Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / A popular method for interpolating multidimensional scattered data is using radial basis functions. In this thesis we present the basic theory of radial basis function interpolation and also regard the solvability and stability of the method. Solving the interpolant directly has a high computational cost for large datasets, hence using numerical methods to approximate the interpolant is necessary. We consider some recent numerical algorithms. Software to implement radial basis function interpolation and to display the 3D interpolants obtained, is developed. We present results obtained from using our implementation for radial basis functions on GIS and 3D face data as well as an image warping application.

Orientador: Silvia Regina Lucas de Souza / Resumo: Em ambientes de produção animal o processo de análise das condições de conforto térmico pode ser realizado mediante o cálculo de índices de conforto térmico e da elaboração de mapas de variabilidade espacial de parâmetros meteorológicos e dos próprios índices de conforto térmico. Os métodos atuais para análise do bem-estar animal, especificamente aqueles por meio de mapas de índices de conforto térmico, são realizados em programas de computador utilizando-se softwares específicos de interpolação de dados. Esses processos, por sua vez, não permitem que avaliações de conforto térmico sejam realizadas facilmente no próprio local de coleta de dados. Assim, esse trabalho foi realizado com o objetivo de desenvolver um sistema para criação de mapas de variabilidade espacial da temperatura do ar e de índices de conforto térmico de animais, utilizando o método interpolador IDW (Inverse Distance Weighted), ou Inverso da Distância Ponderada. O sistema desenvolvido é composto por um aplicativo Android e dispositivos de transmissão e recepção de dados meteorológicos. O aplicativo foi desenvolvido por meio da linguagem de programação orientada a objetos Java, mediante o ambiente de desenvolvimento integrado Android Studio. Já os dispositivos móveis foram criados utilizando-se sensores de baixo custo de temperatura do ar, temperatura de globo negro e umidade relativa, juntamente com microcontroladores de plataforma eletrônica de código aberto – Arduino. O algoritmo desenvolvido para o aplic... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In animal production environments, the process of analysis of thermal comfort conditions may be carried out through the calculation of the thermal comfort index and the creation of maps showing spatial variabilities of meteorological parameters and the thermal comfort indexes. The current methods for analysis of animal well fare, especially those including maps of thermal comfort indexes, are carried out using computer programs with specific software of data interpolation. On the other hand, these processes don’t allow the evaluations of thermal comfort to be easily performed in the data collection site. Therefore, the aim of this study was developing a system for creation of maps with spatial variability of the air temperature and thermal comfort of animals, using the IDW (Inverse Distance Weighted) interpolation method. The system created is formed by an Android app and devices of transmission and reception of meteorological data. The app was developed using the Java programming language, through the Android Studio integrated development environment. The mobile devices were made using low cost air temperature, black globe temperature and relative humidity sensors, and microcontrollers of open source electronic platform – Arduino. The algorithm developed for the Android app was validated by means of comparing the air temperature figures measured, and those interpolated by the app in a broiler shed. On the other hand, the transmission and reception of data were validated via ... (Complete abstract click electronic access below) / Doutor

Analysis and visualization of dynamic scenes is often constrained by the amount of spatio-temporal information available from the environment. In most scenarios, we have to account for incomplete information and sparse motion data, requiring us to employ interpolation and approximation methods to fill for the missing information. Scattered data interpolation and approximation techniques have been widely used for solving the problem of completing surfaces and images with incomplete input data. We introduce approaches for such data interpolation and approximation from limited sensors, into the domain of analyzing and visualizing dynamic scenes. Data from dynamic scenes is subject to constraints due to the spatial layout of the scene and/or the configurations of video cameras in use. Such constraints include: (1) sparsely available cameras observing the scene, (2) limited field of view provided by the cameras in use, (3) incomplete motion at a specific moment, and (4) varying frame rates due to different exposures and resolutions. In this thesis, we establish these forms of incompleteness in the scene, as spatio-temporal uncertainties, and propose solutions for resolving the uncertainties by applying scattered data approximation into a spatio-temporal domain. The main contributions of this research are as follows: First, we provide an efficient framework to visualize large-scale dynamic scenes from distributed static videos. Second, we adopt Radial Basis Function (RBF) interpolation to the spatio-temporal domain to generate global motion tendency. The tendency, represented by a dense flow field, is used to optimally pan and tilt a video camera. Third, we propose a method to represent motion trajectories using stochastic vector fields. Gaussian Process Regression (GPR) is used to generate a dense vector field and the certainty of each vector in the field. The generated stochastic fields are used for recognizing motion patterns under varying frame-rate and incompleteness of the input videos. Fourth, we also show that the stochastic representation of vector field can also be used for modeling global tendency to detect the region of interests in dynamic scenes with camera motion. We evaluate and demonstrate our approaches in several applications for visualizing virtual cities, automating sports broadcasting, and recognizing traffic patterns in surveillance videos.

A presente tese trata da representação e da visualização volumétrica de dados espaciais, através das quais novas possibilidades para visualizar tridimensionalmente as propriedades dos solos são apresentadas. O uso de volumes nos processos avaliativos dos solos permite ver e explorar a estrutura complexa do fenômeno como um corpo contínuo no espaço, incorporando dessa forma a terceira dimensão na cartografia de solos. A metodologia de trabalho adotada compreende a aquisição, organização e codificação dos dados, como também a interpolação, formação e visualização de volumes, através da utilização de um Sistema de Informações Geográficas Tridimensional em conjunto com um programa para visualização de volumes. Os dados de solos são de natureza qualitativa e quantitativa e foram interpolados para sua representação contínua no espaço tridimensional. Os resultados obtidos para as representações volumétricas das propriedades físicas, químicas e morfológicas são uma nova forma de visualizar os solos e se constituem em uma fonte de novos conhecimentos para o estudo deste fenômeno. / This dissertation is on volume representation and visualization of spatial data. It aims at showing new possibilities to visualize soil’s properties on 3D. The use of volumes in soil’s evaluation processes allows one to visualize and to explore the phenomenon as a continuous body in the space bringing third dimension to soil cartography. The methodology of this research entails the acquisition, organization and coding of data, as well as volume interpolation, formation and visualization by means of 3D-GIS environment. Soil’s data are either qualitative or quantitative and they were interpolated in order to be continuously represented in 3D space. The outcome of volume representations of physical, chemical and morphological properties is a new way to visualize the soil and a new source of knowledge to the study of this phenomenon.

A presente tese trata da representação e da visualização volumétrica de dados espaciais, através das quais novas possibilidades para visualizar tridimensionalmente as propriedades dos solos são apresentadas. O uso de volumes nos processos avaliativos dos solos permite ver e explorar a estrutura complexa do fenômeno como um corpo contínuo no espaço, incorporando dessa forma a terceira dimensão na cartografia de solos. A metodologia de trabalho adotada compreende a aquisição, organização e codificação dos dados, como também a interpolação, formação e visualização de volumes, através da utilização de um Sistema de Informações Geográficas Tridimensional em conjunto com um programa para visualização de volumes. Os dados de solos são de natureza qualitativa e quantitativa e foram interpolados para sua representação contínua no espaço tridimensional. Os resultados obtidos para as representações volumétricas das propriedades físicas, químicas e morfológicas são uma nova forma de visualizar os solos e se constituem em uma fonte de novos conhecimentos para o estudo deste fenômeno. / This dissertation is on volume representation and visualization of spatial data. It aims at showing new possibilities to visualize soil’s properties on 3D. The use of volumes in soil’s evaluation processes allows one to visualize and to explore the phenomenon as a continuous body in the space bringing third dimension to soil cartography. The methodology of this research entails the acquisition, organization and coding of data, as well as volume interpolation, formation and visualization by means of 3D-GIS environment. Soil’s data are either qualitative or quantitative and they were interpolated in order to be continuously represented in 3D space. The outcome of volume representations of physical, chemical and morphological properties is a new way to visualize the soil and a new source of knowledge to the study of this phenomenon.

A presente tese trata da representação e da visualização volumétrica de dados espaciais, através das quais novas possibilidades para visualizar tridimensionalmente as propriedades dos solos são apresentadas. O uso de volumes nos processos avaliativos dos solos permite ver e explorar a estrutura complexa do fenômeno como um corpo contínuo no espaço, incorporando dessa forma a terceira dimensão na cartografia de solos. A metodologia de trabalho adotada compreende a aquisição, organização e codificação dos dados, como também a interpolação, formação e visualização de volumes, através da utilização de um Sistema de Informações Geográficas Tridimensional em conjunto com um programa para visualização de volumes. Os dados de solos são de natureza qualitativa e quantitativa e foram interpolados para sua representação contínua no espaço tridimensional. Os resultados obtidos para as representações volumétricas das propriedades físicas, químicas e morfológicas são uma nova forma de visualizar os solos e se constituem em uma fonte de novos conhecimentos para o estudo deste fenômeno. / This dissertation is on volume representation and visualization of spatial data. It aims at showing new possibilities to visualize soil’s properties on 3D. The use of volumes in soil’s evaluation processes allows one to visualize and to explore the phenomenon as a continuous body in the space bringing third dimension to soil cartography. The methodology of this research entails the acquisition, organization and coding of data, as well as volume interpolation, formation and visualization by means of 3D-GIS environment. Soil’s data are either qualitative or quantitative and they were interpolated in order to be continuously represented in 3D space. The outcome of volume representations of physical, chemical and morphological properties is a new way to visualize the soil and a new source of knowledge to the study of this phenomenon.

Deteriorating air quality is a growing concern that has been linked to many health- related issues. Its monitoring is a good first step to understanding the problem. However, it is not always possible to collect air quality data from every location. Various data interpolation techniques are used to assist with populating sparse maps with more context, but many of these algorithms are computationally expensive. This work presents a three- step chain mail algorithm that uses kriging (without any modifications to the kriging algorithm itself) and achieves up to ×100 execution time improvement with minimal accuracy loss (relative RMSE of 3%) by parallelizing the load for the locally tested data sets. This approach can be described as a multiple- step parallel interpolation algorithm that includes specific regional border data manipulation for achieving greater accuracy. It does so by interpolating geographically defined data chunks in parallel and sharing the results with their neighboring nodes to provide context and compensate for lack of knowledge of the surrounding areas. Combined with the cloud serverless function architecture, this approach opens doors to interpolating data sets of huge sizes in a matter of minutes while remaining cost- efficient. The effectiveness of the three- step chain mail approach depends on the equal point distribution among all regions and the resolution of the parallel configuration, but in general, it offers a good balance between execution speed and accuracy. / Försämrad luftkvalitet är en växande oro som har kopplats till många hälsorelaterade frågor. Övervakningen är ett bra första steg för att förstå problemet. Det är dock inte alltid möjligt att samla in luftkvalitetsdata från alla platser. Olika interpolationsmetoder används för att hjälpa till att fylla i glesa kartor med mer sammanhang, men många av dessa algoritmer är beräkningsdyra. Detta arbete presenterar en trestegs ‘kedjepostalgoritm’ som använder kriging (utan några modifieringar av själva krigingsalgoritmen) och uppnår upp till × 100 förbättring av exekveringstiden med minimal noggrannhetsförlust (relativ RMSE på 3%) genom att parallellisera exekveringen för de lokalt testade datamängderna. Detta tillvägagångssätt kan beskrivas som en flerstegs parallell interpoleringsalgoritm som inkluderar regional specifik gränsdatamanipulation för att uppnå större noggrannhet. Det görs genom att interpolera geografiskt definierade databitar parallellt och dela resultaten med sina angränsande noder för att ge sammanhang och kompensera för bristande kunskap om de omgivande områdena. I kombination med den molnserverfria funktionsarkitekturen öppnar detta tillvägagångssätt dörrar till interpolering av datamängder av stora storlekar på några minuter samtidigt som det förblir kostnadseffektivt. Effektiviteten i kedjepostalgorithmen i tre steg beror på lika punktfördelning mellan alla regioner och upplösningen av den parallella konfigurationen, men i allmänhet erbjuder den en bra balans mellan exekveringshastighet och noggrannhet.

Le problème de restauration d'images dégradées par des flous variables connaît un attrait croissant et touche plusieurs domaines tels que l'astronomie, la vision par ordinateur et la microscopie à feuille de lumière où les images sont de taille un milliard de pixels. Les flous variables peuvent être modélisés par des opérateurs intégraux qui associent à une image nette u, une image floue Hu. Une fois discrétisé pour être appliqué sur des images de N pixels, l'opérateur H peut être vu comme une matrice de taille N x N. Pour les applications visées, la matrice est stockée en mémoire avec un exaoctet. On voit apparaître ici les difficultés liées à ce problème de restauration des images qui sont i) le stockage de ce grand volume de données, ii) les coûts de calculs prohibitifs des produits matrice-vecteur. Ce problème souffre du fléau de la dimension. D'autre part, dans beaucoup d'applications, l'opérateur de flou n'est pas ou que partialement connu. Il y a donc deux problèmes complémentaires mais étroitement liés qui sont l'approximation et l'estimation des opérateurs de flou. Cette thèse a consisté à développer des nouveaux modèles et méthodes numériques permettant de traiter ces problèmes. / The restoration of images degraded by spatially varying blurs is a problem of increasing importance. It is encountered in many applications such as astronomy, computer vision and fluorescence microscopy where images can be of size one billion pixels. Variable blurs can be modelled by linear integral operators H that map a sharp image u to its blurred version Hu. After discretization of the image on a grid of N pixels, H can be viewed as a matrix of size N x N. For targeted applications, matrices is stored with using exabytes on the memory. This simple observation illustrates the difficulties associated to this problem: i) the storage of a huge amount of data, ii) the prohibitive computation costs of matrix-vector products. This problems suffers from the challenging curse of dimensionality. In addition, in many applications, the operator is usually unknown or only partially known. There are therefore two different problems, the approximation and the estimation of blurring operators. They are intricate and have to be addressed with a global overview. Most of the work of this thesis is dedicated to the development of new models and computational methods to address those issues.