<p>Periodic signal modeling plays an important role in different fields. The unifying theme of this thesis is using nonlinear techniques to model periodic signals. The suggested techniques utilize the user pre-knowledge about the signal waveform. This gives these techniques an advantage as compared to others that do not consider such priors.</p><p>The technique of Part I relies on the fact that a sine wave that is passed through a static nonlinear function produces a harmonic spectrum of overtones. Consequently, the estimated signal model can be parameterized as a known periodic function (with unknown frequency) in cascade with an unknown static nonlinearity. The unknown frequency and the parameters of the static nonlinearity are estimated simultaneously using the recursive prediction error method (RPEM). A treatment of the local convergence properties of the RPEM is provided. Also, an adaptive grid point algorithm is introduced to estimate the unknown frequency and the parameters of the static nonlinearity in a number of adaptively estimated grid points. This gives the RPEM more freedom to select the grid points and hence reduces modeling errors.</p><p>Limit cycle oscillations problem are encountered in many applications. Therefore, mathematical modeling of limit cycles becomes an essential topic that helps to better understand and/or to avoid limit cycle oscillations in different fields. In Part II, a second-order nonlinear ODE is used to model the periodic signal as a limit cycle oscillation. The right hand side of the ODE model is parameterized using a polynomial function in the states, and then discretized to allow for the implementation of different identification algorithms. Hence, it is possible to obtain highly accurate models by only estimating a few parameters.</p><p>In Part III, different user aspects for the two nonlinear approaches of the thesis are discussed. Finally, topics for future research are presented. </p>

Periodic signal modeling plays an important role in different fields. The unifying theme of this thesis is using nonlinear techniques to model periodic signals. The suggested techniques utilize the user pre-knowledge about the signal waveform. This gives these techniques an advantage as compared to others that do not consider such priors. The technique of Part I relies on the fact that a sine wave that is passed through a static nonlinear function produces a harmonic spectrum of overtones. Consequently, the estimated signal model can be parameterized as a known periodic function (with unknown frequency) in cascade with an unknown static nonlinearity. The unknown frequency and the parameters of the static nonlinearity are estimated simultaneously using the recursive prediction error method (RPEM). A treatment of the local convergence properties of the RPEM is provided. Also, an adaptive grid point algorithm is introduced to estimate the unknown frequency and the parameters of the static nonlinearity in a number of adaptively estimated grid points. This gives the RPEM more freedom to select the grid points and hence reduces modeling errors. Limit cycle oscillations problem are encountered in many applications. Therefore, mathematical modeling of limit cycles becomes an essential topic that helps to better understand and/or to avoid limit cycle oscillations in different fields. In Part II, a second-order nonlinear ODE is used to model the periodic signal as a limit cycle oscillation. The right hand side of the ODE model is parameterized using a polynomial function in the states, and then discretized to allow for the implementation of different identification algorithms. Hence, it is possible to obtain highly accurate models by only estimating a few parameters. In Part III, different user aspects for the two nonlinear approaches of the thesis are discussed. Finally, topics for future research are presented.

Dynamic Linear Models are a state space model framework based on the Kalman filter. We use this framework to do seasonal adjustments of empirical and artificial data. A simple model and an extended model based on Gibbs sampling are used and the results are compared with the results of a standard seasonal adjustment method. The state space approach is then extended to discuss direct and indirect seasonal adjustments. This is achieved by applying a seasonal level model with no trend and some specific input variances that render different signal-to-noise ratios. This is illustrated for a system consisting of two artificial time series. Relative efficiencies between direct, indirect and multivariate, i.e. optimal, variances are then analyzed. In practice, standard seasonal adjustment packages do not support optimal/multivariate seasonal adjustments, so a univariate approach to simultaneous estimation is presented by specifying a Holt-Winters exponential smoothing method. This is applied to two sets of time series systems by defining a total loss function that is specified with a trade-off weight between the individual series’ loss functions and their aggregate loss function. The loss function is based on either the more conventional squared errors loss or on a robust Huber loss. The exponential decay parameters are then estimated by minimizing the total loss function for different trade-off weights. It is then concluded what approach, direct or indirect seasonal adjustment, is to be preferred for the two time series systems. The dynamic linear modeling approach is also applied to Swedish political opinion polls to assert the true underlying political opinion when there are several polls, with potential design effects and bias, observed at non-equidistant time points. A Wiener process model is used to model the change in the proportion of voters supporting either a specific party or a party block. Similar to stock market models, all available (political) information is assumed to be capitalized in the poll results and is incorporated in the model by assimilating opinion poll results with the model through Bayesian updating of the posterior distribution. Based on the results, we are able to assess the true underlying voter proportion and additionally predict the elections. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 3: Manuscript; Paper 4: Manuscripts</p>

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

Simulated mirror displays have a promising prospect in applications, due to its capability for virtual visualization. In most existing mirror displays, cameras are placed on top of the displays and unable to capture the person in front of the display at the highest possible resolution. The lack of a direct frontal capture of the subject's face and the geometric error introduced by image warping techniques make realistic mirror image rendering a challenging problem. The objective of this thesis is to explore the use of a robotic camera in tracking the face of the subject in front of the display to obtain a high-quality image capture. Our system uses a Bislide system to control a camera for face capture, while using a separate color-depth camera for accurate face tracking. We construct an optical device in which a one-way mirror is used so that the robotic camera behind can capture the subject while the rendered images can be displayed by reflecting off the mirror from an overhead projector. A key challenge of the proposed system is the reduction of light due to the one-way mirror. The optimal 2D Wiener filter is selected to enhance the low contrast images captured by the camera.

ABSTRACT INVERSE PROBLEMS FOR A SEMILINEAR HEAT EQUATIONS WITH MEMORY Kaya, M&uuml / jdat Ph.D, Department of Mathematics Supervisor: Prof. Dr. A. Okay &Ccedil / elebi Co-Supervisor: Prof. Dr. Varga Kalantarov May 2005, 79 pages In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.

Zugl.: Rostock, Universiẗat, Magisterarbeit, 2005.

Nous étudions certains processus de Lévy à valeurs dans les groupes d'isométries respectifs des espace-temps de Minkowski, de De Sitter et de Anti-De-Sitter. Le groupe d'isométries est vu comme le fibré des repères de l'espace-temps et les processus de Lévy considérés se projettent sur le fibré unitaire en un processus markovien relativiste ; c'est-à-dire que les trajectoires dans l'espace-temps sont de genre temps et que le générateur est invariant par les isométries. Dans la première partie nous adaptons pour les diffusions hypoelliptiques générales un résultat de Ben Arous et Gradinaru concernant la singularité de la fonction de Green hypoelliptique. Nous déduisons de cela un critère d'effilement de Wiener local pour les diffusions relativistes dans le groupe de Poincaré, groupe des isométries de l'espace-temps de Minkowski. Dans les deux dernières parties nous nous intéressons au comportement asymptotique du flot stochastique associé à ces processus de Lévy dans les différents groupes d'isométries. Sous une condition d'intégrabilité de la mesure de Lévy nous calculons explicitement les coefficients de Lyapounov des processus dans le groupe de Poincaré. Nous effectuons un travail similaire pour les espace-temps de De Sitter et Anti-De-Sitter en nous limitant au cas des diffusions. Nous explicitons de plus la frontière de Poisson pour la diffusion dans le groupe d'isométries de l'espace-temps de De Sitter. / We study some Lévy processes with values in the isometry group of Minkowski, De Sitter and Anti-de-Sitter space-times. The isometry group is seen as the frame bundle of the space-time and the Lévy processes we consider are some lift of relativistic markovian processes with values in the unitary tangent bundle of the space-time. Theses processes are relativistic in the sense that theirs trajectories are time-like and their generators are invariant by the isometries of the space-time. In the first part of this work we adapt to the case of a general hypoelliptic diffusion a result of Ben Arous and Gradinaru concerning the singularity of the hypoelliptic Green function. We deduce of this a local Wiener criterion for the relativistic diffusion in the isometry group of Minkowski space-time. In the two last parts we are interested to the asymptotic behavior of the stochastic flow associated to these Lévy processes in the different considered space-times. Under a integrability condition on the Lévy measure we compute explicitly the Lyapunov coefficient for such flows in the isometry group of Minkowski space-time. Then, we do a similar work in the context of de Sitter and Anti-de-Sitter space-times limiting ourselves to the case of diffusions. In fine, we explicit the Poisson boundary of the diffusion in the isometry group of de Sitter space-time.

Made available in DSpace on 2016-06-02T19:03:57Z (GMT). No. of bitstreams: 1 5096.pdf: 8198780 bytes, checksum: 111ff0c36ae2d9c790f8a8d7129dccba (MD5) Previous issue date: 2013-03-22 / Universidade Federal de Sao Carlos / Computed Tomography (CT) images, in many cases, need to be acquired with low photon counting due to low exposure time to the rays of the CT scanner to reduce the radiation doses to the maximum possible (in Medicine, the ALARA principle As Low As Reasonably Achievable) or even for reasons of cost, obtaining projections corrupted by Poisson noise. Invoking the Central Limit Theorem, the reconstructed images tend to be corrupted by Gaussian noise. Moreover, it was observed that this noise remains signal-dependent after the reconstruction. Thus, this work proposes to denoise the reconstructed images (post-filtering), by adopting an a priori contextual model by using Markov Random Field (MRF), to improve the visual quality of the image. Basically, for contextual filtering two approaches were considered. One uses iterative algorithms for combinatorial optimization such as ICM (Iterated Conditional Modes), GSA (Game Strategy Approach) and MPM (Maximizer of the Posterior Marginals). And the other uses variations of the Wiener filter by considering Fisher Information, Separable MRF and Isotropic MRF. Also, to address the issue of signal-dependent noise, three new methods for its local variance estimation, as well as ways to consider this model in both iterative and those based on Wiener filter methods were investigated. The proposed methods were applied to simulated and real CT images that were reconstructed by Filtered Backprojection (FBP) and Projections Onto Convex Sets (POCS) algorithms. Furthermore, the use of Non Local Means method has been proposed for a better estimate of the noise-free image. Finally, several experiments were conducted and the results were compiled and presented comparing the various methods, including the state-of-the-art Non Local Means method, showing that the context and the consideration of signal-dependent noise can contribute to CT denoising by improving the Signal-to-Noise Ratio and therefore allow a reduction in the radiation dose. / Imagens de tomografia computadorizada (CT), em diversos casos, precisam ser adquiridas com baixa contagem de fótons devido ao baixo tempo de exposição aos raios do tomógrafo para reduzir a dose de radiação ao máximo possível (na Medicina, princípio ALARA As Low As Reasonably Achievable) ou mesmo por questões de custo, fazendo com que as projeções obtidas sejam corrompidas por ruído Poisson. Invocando o Teorema Central do Limite, as imagens reconstruídas tendem a ser corrompidas por ruído Gaussiano. Além disso, observou-se que este ruído continua a ser dependente do sinal, depois da reconstrução. Desta forma, este trabalho propôs a filtragem de ruído da imagem reconstruída (pós-filtragem), adotando um modelo a priori contextual pela utilização de Campos Aleatórios Markovianos (MRF), a fim de melhorar a qualidade visual da imagem. Basicamente, para a filtragem contextual foram consideradas duas abordagens. Uma utilizando algoritmos iterativos de otimização combinatória como ICM (Iterated Conditional Modes), GSA (Game Strategy Approach) e MPM (Maximizer of the Posterior Marginals). E outra, utilizando variações do filtro de Wiener, considerando Informação de Fisher (Generalizado), MRF Separável e MRF Isotrópico. Ainda, para tratar a questão de ruído dependente do sinal, três novos métodos de estimação de suas variâncias locais, como também maneiras de se considerar este modelo tanto nos métodos iterativos quanto nos baseados em Wiener foram investigados. Os métodos foram aplicados em imagens simuladas e reais de CT reconstruídas por Retroprojeção Filtrada e POCS (Projections Onto Convex Sets). Além disso, foi proposto o uso de Non Local Means para uma melhor estimativa da imagem livre de ruído. Finalmente, diversos experimentos foram realizados e os resultados foram compilados e apresentados comparando os diversos métodos, inclusive com o método em estado-da-arte Non Local Means, mostrando que o contexto e a consideração de ruído dependente do sinal podem contribuir para a filtragem de ruído em CT pela melhora na relação Sinal-Ruído e, consequentemente, permitir a redução da dose de radiação.