This thesis addresses a number of problems in computer vision, image processing, and geometry processing, and presents novel solutions to these problems. The overarching theme of the techniques presented here is a multi-scale approach, leveraging mathematical tools to represent images and surfaces at different scales, and methods that can be adapted from one type of domain (eg., the plane) to another (eg., the sphere). The main problem addressed in this thesis is known as stereo reconstruction: reconstructing the geometry of a scene or object from two or more images of that scene. We develop novel algorithms to do this, which work for both planar and spherical images. By developing a novel way to formulate the notion of disparity for spherical images, we are able effectively adapt our algorithms from planar to spherical images. Our stereo reconstruction algorithm is based on a novel application of distance transforms to multi-scale matching. We use matching information aggregated over multiple scales, and enforce consistency between these scales using distance transforms. We then show how multiple spherical disparity maps can be efficiently and robustly fused using visibility and other geometric constraints. We then show how the reconstructed point clouds can be used to synthesize a realistic sequence of novel views, images from points of view not captured in the input images, in real-time. Along the way to this result, we address some related problems. For example, multi-scale features can be detected in spherical images by convolving those images with a filterbank, generating an overcomplete spherical wavelet representation of the image from which the multiscale features can be extracted. Convolution of spherical images is much more efficient in the spherical harmonic domain than in the spatial domain. Thus, we develop a GPU implementation for fast spherical harmonic transforms and frequency domain convolutions of spherical images. This tool can also be used to detect multi-scale features on geometric surfaces. When we have a point cloud of a surface of a particular class of object, whether generated by stereo reconstruction or by some other modality, we can use statistics and machine learning to more robustly estimate the surface. If we have at our disposal a database of surfaces of a particular type of object, such as the human face, we can compute statistics over this database to constrain the possible shape a new surface of this type can take. We show how a statistical spherical wavelet shape prior can be used to efficiently and robustly reconstruct a face shape from noisy point cloud data, including stereo data.

In this thesis, parameterizations of non-linear interactions in quasigeostrophic (QG) flows for severely truncated models (STM) and Large Eddy Simulations (LES) are studied. Firstly, using Direct Numerical Simulations (DNS), atmospheric barotropic flows over topography are examined, and it is established that such flows exhibit multiple equilibrium states for a wide range of parameters. A STM is then constructed, consisting of the large scale zonal flow and a topographic mode. It is shown that, qualitatively, this system behaves similarly to the DNS as far as the interaction between the zonal flow and topography is concerned, and, in particular, exhibits multiple equilibrium states. By fitting the analytical form of the topographic stationary wave amplitude, obtained from the STM, to the results obtained from DNS, renormalized dissipation and rotation parameters are obtained. The usage of renormalized parameters in the STM results in better quantitative agreement with the DNS.¶ In the second type of problem, subgrid-scale parameterizations in LES are investigated with both atmospheric and oceanic parameters. This is in the context of two-level QG flows on the sphere, mostly, but not exclusively, employing a spherical harmonic triangular truncation at wavenumber 63 (T63) or higher. The methodology that is used is spectral, and is motivated by the stochastic representation of statistical closure theory, with the ‘damping’ and forcing covariance, representing backscatter, determined from the statistics of DNS. The damping and forcing covariance are formulated as 2 × 2 matrices for each wavenumber. As well as the transient subgrid tendency, the mean subgrid tendency is needed in the LES when the energy injection region is unresolved; this is also calculated from the statistics of the DNS. For comparison, a deterministic parameterization scheme consisting of 2×2 ‘damping’ parameters, which are calculated from the statistics of DNS, has been constructed. The main difference between atmospheric and oceanic flows, in this thesis, is that the atmospheric LES completely resolves the deformation scale, the energy and enstrophy injection region, and the truncation scale is spectrally distant from it, being well in the enstrophy cascade inertial range. In oceanic flows, however, the truncation scale is in the vicinity of the injection scale, at least for the parameters chosen, and is therefore not in an inertial range. A lower resolution oceanic LES at T15 is also examined, in which case the injection region is not resolved at all.¶ For atmospheric flows, it is found that, at T63, the matrix parameters are practically diagonal so that stratified atmospheric flows at these resolutions may be treated as uncoupled layers as far as subgrid-scale parameterizations are concerned. It is also found that the damping parameters are relatively independent of the (vertical) level, but the backscatter parameters are proportional to the subgrid flux in a given level. The stochastic and deterministic parameterization schemes give comparably good results relative to the DNS. For oceanic flows, it is found that the full matrix structure of the parameters must be used. Furthermore, it is found that there is a strong injection of barotropic energy from the subgrid scales, due to the unresolved, or partially resolved, baroclinic instability injection scales. It is found that the deterministic parameterization is too numerically unstable to be of use in the LES, and instead the stochastic parameterization must be used to obtain good agreement with the DNS. The subgrid tendency of the ensemble mean flow is also needed in some problems, and is found to reduce the available potential energy of the flow.

This thesis addresses a number of problems in computer vision, image processing, and geometry processing, and presents novel solutions to these problems. The overarching theme of the techniques presented here is a multi-scale approach, leveraging mathematical tools to represent images and surfaces at different scales, and methods that can be adapted from one type of domain (eg., the plane) to another (eg., the sphere). The main problem addressed in this thesis is known as stereo reconstruction: reconstructing the geometry of a scene or object from two or more images of that scene. We develop novel algorithms to do this, which work for both planar and spherical images. By developing a novel way to formulate the notion of disparity for spherical images, we are able effectively adapt our algorithms from planar to spherical images. Our stereo reconstruction algorithm is based on a novel application of distance transforms to multi-scale matching. We use matching information aggregated over multiple scales, and enforce consistency between these scales using distance transforms. We then show how multiple spherical disparity maps can be efficiently and robustly fused using visibility and other geometric constraints. We then show how the reconstructed point clouds can be used to synthesize a realistic sequence of novel views, images from points of view not captured in the input images, in real-time. Along the way to this result, we address some related problems. For example, multi-scale features can be detected in spherical images by convolving those images with a filterbank, generating an overcomplete spherical wavelet representation of the image from which the multiscale features can be extracted. Convolution of spherical images is much more efficient in the spherical harmonic domain than in the spatial domain. Thus, we develop a GPU implementation for fast spherical harmonic transforms and frequency domain convolutions of spherical images. This tool can also be used to detect multi-scale features on geometric surfaces. When we have a point cloud of a surface of a particular class of object, whether generated by stereo reconstruction or by some other modality, we can use statistics and machine learning to more robustly estimate the surface. If we have at our disposal a database of surfaces of a particular type of object, such as the human face, we can compute statistics over this database to constrain the possible shape a new surface of this type can take. We show how a statistical spherical wavelet shape prior can be used to efficiently and robustly reconstruct a face shape from noisy point cloud data, including stereo data.

La reconstruction 3D consiste à acquérir des images d’un objet, et à s’en servir pour en estimer un modèle 3D. Dans ce manuscrit, nous développons une méthode de reconstruction basée sur la modélisation par squelette. Cette méthode a l’avantage de renvoyer un modèle 3D qui est un objet virtuel complet (i.e. fermé) et aisément éditable, grâce à la structure du squelette. Enfin, l’objet acquis n’a pas besoin d’être texturé, et entre 3 et 5 images sont suffisantes pour la reconstruction. Dans une première partie, nous étudions les aspects 2D de l’étude. En effet, l’estimation d’un squelette 3D nécessite d’étudier la formation de la silhouette de l’objet à partir de son squelette, et donc les propriétés de sa projection perspective, appelée squelette perspectif. Cette étude est suivie par notre première contribution : un algorithme d’estimation de la projection perspective d’un squelette 3D curviligne, constitué d’un ensemble de courbes. Cet algorithme a toutefois tendance, comme beaucoup d’algorithmes estimant un squelette, à générer des branches peu informatives, notamment sur une image rastérisée. Notre seconde contribution est donc un algorithme d’estimation de squelette 2D, capable de prendre en compte la discrétisation du contour de la forme 2D, et d’éviter ces branches peu informatives. Cet algorithme, d’abord conçu pour estimer un squelette classique, est ensuite généralisé à l’estimation d’un squelette perspectif. Dans une seconde partie, nous estimons le squelette 3D d’un objet à partir de ses projections. Tout d’abord, nous supposons que le squelette de l’objet 3D à reconstruire est curviligne. Ainsi, chaque squelette perspectif estimé correspond à la projection du squelette 3D de l’objet, sous différents points de vue. La topologie du squelette étant affectée par la projection, nous proposons notre troisième contribution, l’estimation de la topologie du squelette 3D à partir de l’ensemble de ses projections. Une fois celle-ci estimée, la projection d’une branche 3D du squelette est identifiée sur chaque image, i.e. sur chacun des squelettes perspectifs. Avec cette identification, nous pouvons trianguler les branches du squelette 3D, ce qui constitue notre quatrième contribution : nous sommes donc en mesure d’estimer un squelette curviligne associé à un ensemble d’images d’un objet. Toutefois, les squelettes 3D ne sont pas tous constitués d’un ensemble de courbes : certains d’entre eux possèdent aussi des parties surfaciques. Notre dernière contribution, pour reconstruire des squelettes 3D surfaciques, est une nouvelle approche pour l’estimation d’un squelette 3D à partir d’images : son principe est de faire grandir le squelette 3D, sous les contraintes données par les images de l’objet. / The principle of 3D reconstruction is to acquire one or more images of an object, and to use it to estimate a 3D model of the object. In this manuscript, we develop a reconstruction method based on a particular model, the skeleton. The main advantages of our reconstruction approach are: we do reconstruct a whole, complete objet, and thanks to the skeleton structure, easily editable. Moreover, the method we propose allows us to free ourselves from constraints related to more classical reconstruction methods: the reconstructed object does not need to be textured, and between 3 and 5 images are sufficient to perform the reconstruction. In the first part, we focus on the 2D aspects of the work. Indeed, before estimating a 3D skeleton, we study the perspective silhouette of the object, and thus evaluate the properties of the perspective projection of the skeleton. Thus, our first contribution is an algorithm estimating the perspective projection of a curvilinear 3D skeleton, consisting of a set of curves. This algorithm, however, like most skeletonisation algorithms, tends to generate non-informative branches, in particular on a rasterized image. Our second contribution is thus an original 2D skeleton estimation algorithm, able to take into account the noise on the contour of the 2D shape, and to avoid uninformative skeleton branches. This algorithm, first designed to estimate a classical skeleton, is then generalized for computing a perspective skeleton. In a second part, we estimate the 3D skeleton of an object from its projections. First, we assume that the skeleton of the considered object is curvilinear. Thus, each estimated perspective skeleton corresponds to the projection of the 3D skeleton, from several viewpoints. The topology of the skeleton is however affected by the perspective projection, so we propose our third contribution: the estimation of the topology of the 3D skeleton based on its projections. Once this topology is estimated, for any 3D branch of the skeleton we indentify its projections on each image, that is a branch on each of the perspective skeletons. From this identification, we triangulate the branches of the 3D skeleton, which is our fourth contribution. Thus, we are able to estimate a curvilinear skeleton associated with a set of images of a 3D object. However, 3D skeletons are not necessarily made up of a set of curves, and some of them also have surface parts. Our last contribution is a new approach for the estimation of a general 3D skeleton (with surface parts) from images, which principle is to increase the 3D skeleton under the constraints given by the different images of the object.

Orientador: Prof. Dr. Vinicius Cifú Lopes / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016. / Neste trabalho apresentaremos uma visão sobre os princípios das curvas planas. Iniciamos o desenvolvimento dos estudos com as cônicas: parábola, elipse e hipérbole que são aplicadas no Ensino Médio normalmente usando equações cartesianas. Abordaremos o assunto destas e outras curvas usando equações paramétricas, com intuito de mostrar a vantagem de utilizá-las. Abrangeremos em nossos estudos a catenária, a cicloide e a curva de Bézier, curvas as quais não são estudadas no Ensino Básico, mas poderiam ser apresentadas como um desafio motivador ao estudo da Matemática, explorando suas várias aplicações que acontecem de maneira natural em nosso cotidiano. Apresentaremos propriedades gerais das curvas como: continuidade, parametrização, comprimento de arco, curva suave, curvatura e outras, além de realizar a demonstração do teorema fundamental das curvas planas e para finalizar estudaremos uma curva exótica, conhecida como curva de preenchimento de espaço, construída pela primeira vez pelo matemático italiano Giuseppe Peano. / In this work we will present an insight into the principles of flat curves. We start with the conics: parabola, ellipse and hyperbole which are applied in high school usually using Cartesian equations. We will discuss those and other curves using parametric equations, in order to show the advantage of using them. We will cover in our studies the catenary, the cycloid and a Bézier curve, curves which are not studied in basic education, but could be presented as a challenging motivation to the study of Mathematics by exploring their various uses that happen naturally in our everyday lives. We will introduce general properties of curves as: continuity, parameterization, arc length, smooth curve, curvature and others, in addition to the proof of the fundamental theorem of plane curves, and finally we will study an exotic curve, known as space-filling curve, built for the first time by the Italian mathematician Giuseppe Peano.

Made available in DSpace on 2014-08-20T14:25:48Z (GMT). No. of bitstreams: 1 dissertacao_ana_carolina_pereira.pdf: 8965039 bytes, checksum: 94bb2b07c5b8788f73621ddfa4a752de (MD5) Previous issue date: 2008-12-18 / In this work a study of the winds in Rio Grande do Sul, during the period 2000-2007 is presented to identify where and when the wind were stronger in the State. The wind data obtained at 10m height, at three time observation (00, 12 and 18UTC) of 15 meteorological stations, well distributed in the State are utilized. The identification of the stations where the winds were stronger and with less spectrum of variation was made using a Weibull distribution to the monthly average values of velocity, calculated on the total period of data from each station (between 4 and 8 years) and for time. The similarity of velocities behavior between the stations with 8 years of data was obtained by applying the method K-means, non-hierarchical clustering technique, setting up 4 groups for homogeneous regions of pentad average velocities in the state, by hour of observation. Sensibility tests with two Planetary Boundary Layer (PBL) parameterizations (Blackadar and MRF) of MM5 model were realized by using two nested grids with 35 sigma levels in vertical, centered in Rio Grande do Sul and with initial and contour data from NCEP (1º resolution in latitude and longitude). The validation of simulated versus observed velocities at 10m height was realized with the calculation of bias error, the mean square root of error and standard deviation of errors. The cities of São Luiz Gonzaga and Santa Vitória do Palmar stood out from the others because they had more often high values of k and c parameters. The homogeneous group, characterized by having the highest values of pentadal avegared velocities was constitued predominantly by the stations of Santa Vitória do Palmar (in the extreme south of the state) and Passo Fundo (located in the northern region). The highest velocities were observed in the month of November, in Santa Vitória do Palmar. The selected case study was the extratropical cyclone that struck the southern Rio Grande do Sul in the period from 9 to November 13, 2002. The errors between observed and simulated velocities in both experiments were relatively large. The comparison between results and observed data in some stations showed that Blackadar PBL parameterization was better than MRF parameterization. / Neste trabalho é mostrado o estudo sobre os ventos no Rio Grande do Sul, durante o período 2000-2007, para identificar onde e quando os ventos foram mais intensos no Estado. Foram utilizados dados de velocidades obtidos a 10m de altura, nos três horários de observação (00, 12 e 18UTC) de 15 estações meteorológicas, bem distribuídas no Estado. A identificação das estações onde os ventos foram mais fortes e com menor espectro de variação foi feita aplicando a distribuição Weibull aos valores médios mensais de velocidades, calculadas sobre o período total de dados de cada estação (entre 4 e 8 anos) e por horário. A similaridade de comportamento das velocidades entre as estações com 8 anos de dados, foi obtida pela aplicação do método K-means, definindo-se 4 grupos para regiões homogêneas de velocidades médias pentadais no Estado, por horário de observação.Testes de sensibilidade com duas parametrizações (Blackadar e MRF) de Camada Limite Planetária (CLP) do modelo MM5 foram realizados usando duas grades aninhadas com 35 níveis sigma na vertical, centradas no Rio Grande do Sul e com dados iniciais e de contorno do NCEP (resolução de 1º em latitude e longitude). A validação das velocidades simuladas versus observadas, a 10m de altura, foi realizada com o cálculo do erro de viés, da raiz do erro quadrático médio e do desvio padrão dos erros. As cidades de São Luiz Gonzaga e de Santa Vitória do Palmar se destacaram das demais por apresentarem mais frequentemente, os valores mais elevados dos parâmetros k e c, respectivamente. O grupo homogêneo, caracterizado por apresentar os maiores valores de velocidades médias pentadais foi constituído predominantemente pelas estações de Santa Vitória do Palmar (no extremo sul do Estado) e Passo Fundo (localizado na região norte). As velocidades mais elevadas foram observadas no mês de novembro, em Santa Vitória do Palmar. O caso de estudo selecionado foi o ciclone extratropical que atingiu o sul do Rio Grande do Sul no período de 9 a 13 de novembro de 2002. Os erros encontrados entre as velocidades observadas e simuladas nos dois experimentos foram relativamente grandes. A comparação entre os resultados e os dados observados em algumas estações mostrou que a parametrização de camada limite planetária Blackadar foi melhor que a parametrização MRF.

This thesis addresses a number of problems in computer vision, image processing, and geometry processing, and presents novel solutions to these problems. The overarching theme of the techniques presented here is a multi-scale approach, leveraging mathematical tools to represent images and surfaces at different scales, and methods that can be adapted from one type of domain (eg., the plane) to another (eg., the sphere). The main problem addressed in this thesis is known as stereo reconstruction: reconstructing the geometry of a scene or object from two or more images of that scene. We develop novel algorithms to do this, which work for both planar and spherical images. By developing a novel way to formulate the notion of disparity for spherical images, we are able effectively adapt our algorithms from planar to spherical images. Our stereo reconstruction algorithm is based on a novel application of distance transforms to multi-scale matching. We use matching information aggregated over multiple scales, and enforce consistency between these scales using distance transforms. We then show how multiple spherical disparity maps can be efficiently and robustly fused using visibility and other geometric constraints. We then show how the reconstructed point clouds can be used to synthesize a realistic sequence of novel views, images from points of view not captured in the input images, in real-time. Along the way to this result, we address some related problems. For example, multi-scale features can be detected in spherical images by convolving those images with a filterbank, generating an overcomplete spherical wavelet representation of the image from which the multiscale features can be extracted. Convolution of spherical images is much more efficient in the spherical harmonic domain than in the spatial domain. Thus, we develop a GPU implementation for fast spherical harmonic transforms and frequency domain convolutions of spherical images. This tool can also be used to detect multi-scale features on geometric surfaces. When we have a point cloud of a surface of a particular class of object, whether generated by stereo reconstruction or by some other modality, we can use statistics and machine learning to more robustly estimate the surface. If we have at our disposal a database of surfaces of a particular type of object, such as the human face, we can compute statistics over this database to constrain the possible shape a new surface of this type can take. We show how a statistical spherical wavelet shape prior can be used to efficiently and robustly reconstruct a face shape from noisy point cloud data, including stereo data.

Tato práce se zabývá Houghovou transformací (HT). HT je nejčastěji používána pro detekci přímek nebo křivek, ale byla zobecněna i pro detekci libovolných tvarů. Hlavní téma této práce jsou parametrizace přímek, speciálně PTLM - zobrazení bodů na přímky. Tyto parametrizace mají tu vlastnost, že bodům v obrázku odpovídají přímky v parametrickém prostoru. Tato práce poskytuje důkazy některých vlastností PTLM. Za zmínku stojí existence páru PTLM vhodného pro detekci a efekt konvoluce v obrázku na obsah parametrického prostoru. V práci jsou prezentovány dvě implementace HT. Obě využívají k akceleraci grafický hardware. Jedna využívá GPGPU API CUDA a druhá zobrazovací API OpenGL. Jako aplikace detekce přímek je uvedena část detekce šachovnicových markerů použitelných pro rozšířenou realitu.

Screen level temperature is measured at 2 meters above the ground. It is one of the most used atmospheric characteristics in various applications in meteorology and other fields related to weather prediction. Essential is not only the knowledge of its current state, but also its prediction. It is forecasted by numerical weather prediction (NWP) models from the atmospheric current state. Its long-term characteristics can be obtained from the integration of climate models. This text discusses fundamental parametriza- tions, mostly related to temperature forecast, used in the NWP model ALADIN and the regional climate model RegCM. Physical processes which influence temperature are studied using ALADIN in several cases which include the presence of low cloudiness, gravity waves and inappropriate thermic coefficient. A detailed description of the most relevant parametrization schemes is given and the results are studied in a form of indi- vidual feedback loops. Most dominant processes are also found. However, the level of 2 meters above the ground is not the model level, so temperature at 2 meters is obtained by interpolation from the surface temperature and the lowest model level temperature. Using RegCM, two differently complex interpolation schemes are compared to each other. 1

Le mélange vertical dans l’océan contribue au maintien de la Cellule Méridienne de Circulation Océanique (MOC) en permettant le renouvellement des eaux profondes. Une coupe transverse d’une partie de la MOC est réalisée par la radiale hydrologique OVIDE qui a lieu tous les 2 ans depuis 2002 entre le Portugal et le Groenland. L’énergie nécessaire au mélange est fournie par les ondes internes générées par le vent et la marée, et des mesures de micro-structure (VMP) en 2008 montrent des valeurs de dissipation Evmp intensifiées dans la thermocline principale et au niveau des topographies. Notre étude se base sur ces observations pour étudier la fine-structure verticale de l’océan et estimer indirectement la dissipation E due aux ondes internes à l’aide de mesures CTD et LADCP. La comparaison au mesures du VMP permet d’optimiser la paramétrisation de E en encadrant les observations par facteur 3 et leurs valeurs moyennes à ±30%. L’application sur l’ensemble des données OVIDE permet d’obtenir une cartographie du mélange a travers le bassin. La distribution géographique de la diffusion verticale K est similaire le long des 5 sections, avec des valeurs de l’ordre de 10−4m2/s dans la thermocline principale, au fond et au niveau des topographies, et de l’ordre de 10−5m2/s dans l’océan intérieur. Des différences régionales sont présentes et K peut être localement proche de 10−3m2/s. L’étude de la section FOUREX 1997 révèle une intensification de K le long de la dorsale médio-Atlantique où les valeurs moyennes sont de 2 à 3 plus fortes que le long des sections OVIDE. La distribution spatiale des échelles de Thorpe LT est corrélée avec celle du mélange. Néanmoins la dissi-pation basée sur LT surestime Evmp d’un facteur 10 à100, ce qui pourrait être dû à une mauvaise représentation de la durée de vie de la turbulence dans l’océan. Certains mécanismes susceptibles de générer des ondes internes sont proposés. Des sites possibles de génération par la marée sont localisés à l’aide d’un modèle simple de la trajectoire des rayons d’ondes. Une corrélation possible entre les mouvements géostrophiques et les ondes internes est envisagée dans la thermocline principale. Enfin l’étude des angles de Turner montre que des instabilités de double-diffusion peuvent être présentes sur une grande partie de la section. / Vertical mixing in the ocean contributes to sustain the Meridionnal Overturning. Circulation (MOC) by allowing the renewal of deep waters. A section across the MOC is performed by the hydrological radial OVIDE repeated every two years between Portugal and Greenland since 2002. The energy required for mixing is provided by internal waves generated by wind and tides and micro-structure measurements(VMP) in 2008 show intensified values of dissipation Evmp in the main thermocline and near topographies. Our study is based on these observations and aims tostudy the vertical fine-scale structure of the ocean. Estimates of the dissipation E due to internal waves are made with CTD and LADCP measurements. The comparison with VMP measurements allow us to optimize the parameterization of E by framing the observations by factor 3 and their mean values at ±30%. The systematic application to the OVIDE dataset provides a mapping of the mixing across the basin. Geographical distribution of the vertical diffusion K is similar along the five sections, with values near10−4m2/s in the main thermocline and at the bottom of topographies, and near 10−5m2/s in the ocean interior. Regional differences are present and K can belocally close to 10−3m2/s. Application to FOUREX1997 datas et reveals an increase of K along the Mid-Atlantic Ridge, where the average values are 2 to 3stronger than along OVIDE sections. The spatial distribution of Thorpe scales LT appears to be correlated with internal waves mixing patterns. Nevertheless dissipation estimates based on LT overestimates Evmp by a 10 to 100 factor, which maybe due to misrepresentation of the stage of turbulence development in the ocean. Some mechanisms that can generate internal waves are proposed. Probable sites where tidal generation could occur are located using a simple model of wave beam trajectory. A possible correlation between geostrophic flows and internal waves is considered in the main thermocline. Finally the study of Turnerangles shows that double-diffusion instabilities may bepresent over a large part of the section.