlingen att hinderproblemet är entydigt lösbart. Ett huvudresultat i avhandlingen är att om våra hinder är kontinuerliga så blir även lösningen kontinuerlig. Vidare visas diverse konvergenssatser som visar hur lösningarna varierar när hindren eller området i vilket problemet löses varierar. Hinderproblem har utöver eget intresse viktiga tillämpningar i potentialteorin, bland annat för att studera motsvarande energiminimeringsproblem utan hinder. / <p>Incorrect series title in colophon.</p>

No description available.

The aim of this thesis is to expand on parts of the work of Björn–Björn–Shanmugalingam [2] and in particular on the construction and properties of hyperbolic fillings of nonempty bounded metric spaces. In light of [2], we introduce two new parameters λ and ξ to the construction while relaxing a specific maximal-condition. With these modifications we obtain a slightly more flexible model that generates a larger family of hyperbolic fillings. We then show that every hyperbolic filling in this family possess the desired property of being Gromov hyperbolic. Next, we uniformize an arbitrary hyperbolic filling of this type and show that, under fairly weak conditions, the boundary of the uniformization is snowflake-equivalent to the completion of the metric space it corresponds to. Finally, we show that this unifomized hyperbolic filling is a uniform space. In summary, our construction generates hyperbolic fillings which satisfy the necessary conditions for it to serve its intended purpose of an analytical tool for further studies in [2, Chapters 9-13 ] or similar. As such, it can be regarded as an improvement to the reference model.

We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn, and some modern generalizations of these concepts to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the assumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verification of the sharpness of a parameter. At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rn if and only if it holds on the corresponding bow-tie Finally, we consider a concrete weight function, show that it is a Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].

Made available in DSpace on 2016-04-27T16:58:45Z (GMT). No. of bitstreams: 1 Luis Carlos de Carvalho.pdf: 2327001 bytes, checksum: 2f70a04711cd873ebfd4e6fc2bab144a (MD5) Previous issue date: 2008-10-07 / Secretaria da Educação do Estado de São Paulo / This work had as objective to investigate which the organization that the didactic books of Mathematics destined to 2ª series of Average Education make, referring to the subject Metric Space Geometry, and if this organization favors the construction of the geometric thought. For in such a way, the subject was searched in available literature, official documents, dissertations and theses defended in Brazil and articles of national and international congresses. They had been analyzed three of eight books of Mathematics sent for 2ª series of Average Education for the National Program of the Book for Average Education, 2006 of the Ministry of the Education and Culture, to the schools of the Direction of Education of São Bernardo do Campo - SP, according to resulted of the research of: Duval (1995), Robert (1998) and Parsysz (2000). The analysis of didactic books had as purpose to verify if the activities for them proposals provide and favor the construction of the knowledge on the part of the pupils, considering activities with use of material concrete, constructions with instruments as ruler and compass or softwares that they facilitate the visualization and they develop the space geometric thought. One concluded that the analyzed didactic books take care of partially to the construction of the geometric thought, because the results of the research indicated to little exploration on the part of the authors of activities that develop the visualization, the representation in the plan of the three-dimensional figures are not stimulated. A balance with regard to the number of considered exercises that demand the levels technician and mobilized and a discrepancy with regard to the available level and to the lack of activities was evidenced that can be developed by educational software, contributing in this way for the diffusion of a vision maken a mistake of the teacher on the education of Metric Space Geometry / Este trabalho teve como objetivo investigar qual a organização que os livros didáticos de Matemática destinados à 2ª série do Ensino Médio fazem referente ao tema Geometria Espacial Métrica, e se essa organização favorece a construção do pensamento geométrico. Para tanto, o tema foi pesquisado na literatura disponível, em documentos oficiais, em dissertações e teses defendidas no Brasil e em artigos de congressos nacionais e internacionais. Três dos oito livros de Matemática enviados para a 2ª série do Ensino Médio pelo Programa Nacional do Livro para o Ensino Médio, 2006, do Ministério da Educação e Cultura, às escolas da Diretoria de Ensino de São Bernardo do Campo SP foram analisados, segundo os resultados das pesquisas de: Duval (1995); Robert (1998) e Parsysz (2000). A análise dos livros didáticos teve como finalidade verificar se as atividades por eles propostas proporcionam e favorecem a construção do conhecimento por parte dos alunos, propondo atividades com uso de material concreto, construções com instrumentos, como régua e compasso ou softwares que facilitam a visualização e desenvolvem o pensamento geométrico espacial. Concluiu-se que os livros didáticos analisados atendem parcialmente à construção do pensamento geométrico espacial, porque os resultados da pesquisa indicaram a pouca exploração por parte dos autores de atividades que desenvolvem a visualização, observou-se que a representação no plano das figuras tridimensionais não é estimulada. Constatou-se um equilíbrio com relação ao número de exercícios propostos que exigem os níveis técnicos e mobilizáveis e uma discrepância com relação ao nível disponível e à falta de atividades que possam ser desenvolvidas por software educacional, contribuindo desta maneira para a difusão de uma visão equivocada do professor sobre o ensino da Geometria Espacial Métrica

Submitted by Alison Vanceto (alison-vanceto@hotmail.com) on 2017-03-17T12:24:22Z No. of bitstreams: 1 DissCOSF.pdf: 3751345 bytes, checksum: 50050f093a497de77a404a0a957ad02c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-04-24T13:10:09Z (GMT) No. of bitstreams: 1 DissCOSF.pdf: 3751345 bytes, checksum: 50050f093a497de77a404a0a957ad02c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-04-24T13:10:17Z (GMT) No. of bitstreams: 1 DissCOSF.pdf: 3751345 bytes, checksum: 50050f093a497de77a404a0a957ad02c (MD5) / Made available in DSpace on 2017-04-24T13:13:58Z (GMT). No. of bitstreams: 1 DissCOSF.pdf: 3751345 bytes, checksum: 50050f093a497de77a404a0a957ad02c (MD5) Previous issue date: 2016-11-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The expansion of different areas of knowledge through many types of information brought the necessity to support complex data (images, sounds, videos, strings, DNA chains, etc.), that do not have a Total Order Relationship and need other management mechanisms, like the contentbased retrieval. In general, they are represented in metric space domains, where we have only the elements and the distances between them. Through the characteristics extracted from them, we perform the similarity search. Considering the necessity to associate temporal information on these data in many applications, this work aims to analyze the temporal evolve of metric data. One alternative for this is embedding them into a multidimensional space to allow trajectories estimates. We studied different methods of embedding and analyzed how this affected the data’s distribution and, consequently, the estimates. Two new methods were purposed to estimate an element’s status on a different time from that available in database, in order to reduce the number of non-relevant elements on search results. These methods are based on radius search reduction (range) and evaluation of retrieved element’s proximity by using an approximation of reverse k- NN. We performed experiments which showed that purposed methods could improve the estimate’s result, that used to be performed only using k-NN searches. / A expansão de diferentes áreas do conhecimento com os diversos tipos de informação tornou necessário o suporte a dados complexos (imagens, sons, vídeos, cadeias de DNA, entre outros), que por não possuírem uma Relação de Ordem Total (ROT), necessitam de outros mecanismos de gerenciamento, como a recuperação por conteúdo. Em geral, esses dados são representados em domínios de espaços métricos, onde apenas se tem os elementos e as distâncias entre eles. Através das características extraídas dos mesmos, realiza-se consultas por similaridade. Considerando a necessidade de associar a informação temporal a esses dados em muitas aplicações, este trabalho visa analisar a evolução temporal dos dados métricos. Para isso, uma alternativa é mapeá-los para um espaço multidimensional, a fim possibilitar a estimativa de trajetórias. Neste trabalho, foram estudados diferentes métodos de mapeamento, sendo também analisado como o mapeamento afetou a distribuição dos mesmos e, por conseguinte, a realização das estimativas. Foram propostos dois novos métodos para estimar o estado de um elemento em um tempo diferente daqueles disponíveis na base de dados, com o objetivo de reduzir no conjunto resposta a quantidade de elementos não relevantes. Os métodos propostos são baseados na redução do raio de consulta na região estimada pela delimitação do raio de consulta (range) e a avaliação da proximidade dos elementos retornados utilizando verificação (aproximação) do k-NN reverso. Foram realizados experimentos que mostraram que os métodos propostos melhoraram o resultado final das estimativas, que anteriormente eram realizadas apenas com consultas aos vizinhos mais próximos.

Orientador: Clésio Luis Tozzi / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-20T02:20:27Z (GMT). No. of bitstreams: 1 Gomes_MarceloMarques_M.pdf: 2920969 bytes, checksum: 0ae0c041b47241b92c32e9792f18d608 (MD5) Previous issue date: 2012 / Resumo: Este trabalho aborda a solução para o problema da recuperação da informação tridimensional somente a partir de imagens da cena ou do objeto que se deseja modelar desconhecidas as informações a respeito das câmeras utilizadas. A solução apresentada divide-se em duas partes: na primeira se estabelece a relação entre pares ou trios de câmeras utilizando matriz fundamental ou tensor trifocal, respectivamente e a partir dessas relações obtém-se por meio de triangulação uma reconstrução tridimensional da cena ou do objeto observado em função de uma transformação projetiva arbitrária. Na segunda parte determina-se uma transformação que leva a cena ou o objeto obtidos no espaço projetivo para o espaço métrico. Essa transformação que leva o objeto recuperado no espaço projetivo para o espaço métrico é encontrada por meio de autocalibração utilizando o plano no infinito e a cônica absoluta que apresentam a propriedade de serem invariantes a rotação, translação e escala, em relação a um referencial escolhido arbitrariamente. Dependendo do número de imagens disponíveis é necessário inserir restrições nos parâmetros intrínsecos das câmeras para viabilizar o cálculo da transformação. Essas restrições são inseridas na forma de suposições a respeito dos parâmetros intrínsecos das câmeras, como pontos principais conhecidos ou constantes entre as câmeras, skew nulo, relação de aspecto unitária etc. Os resultados da reconstrução no espaço projetivo obtidos com o uso da matriz fundamental e do tensor trifocal foram comparados em relação ao erro de reconstrução utilizando protótipo implementado em Matlab e imagens sintéticas. A solução geral foi avaliada em relação ao erro de reprojeção, reconstrução no espaço métrico com base em imagens sintéticas e imagens reais de objetos conhecidos, utilizando um protótipo desenvolvido na plataforma Embarcadero Delphi / Abstract: This work addresses a solution to the problem of recovering three-dimensional information from images of a scene or modeled object based only on images and without any information of the cameras parameters. The presented solution is divided in two parts: in the first part it is established the relationship between a pair or a triple of cameras using the fundamental matrix or trifocal tensor, respectively, and obtained by triangulation a three-dimensional reconstruction of the observed scene or object in function of an arbitrary projective transformation. In the second part it is determined a transformation to covert the obtained scene or object reconstruction from the projective space to the metric space. This transformation is found by auto-calibration using the plane at infinity and the absolute conic which have the property of being invariant to rotation, translation and scale in relation to an arbitrarily chosen reference frame. According to the number of available images, restrictions on the intrinsic parameters may be necessary in order to obtain a valid transformation. These restrictions are inserted in the form of assumptions about the values of the intrinsic parameters of the cameras or relations between then, like known principal points or constant values for the cameras, zero skew, unit aspect ratio and so on. Using synthetic images and a prototype implemented in Matlab, the results of reconstruction in projective space based on the fundamental matrix and trifocal tensor were compared in relation to the error of reconstruction. Based on synthetic images and real images of known objects and a prototype developed in the Embarcadero Delphi platform, the general solution was evaluated in relation to the reprojection error and the error of reconstruction in the metric space / Mestrado / Engenharia de Computação / Mestre em Engenharia Elétrica