Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-02-23Bitstream added on 2014-06-13T20:55:45Z : No. of bitstreams: 1 silva_alo_me_sjrp.pdf: 1157182 bytes, checksum: 08ed86b39eb7aa9014461e7988e01266 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho apresentamos conceitos teþoricos fundamentais como os Príncipios da Boa Continuação de Gestalt e da Conectividade de Kanizsa, os quais estão intimamente relacionados `a percepção visual humana estudada por psicólogos. Tais conceitos são muito importantes no contexto do processamento de imagens, principalmente no que se refere ao processo de Retoque Digital de Imagens, influenciando e auxiliando pesquisadores a criar modelos matemáticos que imitem o sistema visual humano, com a intenção de deixar o processo mais real possþývel. Apresentamos também, diversos modelos matemþaticos propostos para solucionar o problema de retoque digital, bem como técnicas para implementação computacional de tais modelos. / In this work we present fundamental theoretical concepts like the Gestalt s Good Continuation Principle and the Kanizsa s Connectivity Principle, which are closely related to human visual perception studied by psychologists. Such concepts are very important in the context of the image processing, mainly in those related to the inpainting process. These concepts are influencing and helping researchers to create mathematical models that imitate the human visual system, with the purpose to make the process as real as possible. We also present, various mathematical models developed to solve the inpainting problem and techniques for the computational implementation of theses models.

Orientador: Maurílio Boaventura / Banca: Maria Amélia Novais Schleicher / Banca: Elso Drigo Filho / Resumo: Neste trabalho apresentamos conceitos teþoricos fundamentais como os Príncipios da Boa Continuação de Gestalt e da Conectividade de Kanizsa, os quais estão intimamente relacionados 'a percepção visual humana estudada por psicólogos. Tais conceitos são muito importantes no contexto do processamento de imagens, principalmente no que se refere ao processo de Retoque Digital de Imagens, influenciando e auxiliando pesquisadores a criar modelos matemáticos que imitem o sistema visual humano, com a intenção de deixar o processo mais real possþývel. Apresentamos também, diversos modelos matemþaticos propostos para solucionar o problema de retoque digital, bem como técnicas para implementação computacional de tais modelos. / Abstract: In this work we present fundamental theoretical concepts like the Gestalt’s Good Continuation Principle and the Kanizsa’s Connectivity Principle, which are closely related to human visual perception studied by psychologists. Such concepts are very important in the context of the image processing, mainly in those related to the inpainting process. These concepts are influencing and helping researchers to create mathematical models that imitate the human visual system, with the purpose to make the process as real as possible. We also present, various mathematical models developed to solve the inpainting problem and techniques for the computational implementation of theses models. / Mestre

Orientador: Maurílio Boaventura / Banca: Antonio Castelo Filho / Banca: Heloisa Helena Marino Silva / Resumo: O objetivo deste trabalho þe aplicar Mþetodos Numþericos de ordem de precisão mais alta ao problema de Retoque Digital, visando melhorar a qualidade da aproximação quando comparada com o Método de Euler, que þe geralmente utilizado para esse tipo de problema. Para testar a eficiência de tais métodos, utilizamos três modelos de Retoque Digital: o modelo proposto por Bertalmþýo, Sapiro, Ballester e Caselles (BSBC), o modelo de Rudin, Osher e Fatemi conhecido como Variacional Total (TV) e o modelo de Chan e Shen, chamado de Difusão Guiada pela Curvatura (CDD). / Abstract: The purpose of this work is to apply Numerical Methods of higher order to the problem of Digital Inpainting, aiming to improve the quality of the approach when compared with the Euler’s Method which is generally used for this kind of problem. To test the e ciency of these methods we use three models of Digital Inpainting: the model considered by Bertalmþýo, Sapiro, Ballester and Caselles (BSBC), the model of Rudin, Osher and Fatemi known as Total Variation (TV) and the model of Chan and Shen, named Curvature Driven Di usion (CDD) / Mestre

The visualization of a 3D image obtained from computerized tomography examinations has shown itself to be an important factor for increasing the quality of medical diagnoses and, consequently, treatment efficacy. There already exist on the market, several visualization softwares, which use different techniques to show the 3D tomography image. However, to show a high quality 3D image, sophisticated devices must be used to obtain slices, close to one another, thus increasing the incidence of X-ray given to the patient. An interpolation slice method which resulted from the TC examination produces good results, and is able to reduce the X-ray incidence upon the patient. This method must reconstruct the curvature from the patient s internal structures without using slices in close proximity. This work proposes a method of 3D image interpolation, composed of a juxtaposition of the slices from CT examination results. The goal of this method is to increase the quality of 3D visualization through the production of sharp and precise structure contours. This thesis proposes the division of the interpolation method into two steps. In the first step, the goal is to obtain an initial representation of the image in 3D, which is composed of real slices as well as virtual slices which are referred to in this work as initial virtual slices. In the second step, the empty spaces of the structure are recovered by the 3D image inpainting process. This work also proposes a method to obtain the initial virtual slice and two different methods for inpainting the 3D image. These inpainting methods are the transversal slice line prolongation method and the transportation and diffusion of information. Both methods use the differential equation theory. The transportation and diffusion of information method shows better results than other methods proposed in this work, besides this, this method presents better results than the linear interpolation and Goshtasby et al. [1] methods also implemented in this work. Visual and numerical comparisons are used to obtain this conclusion. The numerical measures used are statistical correlation, the PSNR and the Hausdorff distance [2]. The transportation and diffusion of information method shows itself able to produce better results than all the other tested methods. Besides this principal contribution, this work also developed a KIT to implement 2D and 3D CT visualize applications. / A visualização de imagens resultantes de exame de tomografia computadorizada (TC) em 3D ´e um fator importante para o aumento da precisão nos diagnósticos médicos e, consequentemente, na eficácia dos tratamentos. Atualmente existem diversos produtos no mercado, que fazem uso de várias técnicas existentes para apresentação de imagens tomográficas em 3D. Contudo, para se obter maior suavidade e precisão nos contornos das estruturas visualizadas em 3D, utiliza-se equipamentos capazes de produzir fatias paralelas do corpo humano muito próximas uma das outras, aumentando a exposição dos pacientes aos raios X. Um método de interpolação de fatias resultantes de exame de TC que forneça bons resultados, pode reduzir a incidência de raios X no paciente, pois esse método pode recuperar a curvatura das estruturas sem a necessidade de uma grande proximidade entre as fatias. Este trabalho propõe um método para a interpolação de imagem em 3D, formada pela justaposição de fatias de resultados de exames de tomografia computadorizada. O objetivo desse método ´e obter contornos suaves e precisos, melhorando os processos de visualização em 3D. Para isso, esta tese propõe a divisão do processo de interpolação em duas etapas. Na primeira etapa obtém-se uma representação inicial da imagem em 3D composta por fatias reais e por fatias denominadas de fatias virtuais iniciais e, na segunda etapa, restaura-se essas estruturas geradas com um processo de retoque de imagem em 3D. Este trabalho propõe também um método para obtenção da fatia virtual inicial e dois métodos diferentes para a realização do passo de retoque da imagem em 3D resultante da justaposição das fatias reais e virtuais iniciais. Esses métodos são o prolongamento de linhas nas fatias transversais e transporte e difusão de informações. Ambos os métodos utilizam a teoria de equações diferenciais. O método de transporte e difusão de informações demonstrou melhores resultados do que outro método proposto neste trabalho, além de obter melhores resultados do que os métodos de interpolação linear e Goshtasby e outros [1] implementados neste trabalho. Comparações visuais e comparações numéricas utilizando a correlação estatística, a PSNR e a distância de Haussdorff [2] foram realizadas para se obter essas conclusões. O método de transporte e difusão de informações é capaz de gerar contornos mais suaves e precisos que esses outros métodos testados. Além dessa contribuição principal, este trabalho também desenvolveu um KIT para a construção de aplicações visualizadoras de tomografias computadorizadas em 2D e em 3D. / Mestre em Ciências

Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed.

Cette thèse se situe dans le cadre de l'analyse théorique et numérique de quelques généralisations de l'équation de Cahn-Hilliard. On étudie l'existence, l'unicité et la régularité de la solution de ces modèles ainsi que son comportement asymptotique en terme d'existence d'un attracteur global de dimension fractale finie. La première partie de la thèse concerne des modèles appliqués à la retouche d'images. D'abord, on étudie la dynamique de l'équation de Bertozzi-Esedoglu-Gillette-Cahn-Hilliard avec des conditions de type Neumann sur le bord et une nonlinéarité régulière de type polynomial et on propose un schéma numérique avec une méthode de seuil efficace pour le problème de la retouche et très rapide en terme de temps de convergence. Ensuite, on étudie ce modèle avec des conditions de type Neumann sur le bord et une nonlinéarité singulière de type logarithmique et on donne des simulations numériques avec seuil qui confirment que les résultats obtenus avec une nonlinéarité de type logarithmique sont meilleurs que ceux obtenus avec une nonlinéarité de type polynomial. Finalement, on propose un modèle basé sur le système de Cahn-Hilliard pour la retouche d'images colorées. La deuxième partie de la thèse est consacrée à des applications en biologie et en chimie. On étudie la convergence de la solution d'une généralisation de l'équation de Cahn-Hilliard avec un terme de prolifération, associée à des conditions aux limites de type Neumann et une nonlinéarité régulière. Dans ce cas, on démontre que soit la solution explose en temps fini soit elle existe globalement en temps. Par ailleurs, on donne des simulations numériques qui confirment les résultats théoriques obtenus. On termine par l'étude de l'équation de Cahn-Hilliard avec un terme source et une nonlinéarité régulière. Dans cette étude, on considère le modèle à la fois avec des conditions aux limites de type Neumann et de type Dirichlet. / This thesis is situated in the context of the theoretical and numerical analysis of some generalizations of the Cahn-Hilliard equation. We study the well-possedness of these models, as well as the asymptotic behavior in terms of the existence of finite-dimenstional (in the sense of the fractal dimension) attractors. The first part of this thesis is devoted to some models which, in particular, have applications in image inpainting. We start by the study of the dynamics of the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with Neumann boundary conditions and a regular nonlinearity. We give numerical simulations with a fast numerical scheme with threshold which is sufficient to obtain good inpainting results. Furthermore, we study this model with Neumann boundary conditions and a logarithmic nonlinearity and we also give numerical simulations which confirm that the results obtained with a logarithmic nonlinearity are better than the ones obtained with a polynomial nonlinearity. Finally, we propose a model based on the Cahn-Hilliard system which has applications in color image inpainting. The second part of this thesis is devoted to some models which, in particular, have applications in biology and chemistry. We study the convergence of the solution of a Cahn-Hilliard equation with a proliferation term and associated with Neumann boundary conditions and a regular nonlinearity. In that case, we prove that the solutions blow up in finite time or exist globally in time. Furthermore, we give numericial simulations which confirm the theoritical results. We end with the study of the Cahn-Hilliard equation with a mass source and a regular nonlinearity. In this study, we consider both Neumann and Dirichlet boundary conditions.

In this thesis we study novel higher order total variation-based variational methods for digital image reconstruction. These methods are formulated in the context of Tikhonov regularisation. We focus on regularisation techniques in which the regulariser incorporates second order derivatives or a sophisticated combination of first and second order derivatives. The introduction of higher order derivatives in the regularisation process has been shown to be an advantage over the classical first order case, i.e., total variation regularisation, as classical artifacts such as the staircasing effect are significantly reduced or totally eliminated. Also in image inpainting the introduction of higher order derivatives in the regulariser turns out to be crucial to achieve interpolation across large gaps. First, we introduce, analyse and implement a combined first and second order regularisation method with applications in image denoising, deblurring and inpainting. The method, numerically realised by the split Bregman algorithm, is computationally efficient and capable of giving comparable results with total generalised variation (TGV), a state of the art higher order method. An additional experimental analysis is performed for image inpainting and an online demo is provided on the IPOL website (Image Processing Online). We also compute and study properties of exact solutions of the one dimensional total generalised variation problem with L^{2} data fitting term, for simple piecewise affine data functions, with or without jumps . This gives an insight on how this type of regularisation behaves and unravels the role of the TGV parameters. Finally, we introduce, study and analyse a novel non-local Hessian functional. We prove localisations of the non-local Hessian to the local analogue in several topologies and our analysis results in derivative-free characterisations of higher order Sobolev and BV spaces. An alternative formulation of a non-local Hessian functional is also introduced which is able to produce piecewise affine reconstructions in image denoising, outperforming TGV.

The master's thesis deals with the problem of restoring a block of missing samples in a digital audio signal. This problem is formulated as an optimization task, which seeks the sparsest time-frequency representation of a signal within the set of feasible reconstructed signals. Several particular formulations are discussed, namely the analyzing and the synthesizing model, both for convex and non-convex approaches. Suitable algorithms are proposed for solving these formulations, and in the convex case, the method is further enhanced by various procedures to compensate for the energy drop in the inpainted signal segment. The proposed algorithms are tested on real recordings, and their performance is shown to be competitive with the state-of-the-art.

Tato práce se zabývá doplňováním chybějících dat do audio signálů a algoritmy řešícími problém založenými na řídké reprezentaci audio signálu. Práce se zaměřuje na některé algoritmy, které řeší doplňování chybějících dat do audio signálů pomocí řídké reprezentace signálů. Součástí práce je také návrh algoritmu, který používá řídkou reprezentaci signálu a také nízkou hodnost signálu ve spektrogramu audio signálu. Dále práce uvádí implementaci tohoto algoritmu v programu Matlab a jeho vyhodnocení.

The aim of this work was to develop an automated methods for removing unwanted objects from video sequences. The proposed method is able to autonomously tackle the static and the moving object with no user intervention into the process. The user only determines the object to deleted.