abstract: Divergence-free vector field interpolants properties are explored on uniform and scattered nodes, and also their application to fluid flow problems. These interpolants may be applied to physical problems that require the approximant to have zero divergence, such as the velocity field in the incompressible Navier-Stokes equations and the magnetic and electric fields in the Maxwell's equations. In addition, the methods studied here are meshfree, and are suitable for problems defined on complex domains, where mesh generation is computationally expensive or inaccurate, or for problems where the data is only available at scattered locations. The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral method. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Esta tese investiga uma abordagem supervisionada para o problema da detecção de mudanças em imagens multitemporais de sensoriamento remoto empregando Support Vector Machines (SVM) com o uso dos kernels polinomial e gaussiano (RBF). A proposta metodológica está baseada na diferença das imagens-fração produzidas para cada data. Em imagens de cenas naturais a diferença nas frações de solo e vegetação tendem a apresentar uma distribuição simétrica em torno da origem. Esse fato pode ser usado para modelar duas distribuições normais multivariadas: mudança e não-mudança. O algoritmo Expectation-Maximization (EM) é implementado para estimar os parâmetros (vetor de médias, matriz de covariância e probabilidade a priori) associados a essas duas distribuições. Amostras aleatórias são extraídas dessas distribuições e usadas para treinar o classificador SVM nesta abordagem supervisionada. A metodologia proposta realiza testes com o uso de conjuntos de dados multitemporais de imagens multiespectrais TM-Landsat, que cobrem a mesma cena em duas datas diferentes. Os resultados são comparados com outros procedimentos, incluindo trabalhos anteriores, um conjunto de dados sintéticos e o classificador SVM One-Class. / In this thesis, we investigate a supervised approach to change detection in remote sensing multi-temporal image data by applying Support Vector Machines (SVM) technique using polynomial kernel and Gaussian kernel (RBF). The methodology is based on the difference-fraction images produced for two dates. In natural scenes, the difference in the fractions such as vegetation and bare soil occurring in two different dates tend to present a distribution symmetric around the origin of the coordinate system. This fact can be used to model two normal multivariate distributions: class change and no-change. The Expectation-Maximization algorithm (EM) is implemented to estimate the parameters (mean vector, covariance matrix and a priori probability) associated with these two distributions. Random samples are drawn from these distributions and used to train the SVM classifier in this supervised approach.The proposed methodology performs tests using multi-temporal TMLandsat multispectral image data covering the same scene in two different dates. The results are compared to other procedures including previous work, a synthetic data set and SVM One-Class.

Esta tese investiga uma abordagem não supervisionada para o problema da detecção de mudanças em imagens multiespectrais e multitemporais de sensoriamento remoto empregando Support Vector Machines (SVM) com o uso dos kernels polinomial e RBF e de uma nova métrica de pertinência de pixels. A proposta metodológica está baseada na diferença das imagens-fração produzidas para cada data. Em imagens de cenas naturais essa diferença nas frações de solo e vegetação tendem a apresentar uma distribuição simétrica próxima à origem. Essa caracteristica pode ser usada para modelar as distribuições normais multivariadas das classes mudança e não-mudança. O algoritmo Expectation-Maximization (EM) é implementado com a finalidade de estimar os parâmetros (vetor de médias, matriz de covariância e probabilidade a priori) associados a essas duas distribuições. A seguir, amostras aleatórias e normalmente distribuidas são extraídas dessas distribuições e rotuladas segundo sua pertinência em uma das classes. Essas amostras são então usadas no treinamento do classificador SVM. A partir desta classificação é estimada uma nova métrica de pertinência de pixels. A metodologia proposta realiza testes com o uso de conjuntos de dados multitemporais de imagens multiespectrais Landsat-TM que cobrem a mesma cena em duas datas diferentes. A métrica de pertinência proposta é validada através de amostras de teste controladas obtidas a partir da técnica Change Vetor Analysis, além disso, os resultados de pertinência obtidos para a imagem original com essa nova métrica são comparados aos resultados de pertinência obtidos para a mesma imagem pela métrica proposta em (Zanotta, 2010). Baseado nos resultados apresentados neste trabalho que mostram que a métrica para determinação de pertinência é válida e também apresenta resultados compatíveis com outra técnica de pertinência publicada na literatura e considerando que para obter esses resultados utilizou-se poucas amostras de treinamento, espera-se que essa métrica deva apresentar melhores resultados que os que seriam apresentados com classificadores paramétricos quando aplicado a imagens multitemporais e hiperespectrais. / This thesis investigates a unsupervised approach to the problem of change detection in multispectral and multitemporal remote sensing images using Support Vector Machines (SVM) with the use of polynomial and RBF kernels and a new metric of pertinence of pixels. The methodology is based on the difference-fraction images produced for each date. In images of natural scenes. This difference in the fractions of bare soil and vegetation tend to have a symmetrical distribution close to the origin. This feature can be used to model the multivariate normal distributions of the classes change and no-change. The Expectation- Maximization algorithm (EM) is implemented in order to estimate the parameters (mean vector, covariance matrix and a priori probability) associated with these two distributions. Then random and normally distributed samples are extracted from these distributions and labeled according to their pertinence to the classes. These samples are then used in the training of SVM classifier. From this classification is estimated a new metric of pertinence of pixel. The proposed methodology performs tests using multitemporal data sets of multispectral Landsat-TM images that cover the same scene at two different dates. The proposed metric of pertinence is validated via controlled test samples obtained from Change Vector Analysis technique. In addition, the results obtained at the original image with the new metric are compared to the results obtained at the same image applying the pertinence metric proposed in (Zanotta, 2010). Based on the results presented here showing that the metric of pertinence is valid, and also provides results consistent with other published in the relevant technical literature, and considering that to obtain these results was used a few training samples, it is expected that the metric proposed should present better results than those that would be presented with parametric classifiers when applied to multitemporal and hyperspectral images.

Esta tese investiga uma abordagem supervisionada para o problema da detecção de mudanças em imagens multitemporais de sensoriamento remoto empregando Support Vector Machines (SVM) com o uso dos kernels polinomial e gaussiano (RBF). A proposta metodológica está baseada na diferença das imagens-fração produzidas para cada data. Em imagens de cenas naturais a diferença nas frações de solo e vegetação tendem a apresentar uma distribuição simétrica em torno da origem. Esse fato pode ser usado para modelar duas distribuições normais multivariadas: mudança e não-mudança. O algoritmo Expectation-Maximization (EM) é implementado para estimar os parâmetros (vetor de médias, matriz de covariância e probabilidade a priori) associados a essas duas distribuições. Amostras aleatórias são extraídas dessas distribuições e usadas para treinar o classificador SVM nesta abordagem supervisionada. A metodologia proposta realiza testes com o uso de conjuntos de dados multitemporais de imagens multiespectrais TM-Landsat, que cobrem a mesma cena em duas datas diferentes. Os resultados são comparados com outros procedimentos, incluindo trabalhos anteriores, um conjunto de dados sintéticos e o classificador SVM One-Class. / In this thesis, we investigate a supervised approach to change detection in remote sensing multi-temporal image data by applying Support Vector Machines (SVM) technique using polynomial kernel and Gaussian kernel (RBF). The methodology is based on the difference-fraction images produced for two dates. In natural scenes, the difference in the fractions such as vegetation and bare soil occurring in two different dates tend to present a distribution symmetric around the origin of the coordinate system. This fact can be used to model two normal multivariate distributions: class change and no-change. The Expectation-Maximization algorithm (EM) is implemented to estimate the parameters (mean vector, covariance matrix and a priori probability) associated with these two distributions. Random samples are drawn from these distributions and used to train the SVM classifier in this supervised approach.The proposed methodology performs tests using multi-temporal TMLandsat multispectral image data covering the same scene in two different dates. The results are compared to other procedures including previous work, a synthetic data set and SVM One-Class.

Esta tese investiga uma abordagem não supervisionada para o problema da detecção de mudanças em imagens multiespectrais e multitemporais de sensoriamento remoto empregando Support Vector Machines (SVM) com o uso dos kernels polinomial e RBF e de uma nova métrica de pertinência de pixels. A proposta metodológica está baseada na diferença das imagens-fração produzidas para cada data. Em imagens de cenas naturais essa diferença nas frações de solo e vegetação tendem a apresentar uma distribuição simétrica próxima à origem. Essa caracteristica pode ser usada para modelar as distribuições normais multivariadas das classes mudança e não-mudança. O algoritmo Expectation-Maximization (EM) é implementado com a finalidade de estimar os parâmetros (vetor de médias, matriz de covariância e probabilidade a priori) associados a essas duas distribuições. A seguir, amostras aleatórias e normalmente distribuidas são extraídas dessas distribuições e rotuladas segundo sua pertinência em uma das classes. Essas amostras são então usadas no treinamento do classificador SVM. A partir desta classificação é estimada uma nova métrica de pertinência de pixels. A metodologia proposta realiza testes com o uso de conjuntos de dados multitemporais de imagens multiespectrais Landsat-TM que cobrem a mesma cena em duas datas diferentes. A métrica de pertinência proposta é validada através de amostras de teste controladas obtidas a partir da técnica Change Vetor Analysis, além disso, os resultados de pertinência obtidos para a imagem original com essa nova métrica são comparados aos resultados de pertinência obtidos para a mesma imagem pela métrica proposta em (Zanotta, 2010). Baseado nos resultados apresentados neste trabalho que mostram que a métrica para determinação de pertinência é válida e também apresenta resultados compatíveis com outra técnica de pertinência publicada na literatura e considerando que para obter esses resultados utilizou-se poucas amostras de treinamento, espera-se que essa métrica deva apresentar melhores resultados que os que seriam apresentados com classificadores paramétricos quando aplicado a imagens multitemporais e hiperespectrais. / This thesis investigates a unsupervised approach to the problem of change detection in multispectral and multitemporal remote sensing images using Support Vector Machines (SVM) with the use of polynomial and RBF kernels and a new metric of pertinence of pixels. The methodology is based on the difference-fraction images produced for each date. In images of natural scenes. This difference in the fractions of bare soil and vegetation tend to have a symmetrical distribution close to the origin. This feature can be used to model the multivariate normal distributions of the classes change and no-change. The Expectation- Maximization algorithm (EM) is implemented in order to estimate the parameters (mean vector, covariance matrix and a priori probability) associated with these two distributions. Then random and normally distributed samples are extracted from these distributions and labeled according to their pertinence to the classes. These samples are then used in the training of SVM classifier. From this classification is estimated a new metric of pertinence of pixel. The proposed methodology performs tests using multitemporal data sets of multispectral Landsat-TM images that cover the same scene at two different dates. The proposed metric of pertinence is validated via controlled test samples obtained from Change Vector Analysis technique. In addition, the results obtained at the original image with the new metric are compared to the results obtained at the same image applying the pertinence metric proposed in (Zanotta, 2010). Based on the results presented here showing that the metric of pertinence is valid, and also provides results consistent with other published in the relevant technical literature, and considering that to obtain these results was used a few training samples, it is expected that the metric proposed should present better results than those that would be presented with parametric classifiers when applied to multitemporal and hyperspectral images.

Dans le cadre de l'apprentissage automatique, l'utilisation de représentations alternatives, ou descripteurs, pour les données est un problème fondamental permettant d'améliorer sensiblement les résultats des algorithmes. Parmi eux, les descripteurs topologiques calculent et encodent l'information de nature topologique contenue dans les données géométriques. Ils ont pour avantage de bénéficier de nombreuses bonnes propriétés issues de la topologie, et désirables en pratique, comme par exemple leur invariance aux déformations continues des données. En revanche, la structure et les opérations nécessaires à de nombreuses méthodes d'apprentissage, comme les moyennes ou les produits scalaires, sont souvent absents de l'espace de ces descripteurs. Dans cette thèse, nous étudions en détail les propriétés métriques et statistiques des descripteurs topologiques les plus fréquents, à savoir les diagrammes de persistance et Mapper. En particulier, nous montrons que le Mapper, qui est empiriquement un descripteur instable, peut être stabilisé avec une métrique appropriée, que l'on utilise ensuite pour calculer des régions de confiance et pour régler automatiquement ses paramètres. En ce qui concerne les diagrammes de persistance, nous montrons que des produits scalaires peuvent être utilisés via des méthodes à noyaux, en définissant deux noyaux, ou plongements, dans des espaces de Hilbert en dimension finie et infinie. / In the context of supervised Machine Learning, finding alternate representations, or descriptors, for data is of primary interest since it can greatly enhance the performance of algorithms. Among them, topological descriptors focus on and encode the topological information contained in geometric data. One advantage of using these descriptors is that they enjoy many good and desireable properties, due to their topological nature. For instance, they are invariant to continuous deformations of data. However, the main drawback of these descriptors is that they often lack the structure and operations required by most Machine Learning algorithms, such as a means or scalar products. In this thesis, we study the metric and statistical properties of the most common topological descriptors, the persistence diagrams and the Mappers. In particular, we show that the Mapper, which is empirically instable, can be stabilized with an appropriate metric, that we use later on to conpute confidence regions and automatic tuning of its parameters. Concerning persistence diagrams, we show that scalar products can be defined with kernel methods by defining two kernels, or embeddings, into finite and infinite dimensional Hilbert spaces.

kernlab is an extensible package for kernel-based machine learning methods in R. It takes advantage of R's new S4 object model and provides a framework for creating and using kernel-based algorithms. The package contains dot product primitives (kernels), implementations of support vector machines and the relevance vector machine, Gaussian processes, a ranking algorithm, kernel PCA, kernel CCA, and a spectral clustering algorithm. Moreover it provides a general purpose quadratic programming solver, and an incomplete Cholesky decomposition method.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24257号 / 情博第801号 / 新制||情||135(附属図書館) / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 鹿島 久嗣, 教授 山本 章博, 教授 下平 英寿 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Continuous monitoring of coastal ecosystems aids in better understanding of their dynamics and inherent harmful effects. As many of these ecosystems prevail over space and time, there is a need for mining this spatio-temporal information for building accurate monitoring and forecast systems. Harmful Algal Blooms (HABs) pose an enormous threat to the U.S. marine habitation and economy in the coastal waters. Federal and state coastal administrators have been devising a state-of-the-art monitoring and forecasting systems for these HAB events. The efficacy of a monitoring and forecasting system relies on the performance of HAB detection. A Machine Learning based Spatio-Temporal data mining approach for the detection of HAB (STML-HAB) events in the region of Gulf of Mexico is proposed in this work. The spatio-temporal cubical neighborhood around the training sample is considered to retrieve relevant spectral information pertaining to both HAB and Non-HAB classes. A unique relevant feature subset combination is derived through evolutionary computation technique towards better classification of HAB from Non-HAB. Kernel based feature transformation and classification is used in developing the model. STML-HAB model gave significant performance improvements over the current optical detection based techniques by highly reducing the false alarm rate with an accuracy of 0.9642 on SeaWiFS data. The developed model is used for prediction on new datasets for further spatio-temporal analyses such as the seasonal variations of HAB, and sequential occurrence of algal blooms. New variability visualizations are introduced to illustrate the dynamic behavior and seasonal variations of HABs from large spatiotemporal datasets. The results outperformed the ensemble of the currently available empirical methods for HAB detection. The ensemble method is implemented by a new approach for combining the empirical models using a probabilistic neural network model. The model is also compared with the results obtained using various feature extraction techniques, spatial neighborhoods and classifiers.

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