Path tracing is a basic, statistically unbiased method for calculating the global illumination in 3D scenes. For practical purposes, the algorithm is too slow, so it is used mainly for theoretical purposes or as a base for more advanced algorithms. This thesis explores the possibility of improving this algorithm by augmenting the sampling part, which computes outgoing directions during ray traversal through the scene. This optimization is accomplished by creating a special data structure in a preprocess step, which describes approximate light distribution in the scene and which then aids the sampling process. The presented algorithm is implemented in the PBRT library.

Neste trabalho estudamos diferentes noções de diferenciabilidade para funções definidas na esfera unitária S^n-1 de R^n, n>=2. Em relação à derivada usual, encontramos condições necessárias e/ou suficientes para que uma função seja diferenciável até uma ordem fixada. Para as outras duas, a derivada forte de Laplace-Beltrami e a derivada fraca, apresentamos algumas propriedades básicas e procuramos condições que garantam a equivalência destas com a diferenciabilidade usual. / In this work we study different notions of differentiability for functions defined on the unit sphere S^n-1 of R^n, n>=2. With respect to the usual derivative, we find necessary and/or sufficient conditions in order that a function be differentiable up to a fixed order. As for the other two, the strong Laplace-Beltrami derivative and the weak derivative, we present some basic properties about them and search for conditions that guarantee the equivalence of them with the previous one.

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.

Ce travail de thèse à été financé par le projet LICORVE (Développement de garnitures légères, innovantes, recyclables et poly-sensorielles pour les applications de coffres de véhicule). Il consiste à développer une méthodologie de mesure pour localiser et caractériser les sources de bruit dans un coffre de voiture. L'environnement de mesure se caractérise par un petit volume où les réflexions de la source d'intérêt et des sources perturbatrices sur les parois ne peuvent être négligées. La méthode doit donc permettre de séparer les différents contributions pour estimer le plus précisément possible les sources étudiées (déconfinement). Dans un premier temps, deux méthodes de séparation : Double Layer SONAH (Statistically Optimal Near el Acoustical Holography) et Field Separation Method (FSM) sont étudiées numériquement. Les limites et avantages de chacune ont été déterminés dans un environnement de mesure confiné. Cela a permis de choisir la méthode la plus adaptée à notre problématique. Dans un deuxième temps les principales conclusions de l'étude numérique sont validées expérimentalement. Dans ce cadre, un ensemble de mesures sont réalisées dans une maquette avec la méthode FSM pour localiser et caractériser des sources maitrisées. Après avoir validée la fiabilité de la méthode de séparation, FSM a été mise en œuvre dans le coffre d'une Peugeot 508 sw en condition de roulement. Les résultats obtenus ont permis d'orienter le choix des garnitures pour un traitement acoustique optimal. / This thesis consists in developing, through the LICORVE project (light garnitures, innovative, recyclable and multi-sensorial for vehicle boots applications), a measurement method for localizing and characterizing noise sources in a vehicle trunk. The measuring environment is distinguished by a small volume where the reflections on the partitions generated by the source of interest and the interfering sources cannot be neglected. Therefore, the method must allow the separation of the different contributions in order to assess accurately the studied sources. As a first step, two separation methods : Double Layer SONAH (Statistically Optimized Near-Field Acoustical Holography) and Field Separation Method (FSM) are numerically studied. The limitations and advantages of each of them are determined in a confined measuring environment; this allowed to select the most appropriate method to tackle our problem. As a second step, the main conclusions of the numerical study are confirmed experimentally. In this context, measurements are performed, using the FSM method, in a trunk mock-up to localize and characterize the controlled sources. So confirmed the reliability of the separation method, it has been tested in the boot of a Peugeot 508 SW on a roller bench. The obtained results allowed guiding the selection of garniture for the acoustic treatment.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In the last two decades, multiple input multiple output (MIMO) wireless systems have been subject of intense research due to the theoretical promise of the proportional increase of the communications channel capacity as the number of antennas increases. This outstanding property supposes an efficient use of spatial diversity at both the transmitter and receiver. An important and not well explored path towards improving MIMO system performance using spatial diversity takes into account the interactions among the antennas and the (physical) propagation medium. By understanding these interactions, the transmit and receive antenna arrays can be designed to best âmatchâ the propagation medium so that the link quality and capacity can be further improved in a MIMO system. In this work, we consider the use of spherical harmonics and tensor decompositions in the problem of MIMO channel modeling and estimation. The use of spherical harmonics allows to represent the radiation patterns of antennas in terms of coefficients of an expansion, thus decoupling the transmit and receive antenna array responses from the physical propagation medium. By translating simple propagation-motivated channel models with polarization information into the spherical harmonics domain, we study how propagation parameters themselves and antenna configurations affect MIMO performance in terms of capacity and correlation. A second part of this work addresses the problem of estimating directional MIMO channels in the spherical harmonics domain using tensor decompositions. Considering both single-scattering and double-scattering propagation scenarios, we make use of the parallel factor (PARAFAC) and PARATUCK-2 decompositions, respectively, to estimate the propagating spherical modes, from which the directions of arrival (DoA) and directions of departure (DoD) can be extracted. Finally, we propose and compare two methods for optimizing the coefficients of the spherical harmonics expansion of an antenna array for a prespecified MIMO channel response. / Nas Ãltimas dÃcadas, sistemas de comunicaÃÃo sem fio de mÃltiplas antenas (MIMO - Multiple Input Multiple Output) tÃm sido objetos de intensas pesquisas devido à promessa teÃrica do aumento proporcional da capacidade com o aumento do nÃmero de antenas. Esta propriedade excepcional supÃe um uso eficiente da diversidade espacial no transmissor e receptor. Um caminho importante e nÃo bem explorado no sentido de melhorar o desempenho de sistemas MIMO usando diversidade espacial leva em conta a interaÃÃo entre as antenas e meio de propagaÃÃo (fÃsico). AtravÃs da compreensÃo dessas interaÃÃes, arranjos de antenas de recepÃÃo e transmissÃo podem ser projetados para melhor "casar" com o meio de propagaÃÃo, tal que a qualidade do link de comunicaÃÃo e capacidade possam ser melhoradas em um sistema MIMO. Neste trabalho, consideramos o uso de harmÃnicos esfÃricos e decomposiÃÃes tensoriais no problema de modelagem de canal MIMO e estimaÃÃo. O uso de harmÃnicos esfÃricos permite representar os padrÃes de radiaÃÃo de antenas em termos de coeficientes de uma expansÃo, assim desacoplando as respostas dos arranjos de antenas (transmissoras e receptoras) do meio de propagaÃÃo fÃsica. Traduzindo modelos simples de canais baseados em propagaÃÃo, com informaÃÃes de polarizaÃÃo, para o domÃnio dos harmÃnicos esfÃricos, estudamos como os parÃmetros de propagaÃÃo si e configuraÃÃes especÃficas de antenas afetam o desempenho do sistema MIMO em termos de capacidade e de correlaÃÃo. A segunda parte deste trabalho aborda o problema de estimar canais direcionais MIMO no domÃnio dos harmÃnicos esfÃricos usando decomposiÃÃes por tensores. Considerando tanto cenos de espalhamento simples e de duplo espalhamento, fazemos uso das decomposiÃÃes PARAFAC e PARATUCK2, respectivamente, para estimar os modos esfÃricos propagantes, a partir das quais as direÃÃes de chegada (DoA) e as direÃÃes de saÃda (DoD) podem ser extraÃdas. Finalmente, propomos e comparamos dois mÃtodos de otimizaÃÃo dos coeficientes da expansÃo em harmÃnicos esfÃricos de arranjos de antenas para respostas de canais MIMO prÃ-especificados .

Neste trabalho estudamos diferentes noções de diferenciabilidade para funções definidas na esfera unitária S^n-1 de R^n, n>=2. Em relação à derivada usual, encontramos condições necessárias e/ou suficientes para que uma função seja diferenciável até uma ordem fixada. Para as outras duas, a derivada forte de Laplace-Beltrami e a derivada fraca, apresentamos algumas propriedades básicas e procuramos condições que garantam a equivalência destas com a diferenciabilidade usual. / In this work we study different notions of differentiability for functions defined on the unit sphere S^n-1 of R^n, n>=2. With respect to the usual derivative, we find necessary and/or sufficient conditions in order that a function be differentiable up to a fixed order. As for the other two, the strong Laplace-Beltrami derivative and the weak derivative, we present some basic properties about them and search for conditions that guarantee the equivalence of them with the previous one.

The aim for this thesis is to develop a foundation for a compression system for animated mesh sequences, specifically under dynamic change of mesh geometry and topology. Compression of mesh sequences is of special interest in the game industry and this particular thesis is a part of an ongoing series of projects at EA DICE. One of the primary challenges when creating a mesh compression system is creating a matching bijective subset of the mesh surfaces between two subsequent frames in the animation to guide remeshing of the sequence. This thesis describes a method for producing a bijective set of matching mesh patches between two meshes along with an error metric that captures the quality of the matching in terms of shape similarity and distortion. Theory of mathematical topology and tensor algebra used in methods for high performance scientific digital 3D-image recognition are here adopted to extract similar local features between meshes. Techniques for creating parametrizations of mesh patches are combined with techniques for matching point clouds and deforming mesh geometry under energy minimization in order to produce a matching set of patches. The presented algorithm successfully creates bijective sets of matched patches for subsequent meshes in a sequence as well as measures the error for the matchings. Results show an average matching set size of approximately 25% of the mesh areas over a sequence of meshes. This suggests that the data size of such a sequence could potentially be reduced by 25%.

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.

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) method must be employed. In this work an h-p Finite Element Method (FEM) is used to obtain the dominant Lambda mode associated with a configuration of a reactor core using the SPN approximation. The performance of the SPN (N= 1, 3, 5) approximations has been tested for different reactor benchmarks. / [ES] La ecuación de la difusión neutrónica es una aproximación de la ecuación del transporte de neutrones que describe la población de neutrones en el núcleo de un reactor nuclear. En particular, consideraremos reactores de tipo VVER y para simular su comportamiento se utilizará la ecuación de la difusión neutrónica para cuya discretización se hace uso de mallas hexagonales. La mayoría de los códigos de simulación de reactores nucleares utilizan aproximación multigrupo de energía de la ecuación de la difusión neutrónica para describir la distribución de neutrones en el interior del núcleo del reactor. Para estudiar el estado estacionario del reactor, es posible forzar la criticidad del reactor de forma artificial modificando las secciones eficaces de forma que se obtiene un problema de valores propios diferencial, conocido como el problema de los Modos Lambda, que se resuelve para obtener los valores propios dominantes del reactor y sus correspondientes funciones propias. Para discretizar este modelo se ha hecho uso de un método de elementos finitos con adaptabilidad h-p. Este método permite el uso de mallas heterogéneas, y de diferentes refinamientos como el uso mallas h-adaptativas, reduciendo el tamaño de los distintos nodos, y el p-refinado, aumentando el grado del polinomio de las funciones básicas utilizado en los desarrollos de la solución en los diferentes nodos. Se ha desarrollado un código basado en un método de elementos finitos de alto orden para resolver el problema de los Modos Lambda en un reactor con geometría hexagonal y se han obtenido los Modos dominantes para distintos problemas de referencia. Una vez que se ha obtenido la solución para la distribución de neutrones en estado estacionario, ésta se utiliza como condición inicial para la integración de la ecuación de difusión neutrónica dependiente del tiempo. Para simular el comportamiento de un reactor nuclear para un determinado transitorio, es necesario ser capaz de integrar la ecuación de la difusión neutrónica dependiente del tiempo en el interior del núcleo del reactor. La discretización espacial de esta ecuación se hace usando un método de elementos finitos de alto orden que permite refinados de tipo h-p para distintas geometrías. Los transitorios que implican el movimiento de los bancos de las barras de control tienen el problema conocido como el efecto 'rod-cusping'. Estudios anteriores, por lo general, han abordado este problema utilizando una malla fija y definiendo propiedades promedio para los materiales correspondientes a las celdas donde se tiene la barra de control parcialmente insertada. En el presente trabajo se propone el uso de un esquema de malla móvil, de forma que en mallado espacial va cambiando con el movimiento de la barra de control, evitando la necesidad de utilizar secciones eficaces equivalentes para las celdas parcialmente insertadas. El funcionamiento de este esquema de malla móvil propuesto se estudia resolviendo distintos problemas tipo. La precisión obtenida mediante de la teoría de la difusión en los cálculos de reactores es limitada cuando se tienen elementos de combustible complejos o se pretenden realizar cálculos en malla fina. Para mejorar estos resultados, es necesario disponer de un método que incorpore aproximaciones de orden superior de la ecuación del transporte de neutrones. Una posibilidad es hacer uso de las ecuaciones PN simplificadas (SPN ). En este trabajo se utiliza un método de elementos finitos h-p para obtener los modos dominantes asociados con una configuración dada del núcleo de un reactor nuclear con geometría hexagonal usando la aproximación SPN . El funcionamiento de las aproximaciones SPN (N = 1, 3, 5) se ha estudiado para distintos problemas de referencia. / [CAT] L'equació de la difusió neutrònica és una aproximació de l'equació del transport de neutrons que descriu la població de neutrons en el nucli de un reactor nuclear. En particular, considerarem reactors de tipus VVER i per a simular el seu comportament s'utilitzarà l'equació de la difusió neutrónica que es discretitza fent ús de malles hexagonals. La majoria dels codis de simulació de reactors nuclears utilitzen l'aproximació multigrup d'energia de l'equació de la difusió neutrónica per a descriure la distribució de neutrons a l'interior del nucli del reactor. Per a estudiar l'estat estacionari del reactor, és possible forçar la seua criticitat de forma artificial modificant les seccions eficaces de manera que s'obté un problema de valors propis diferencial, conegut com el problema dels Modes Lambda, que es resol per a obtenir els valors propis dominants del reactor i les seues corresponents funcions pròpies. Per a discretitzar aquest model s'ha fet ús d'un mètode d'elements finits amb adaptabilitat h-p. Aquest mètode permet l'ús de malles heterogènies, i de diferents refinaments com l'ús malles h-adaptatives, reduint la grandària dels diferents nodes, i el p-refinat, augmentant el grau del polinomi de les funcions bàsiques utilitzat en els desenvolupaments de la solució en els diferents nodes. S'ha desenvolupat un codi basat en un mètode d'elements finits d'alt ordre per a resoldre el problema dels Modes Lambda en un reactor amb geometria hexagonal i s'han obtingut els Modes dominants per a diferents problemes de referència. Una vegada que s'ha obtingut la solució per a la distribució de neutrons en estat estacionari, aquesta s'utilitza com a condició inicial per a la integració de l'equació de difusió neutrònica depenent del temps. Per a simular el comportament d'un reactor nuclear per a un determinat transitori, és necessari ser capaç d'integrar l'equació de la difusió neutrónica depenent del temps a l'interior del nucli del reactor. La discretitzación espacial d'aquesta equació es fa usant un mètode d'elements finits d'alt ordre que permet refinats de tipus h-p per a diferents geometries. Els transitoris que impliquen el moviment dels bancs de les barres de control tenen el problema conegut com l'efecte 'rod-cusping'. Estudis anteriors, en general, han abordat aquest problema utilitzant una malla fixa i definint propietats equivalents per als materials corresponents a les cel·les on es té la barra de control parcialment inserida. En el present treball es proposa l'ús d'un esquema de malla mòbil, de manera que en mallat espacial va canviant amb el moviment de la barra de control, evitant la necessitat d'utilitzar seccions eficaces equivalents per a les cel·les parcialment inserides. El funcionament de aquest esquema de malla mòbil s'estudia resolent diferents problemes tipus. La precisió obtinguda mitjançant de la teoria de la difusió en els càlculs de reactors és limitada quan es tenen elements de combustible complexos o es pretenen realitzar càlculs en malla fina. Per a millorar aquests resultats, és necessari disposar d'un mètode que incorpore aproximacions d'ordre superior de l'equació del transport de neutrons. Una possibilitat és fer ús de les equacions PN simplificades (SPN ). En aquest treball s'utilitza un mètode d'elements finits h- p per a obtenir els modes dominants associats amb una configuració donada del nucli de un reactor amb geometria hexagonal usant l'aproximació SPN . El funcionament de les aproximacions SPN (N = 1, 3, 5) s'ha estudiat per a diferents problemes de referència. / Fayez Moustafa Moawad, R. (2016). Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/65353 / TESIS

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application.