This work investigate two different approaches for the parametrization of a special moment problem of Stieltjes-type. On the one hand we deal with systems of Potapov's fundamental matrix inequalities. Thereby, we examine certain invariant subspaces, so-called Dubovoj subspaces, and special matrix polynomials as wells as their associated J- forms. On the other hand we consider a Schur-analytic approach and present a special one-step algorithm. Moreover, considerations on linear fractional transformations of matrices serve as an important tool for the development of the algorithm. Both representations aim at a description of the solution in the non-degenerate case as well as in the different degenerate cases.

L’objectif général de cette thèse est d’appliquer une méthode d’optimisation polynomiale basée sur la théorie des moments à certains problèmes de vision artificielle. Ces problèmes sont en général non convexes et classiquement résolus à l’aide de méthodes d’optimisation locales Ces techniques ne convergent généralement pas vers le minimum global et nécessitent de fournir une estimée initiale proche de la solution exacte. Les méthodes d’optimisation globale permettent d’éviter ces inconvénients. L’optimisation polynomiale basée sur la théorie des moments présente en outre l’avantage de prendre en compte des contraintes. Dans cette thèse nous étendrons cette méthode aux problèmes de minimisation d’une somme d’un grand nombre de fractions rationnelles. De plus, sous certaines hypothèses de "faible couplage" ou de "parcimonie" des variables du problème, nous montrerons qu’il est possible de considérer un nombre important de variables tout en conservant des temps de calcul raisonnables. Enfin nous appliquerons les méthodes proposées aux problèmes de vision par ordinateur suivants : minimisation des distorsions projectives induites par le processus de rectification d’images, estimation de la matrice fondamentale, reconstruction 3D multi-vues avec et sans distorsions radiales. / The overall objective of this thesis is to apply a polynomial optimization method, based on moments theory, on some vision problems. These problems are often nonconvex and they are classically solved using local optimization methods. Without additional hypothesis, these techniques don’t converge to the global minimum and need to provide an initial estimate close to the exact solution. Global optimization methods overcome this drawback. Moreover, the polynomial optimization based on moments theory could take into account particular constraints. In this thesis, we extend this method to the problems of minimizing a sum of many rational functions. In addition, under particular assumptions of "sparsity", we show that it is possible to deal with a large number of variables while maintaining reasonable computation times. Finally, we apply these methods to particular computer vision problems: minimization of projective distortions due to image rectification process, Fundamental matrix estimation, and multi-view 3D reconstruction with and without radial distortions.

Neste trabalho, são caracterizadas as soluções do tipo ondas planas e modais de modelos matemáticos referentes à teoria de linhas de transmissão, com e sem perdas, e à teoria de vigas, modelo de Timoshenko e modelo não local de Eringen. Os modelos são formulados matricialmente, e as ondas em questão são determinadas em termos da base gerada pela resposta matricial fundamental de sistemas de equações diferenciais ordinárias de primeira, segunda e quarta ordem. A resposta matricial fundamental é utilizada numa forma fechada que envolve o acoplamento de um número finito de matrizes e uma função escalar geradora e suas derivadas. A função escalar geradora é bem comportada para mudanças em torno de frequências críticas e sua robustez é exibida através da técnica de Liouville. As ondas modais são decompostas em termos de uma parte que viaja para frente e uma parte que viaja para trás. Essa decomposição é utilizada para fornecer matrizes de reflexão e transmissão em descontinuidades e condições de contorno. No contexto das linhas de transmissão são consideradas uma junção de linhas com impedâncias características diferentes ou uma carga em uma extremidade da linha. Na teoria de Timoshenko são consideradas uma fissura ou condições de contorno em uma das extremidades. Exemplos numéricos com descontinuidade são considerados na viga. Na teoria de linhas de transmissão exemplos com multicondutores são considerados e observações são realizadas sobre a decomposição das ondas modais. No modelo não local de Eringen, para vigas bi-apoiadas é discutida a existência do segundo espectro de frequências. / Plane type solutions and modal waves of mathematical models, which refer to transmission lines theory, both lossless and lossy, and to beam theory, using both Timoshenko and nonlocal Eringen models, are being characterized in this work. The models are formulated in matrix form, and the waves are determined in terms of matrix basis generated by fundamental matrix response of systems of ordinary differential equations of first, second and fourth order. The fundamental matrix response is used in the closed-form, which involve the coupling between a number finite of matrices of a generating scalar function and its derivatives. The generating scalar function is well behaved for changes around critical frequencies and its robustness is exhibited through the Liouville technique. Modal waves are decomposed in forward and backward parts. This decomposition is used for providing reflection and transmission matrices when dealing with discontinuities and boundary conditions. In the context of transmission lines junction of lines with different characteristic impedances or a load at one end of the line are being considered. In Timoshenko’s theory the crack problem or boundary conditions at one end are also being considered. Numerical examples with discontinuities are considered in the context of beams. Numerical examples with discontinuities and boundary value problems were approached using modal wave decomposition. In transmission line theory examples with multiconductors are considered and observations are made about decomposition of the modal waves. In the nonlocal of Eringen model, for bi-supported beams, the existence of the second frequency spectrum is discussed.

This work investigates the process of automating reconstruction of buildings from video imagery. New metrics were developed to detect the least blurred images in a sequence for further processing. Phase correlation for point matching was investigated and new metrics were developed to identify successful matches. Direct relative orientation algorithms were investigated in-depth. A significant finding was a new 6-point algorithm which outperformed previously published algorithms for a number of calibrated camera and target geometries. The development of the new metrics and the outcomes from the comprehensive investigations conducted have contributed to a better understanding of the challenging problem of automatically reconstructing 3D objects from image sequences.

Neste trabalho, são caracterizadas as soluções do tipo ondas planas e modais de modelos matemáticos referentes à teoria de linhas de transmissão, com e sem perdas, e à teoria de vigas, modelo de Timoshenko e modelo não local de Eringen. Os modelos são formulados matricialmente, e as ondas em questão são determinadas em termos da base gerada pela resposta matricial fundamental de sistemas de equações diferenciais ordinárias de primeira, segunda e quarta ordem. A resposta matricial fundamental é utilizada numa forma fechada que envolve o acoplamento de um número finito de matrizes e uma função escalar geradora e suas derivadas. A função escalar geradora é bem comportada para mudanças em torno de frequências críticas e sua robustez é exibida através da técnica de Liouville. As ondas modais são decompostas em termos de uma parte que viaja para frente e uma parte que viaja para trás. Essa decomposição é utilizada para fornecer matrizes de reflexão e transmissão em descontinuidades e condições de contorno. No contexto das linhas de transmissão são consideradas uma junção de linhas com impedâncias características diferentes ou uma carga em uma extremidade da linha. Na teoria de Timoshenko são consideradas uma fissura ou condições de contorno em uma das extremidades. Exemplos numéricos com descontinuidade são considerados na viga. Na teoria de linhas de transmissão exemplos com multicondutores são considerados e observações são realizadas sobre a decomposição das ondas modais. No modelo não local de Eringen, para vigas bi-apoiadas é discutida a existência do segundo espectro de frequências. / Plane type solutions and modal waves of mathematical models, which refer to transmission lines theory, both lossless and lossy, and to beam theory, using both Timoshenko and nonlocal Eringen models, are being characterized in this work. The models are formulated in matrix form, and the waves are determined in terms of matrix basis generated by fundamental matrix response of systems of ordinary differential equations of first, second and fourth order. The fundamental matrix response is used in the closed-form, which involve the coupling between a number finite of matrices of a generating scalar function and its derivatives. The generating scalar function is well behaved for changes around critical frequencies and its robustness is exhibited through the Liouville technique. Modal waves are decomposed in forward and backward parts. This decomposition is used for providing reflection and transmission matrices when dealing with discontinuities and boundary conditions. In the context of transmission lines junction of lines with different characteristic impedances or a load at one end of the line are being considered. In Timoshenko’s theory the crack problem or boundary conditions at one end are also being considered. Numerical examples with discontinuities are considered in the context of beams. Numerical examples with discontinuities and boundary value problems were approached using modal wave decomposition. In transmission line theory examples with multiconductors are considered and observations are made about decomposition of the modal waves. In the nonlocal of Eringen model, for bi-supported beams, the existence of the second frequency spectrum is discussed.

Neste trabalho, são caracterizadas as soluções do tipo ondas planas e modais de modelos matemáticos referentes à teoria de linhas de transmissão, com e sem perdas, e à teoria de vigas, modelo de Timoshenko e modelo não local de Eringen. Os modelos são formulados matricialmente, e as ondas em questão são determinadas em termos da base gerada pela resposta matricial fundamental de sistemas de equações diferenciais ordinárias de primeira, segunda e quarta ordem. A resposta matricial fundamental é utilizada numa forma fechada que envolve o acoplamento de um número finito de matrizes e uma função escalar geradora e suas derivadas. A função escalar geradora é bem comportada para mudanças em torno de frequências críticas e sua robustez é exibida através da técnica de Liouville. As ondas modais são decompostas em termos de uma parte que viaja para frente e uma parte que viaja para trás. Essa decomposição é utilizada para fornecer matrizes de reflexão e transmissão em descontinuidades e condições de contorno. No contexto das linhas de transmissão são consideradas uma junção de linhas com impedâncias características diferentes ou uma carga em uma extremidade da linha. Na teoria de Timoshenko são consideradas uma fissura ou condições de contorno em uma das extremidades. Exemplos numéricos com descontinuidade são considerados na viga. Na teoria de linhas de transmissão exemplos com multicondutores são considerados e observações são realizadas sobre a decomposição das ondas modais. No modelo não local de Eringen, para vigas bi-apoiadas é discutida a existência do segundo espectro de frequências. / Plane type solutions and modal waves of mathematical models, which refer to transmission lines theory, both lossless and lossy, and to beam theory, using both Timoshenko and nonlocal Eringen models, are being characterized in this work. The models are formulated in matrix form, and the waves are determined in terms of matrix basis generated by fundamental matrix response of systems of ordinary differential equations of first, second and fourth order. The fundamental matrix response is used in the closed-form, which involve the coupling between a number finite of matrices of a generating scalar function and its derivatives. The generating scalar function is well behaved for changes around critical frequencies and its robustness is exhibited through the Liouville technique. Modal waves are decomposed in forward and backward parts. This decomposition is used for providing reflection and transmission matrices when dealing with discontinuities and boundary conditions. In the context of transmission lines junction of lines with different characteristic impedances or a load at one end of the line are being considered. In Timoshenko’s theory the crack problem or boundary conditions at one end are also being considered. Numerical examples with discontinuities are considered in the context of beams. Numerical examples with discontinuities and boundary value problems were approached using modal wave decomposition. In transmission line theory examples with multiconductors are considered and observations are made about decomposition of the modal waves. In the nonlocal of Eringen model, for bi-supported beams, the existence of the second frequency spectrum is discussed.

The main goal of this master’s thesis was get up used methods for observation the stereoscopic scene with one couple of cameras and find out good solving for processing these resulting pictures for two-view and multiple-view autostereoscopic displays for three-dimensional perception. For methods for acquisition video was introduced two methods. They were method “Off-axis” with parallel camera axis and method “Toe in” with intersections axis. For fit method was choice the method named as “Off-axis“. It was not produces the vertical parallax and in detail was in this work described principle of this method. Further were describe principles off used methods for three-dimensional perception namely from the oldest method named anaglyph after methods for viewing at autostereoscopic displays. The Autostereoscopic displays were main goal of this thesis and so their principles were described in details. For production the result image for autostereoscopic displays was used generation intermediate images between left and right camera. Resulting videos were acquisition for testing scene in created in 3D studio Blender, where was possible setting system of cameras exactly parallel axis. Then were introduce principles processing video where was extract from the couple of cameras where were connected to PC for help digitizing card and next time with two web cameras. Here is not guaranteed exact parallel axis system. Therefore this work try for real cameras achieve exactly parallel axis system by the help of transformations of frames with stereo rectification. Stereo rectification was solving with OpenCV libraries and was used two methods. Both methods work from principles epipolar geometry that was described in this work also in detail. First method rectifies pictures on the basis fundamental matrix and found correspondences points in two images of the scene and second method rectifies pictures from knowledge intrinsic and extrinsic parameters of stereoscopic system of cameras. In the end of this work was described application for implementation introduced methods.

Existing systems of scene reconstruction and theorethical basics necessary for scene reconstruction from images data are described in this work. System of scene reconstruction from video was designed and implemented. Its results were analyzed and possible future work was proposed. OpenCV, ArtToolKit and SIFT libraries which were used in this project are also described.

基礎矩陣在影像處理是非常重要的參數，舉凡不同影像間對應點之計算、座標系統轉換、乃至重建物體三維模型等問題，都有賴於基礎矩陣之精確與否。本論文中，我們提出一個機制，透過粒子群最佳化的觀念來求取基礎矩陣，我們的方法不但能提高基礎矩陣的精確度，同時能降低計算成本。 我們從多視角影像出發，以SIFT取得大量對應點資料後，從中選取8點進行粒子群最佳化。取樣時，我們透過分群與隨機挑選以避免選取共平面之點。然後利用最小平方中值表來估算初始評估值，並遵循粒子群最佳化演算法，以最小疊代次數為收斂準則，計算出最佳之基礎矩陣。 實作中我們以不同的物體模型為標的，以粒子群最佳化與最小平方中值法兩者結果比較。實驗結果顯示，疊代次數相同的實驗，粒子群最佳化演算法估測基礎矩陣所需的時間，約為最小平方中值法來估測所需時間的八分之一，同時粒子群最佳化演算法估測出來的基礎矩陣之平均誤差值也優於最小平方中值法所估測出來的結果。 / Fundamental matrix is a very important parameter in image processing. In corresponding point determination, coordinate system conversion, as well as three-dimensional model reconstruction, etc., fundamental matrix always plays an important role. Hence, obtaining an accurate fundamental matrix becomes one of the most important issues in image processing. In this paper, we present a mechanism that uses the concept of Particle Swarm Optimization (PSO) to find fundamental matrix. Our approach not only can improve the accuracy of the fundamental matrix but also can reduce computation costs. After using Scale-Invariant Feature Transform (SIFT) to get a large number of corresponding points from the multi-view images, we choose a set of eight corresponding points, based on the image resolutions, grouping principles, together with random sampling, as our initial starting points for PSO. Least Median of Squares (LMedS) is used in estimating the initial fitness value as well as the minimal number of iterations in PSO. The fundamental matrix can then be computed using the PSO algorithm. We use different objects to illustrate our mechanism and compare the results obtained by using PSO and using LMedS. The experimental results show that, if we use the same number of iterations in the experiments, the fundamental matrix computed by the PSO method have better estimated average error than that computed by the LMedS method. Also, the PSO method takes about one-eighth of the time required for the LMedS method in these computations.

Virtual reality systems are nowadays common part of many research institutes due to its low cost and effective visualization of data. They mostly allow visualization and exploration of virtual worlds, but many lack user interaction. In this paper we suggest multi-camera optical system, which allows effective user interaction, thereby increasing immersion of virtual system. This paper describes the calibration process of multiple cameras using point correspondences.