Computing correspondences or maps between shapes is one of the oldest problems in Computer Graphics and Geometry Processing with a wide range of applications from deformation transfer, statistical shape analysis, to co-segmentation and exploration among a myriad others. A good map is supposed to be continuous, as-bijective-as-possible, accurate if there are ground-truth corresponding landmarks given, and lowdistortionw.r.t. different measures, for example as-conformal-as-possible to preserve the angles. This thesis contributes to the area of non-rigid shape matching and map space exploration in Geometry Processing. Specifically, we consider the discrete setting, where the shapes are discretized as amesh structure consisting of vertices, edges, and polygonal faces. In the simplest case, we only consider the graph structure with vertices and edges only. In this thesis, we design algorithms to compute soft correspondences between discrete shapes. Specifically, (1)we propose different regularizers, including orientation-preserving operator and the Resolvent Laplacian Commutativity operator, to promote the shape correspondences in the functional map framework. (2) We propose two refinement methods, namely BCICP and ZoomOut, to improve the accuracy, continuity, bijectivity and the coverage of given point-wisemaps. (3)We propose a tree structure and an enumeration algorithm to explore the map space between a pair of shapes that can update multiple high-quality dense correspondences.

Em simulações físicas baseadas em partículas, a informação sobre quais partículas pertencem à fronteira do sistema e quais são consideradas internas é, em geral, uma informação útil porém difícil de ser obtida eficientemente. Esta informação pode ser usada na geração da superfície livre de um fluido ou no cálculo da tensão superficial o mesmo, entre outras aplicações. Técnicas encontradas na literatura podem apresentar resultados satisfatórios, mas em geral são sensíveis à escala do problema, distribuição das partículas e envolvem operações computacionalmente caras como inversão de matrizes. O objetivo deste trabalho é estudar os métodos existentes e apresentar uma alternativa com custo computacional mais baixo e que seja capaz de lidar com problemas de diferentes escalas e naturezas de forma mais simples que os métodos existentes. / In particle-based physics simulations, the information about which particles belong to the boundary of the system and which are considered internal is, in general, an information that is useful but hard to obtain in an efficient way. This information can be applied to generate the free surface of the fluid or to compute the surface tension, among other applications. Techniques found in the literature may present satisfactory results, but in general they are sensible to the problem scale, particle distribution and involve computationally expensive operations such as matrix inversion. The goal of this study is to analyze the existing methods and present an alternative with lower computational cost and which is capable of handling problems with different scales and nature in a simpler way than the existing methods.

Em simulações físicas baseadas em partículas, a informação sobre quais partículas pertencem à fronteira do sistema e quais são consideradas internas é, em geral, uma informação útil porém difícil de ser obtida eficientemente. Esta informação pode ser usada na geração da superfície livre de um fluido ou no cálculo da tensão superficial o mesmo, entre outras aplicações. Técnicas encontradas na literatura podem apresentar resultados satisfatórios, mas em geral são sensíveis à escala do problema, distribuição das partículas e envolvem operações computacionalmente caras como inversão de matrizes. O objetivo deste trabalho é estudar os métodos existentes e apresentar uma alternativa com custo computacional mais baixo e que seja capaz de lidar com problemas de diferentes escalas e naturezas de forma mais simples que os métodos existentes. / In particle-based physics simulations, the information about which particles belong to the boundary of the system and which are considered internal is, in general, an information that is useful but hard to obtain in an efficient way. This information can be applied to generate the free surface of the fluid or to compute the surface tension, among other applications. Techniques found in the literature may present satisfactory results, but in general they are sensible to the problem scale, particle distribution and involve computationally expensive operations such as matrix inversion. The goal of this study is to analyze the existing methods and present an alternative with lower computational cost and which is capable of handling problems with different scales and nature in a simpler way than the existing methods.

La création et la compréhension des déformations de surfaces sont des thèmes récurrent pour le traitement de géométrie 3D. Comme les surfaces lisses peuvent être représentées de multiples façon allant du nuage ​​de points aux maillages polygonales, un enjeu important est de pouvoir comparer ou déformer des formes discrètes indépendamment de leur représentation. Une réponse possible est de choisir une représentation flexible des surfaces déformables qui peut facilement être transportées d'une structure de données à une autre.Dans ce but, les "functional map" proposent de représenter des applications entre les surfaces et, par extension, des déformations comme des opérateurs agissant sur des fonctions. Cette approche a été introduite récemment pour le traitement de modèle 3D, mais a été largement utilisé dans d'autres domaines tels que la géométrie différentielle, la théorie des opérateurs et les systèmes dynamiques, pour n'en citer que quelques-uns. Le principal avantage de ce point de vue est de détourner les problèmes encore non-résolus, tels que la correspondance forme et le transfert de déformations, vers l'analyse fonctionnelle dont l'étude et la discrétisation sont souvent mieux connues. Cette thèse approfondit l'analyse et fournit de nouvelles applications à ce cadre d'étude. Deux questions principales sont discutées.Premièrement, étant donné deux surfaces, nous analysons les déformations sous-jacentes. Une façon de procéder est de trouver des correspondances qui minimisent la distorsion globale. Pour compléter l'analyse, nous identifions les parties les moins fiables du difféomorphisme grâce une méthode d'apprentissage. Une fois repérés, les défauts peuvent être éliminés de façon différentiable à l'aide d'une représentation adéquate des champs de vecteurs tangents.Le deuxième développement concerne le problème inverse : étant donné une déformation représentée comme un opérateur, comment déformer une surface en conséquence ? Dans une première approche, nous analysons un encodage de la structure intrinsèque et extrinsèque d'une forme en tant qu'opérateur fonctionnel. Dans ce cadre, l'objet déformé peut être obtenu, à rotations et translations près, en résolvant une série de problèmes d'optimisation convexe. Deuxièmement, nous considérons une version linéarisée de la méthode précédente qui nous permet d'appréhender les champs de déformation comme agissant sur la métrique induite. En conséquence la résolution de problèmes difficiles, tel que le transfert de déformation, sont effectués à l'aide de simple systèmes linéaires d'équations. / Creating and understanding deformations of surfaces is a recurring theme in geometry processing. As smooth surfaces can be represented in many ways from point clouds to triangle meshes, one of the challenges is being able to compare or deform consistently discrete shapes independently of their representation. A possible answer is choosing a flexible representation of deformable surfaces that can easily be transported from one structure to another.Toward this goal, the functional map framework proposes to represent maps between surfaces and, to further extents, deformation of surfaces as operators acting on functions. This approach has been recently introduced in geometry processing but has been extensively used in other fields such as differential geometry, operator theory and dynamical systems, to name just a few. The major advantage of such point of view is to deflect challenging problems, such as shape matching and deformation transfer, toward functional analysis whose discretization has been well studied in various cases. This thesis investigates further analysis and novel applications in this framework. Two aspects of the functional representation framework are discussed.First, given two surfaces, we analyze the underlying deformation. One way to do so is by finding correspondences that minimize the global distortion. To complete the analysis we identify the least and most reliable parts of the mapping by a learning procedure. Once spotted, the flaws in the map can be repaired in a smooth way using a consistent representation of tangent vector fields.The second development concerns the reverse problem: given a deformation represented as an operator how to deform a surface accordingly? In a first approach, we analyse a coordinate-free encoding of the intrinsic and extrinsic structure of a surface as functional operator. In this framework a deformed shape can be recovered up to rigid motion by solving a set of convex optimization problems. Second, we consider a linearized version of the previous method enabling us to understand deformation fields as acting on the underlying metric. This allows us to solve challenging problems such as deformation transfer are solved using simple linear systems of equations.

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[pt] A aquisição de malhas triangulares normalmente introduz ruídos indesejados. A remoção de ruído de malhas é uma tarefa da área de processamento geométrico que serve para remover esse tipo de distorção. Para preservar a fidelidade em relação à malha desejada, um algoritmo de remoção de ruído de malha deve preservar detalhes enquanto remove altas frequências indesejadas sobre a superfície. Vários algoritmos foram propostos para resolver este problema usando um esquema de filtragem bilateral. Neste trabalho, propomos um algoritmo de dois passos que usa vizinhança adaptativa e filtragem bilateral para remover ruído do campo normal e, em seguida, atualizar as posições dos vértices ajustando os triângulos às novas normais. A nossa contribuição principal é a computação da vizinhança adaptativa. Essa computação é formulada como problemas locais de otimização quadrática que podem ser controlados para obter o comportamento desejado da vizinhança. A proposta é comparada visual e quantitativamente com vários algoritmos propostos na literatura, usando dados sintéticos e reais. / [en] The acquisition of triangular meshes typically introduces undesired noise. Mesh denoising is a geometry processing task to remove this kind of distortion. To preserve the geometric fidelity of the desired mesh, a mesh denoising algorithm must preserve true details while removing artificial high-frequencies from the surface. Several algorithms were proposed to address this problem using a bilateral filtering scheme. In this work, we propose a two-step algorithm which uses adaptive patches and bilateral filtering to denoise the normal field, and then update vertex positions fitting the faces to the denoised normals. The computation of the adaptive patches is our main contribution. We formulate this computation as local quadratic optimization problems that we can control to obtain a desired behavior of the patch. We compared our proposal with several algorithms proposed in the literature using synthetic and real data.

La simulation numérique des écoulements est une étape essentielle de la conception des turbines à gaz équipant les hélicoptères. La recherche permanente de la performance a conduit à des géométries de turbines très complexes et il devient de plus en plus difficile de modéliser des grilles de simulation qui épousent parfaitement la CAO des moteurs. La technique chimère permet de s’affranchir des contraintes de recollement parfait des différentes grilles en autorisant leur chevauchement. Cependant elle soulève de nouveaux problèmes lors de la phase de post-traitement, lorsqu’il s’agit d’exploiter les résultats de simulation afin de faire de nouveaux calculs ou de les visualiser, parce que les outils usuels ne sont pas adaptés à ces configurations particulières. Dans le cadre des deux premiers projets du programme MOSART du pôle de compétitivité Aerospace Valley, respectivement MACAO et OSMOSES, nous avons travaillé en collaboration avec l’entreprise Turbomeca à la conception d’une méthode de reconstruction volumique afin de traiter les résultats de simulations à base chimère. Nous avons ainsi proposé une méthode innovante permettant de reconstruire une partition de l’espace de simulation exempte de chevauchement entre grilles. La nouvelle partition conserve le maximum de propriétés des grilles d’origine et assure en tout point la conformité aux bords. La complexité théorique est linéaire avec la taille des grilles d’origine et nous permet d’obtenir des temps de traitement de l’ordre de la seconde pour des grilles de plusieurs centaines de milliers de mailles. Le principal intérêt de ce travail est de rendre exploitables les résultats de simulations à base chimère par les outils de post-traitement, qu’il s’agisse d’outils maison ou des nombreux logiciels commerciaux ou OpenSource disponibles, condition indispensable pour l’adoption de la méthode chimère par les bureaux d’études. / Computationnal fluid dynamics is an essential step in gas turbine modelling. Continuous optimization of turbines has led to sophisticated geometries, which raises severe issues for the design of adapted simulation grids. The chimera technique aims at relaxing geometry matching constraints by allowing grids overlap. However, post-processing of simulation results performed over chimera grids raises new issues because usual tools are not tuned for this particular geometricconfigurations. In the framework of the MOSART programme of the world competitiveness cluster Aerospace Valley, we have been working in collaboration with Turbomeca in order to develop a technique for the volumetric reconstruction of chimerasimulation results. We propose an innovative method that allows us to build a collection of non-overlapping grids while preserving the main properties of the former simulation grids and featuring boundary conforming property everywhere.The theorical complexity of our algorithms has proved to be linear in the size of the former grids and leads to computation times of a few seconds for grids of hundreds of thousands of cells. The main impact of this work leads in the possibility of using any post-processing tool, including a large number of OpenSource solutions, for post-processing chimera simulation results, which is a mandatory condition for the wide acceptance of this method by industry actors.

3D scanning has grown to become an important tool in a wide variety of applications. Still, acquiring high-quality 3D models using scanning technology is a challenging task. In this thesis, we present various ways that reduce the hurdles of existing 3D scanning pipelines with the ultimate goal of bringing this technology closer to the end user. To achieve this goal, we focus on three sub problems of traditional scanning pipelines: First, we present a novel algorithm that can be used to consistently orient the normals of huge point clouds. We achieve this by formulating the orientation problem as a graph-based energy minimization problem and applying out-of core methods. Second, we introduce interactivity into the scanning pipeline by presenting an online surface reconstruction method that produces high-quality semi-regular meshes. The resulting interactive pipeline is highly efficient because it reduces the turn-around between presentation of the final result and possible corrections by the user. Third, we develop a robust method to texture-map semi-regular meshes. This approach is based on a generalization of motorcycle graphs, which partitions arbitrary meshes into quadrilateral patches. These patches then serve as the parametrization domains of the texture atlas. Finally, we present an application from the area of cave science. The application is targeted at quantitatively and objectively assessing a cave's size. To achieve this goal, we present methods to analyze the structure of caves, especially to distinguish chambers from passages.

Afin de convertir un maillage triangulaire en une surface spline de CAGD/CAM, cette thèse adresse l’un des problèmes les plus cruciaux du processus de conversion : extraire un “bon” maillage de contrôle quadrilatéral de la surface. Ce que nous entendons par “bon” est que les arêtes du maillage de contrôle se croisent perpendiculairement et sont alignées avec les principales directions de la courbure de la surface. Ces deux propriétés du maillage de contrôle permettent de fournir une bonne approximation de la surface avec peu de points de contrôles. D’ailleurs, ils aident considérablement à réduire des oscillations non désirées sur la surface spline finale. Pour résoudre ce problème, nous proposons un nouvel algorithme automatique, appelé paramétrisation globale périodique. L’idée fondamentale de cet algorithme est de trouver une paramétrisation qui ait un “sens d’un point de vue géométrique”, pour ce faire, elle doit être guidée par la courbure de la surface, représentée par une paire de champs de direction orthogonaux. Les iso-lignes de cette paramétrisation sont ensuite extraites pour définir un maillage de contrôle qui ait les propriétés requises. Ce maillage de contrôle, nous permet de construire une approximation en surface T-spline de la surface triangulée initiale. Nous exposons plusieurs résultats de cette conversion d’un maillage triangulée en surface spline. Les résultats montrent que, grâce aux maillages de contrôle anisotropes, les surfaces spline finales ont beaucoup moins d’oscillations que celles construites par les méthodes précédentes qui ne tiennent pas compte de l’anisotropie de la surface / Aiming at converting a triangular mesh into a CAGD/CAM spline surface, this thesis focuses on one of the most crucial problems of the conversion process, i.e. extracting a “good” quadrilateral control mesh of the surface. What we mean by good is that the edges of the control mesh should be orthogonal and aligned with the principal directions of curvature of the surface. These two properties make the control mesh optimum in an approximation point of view, and greatly help to reduce unwanted oscillations on the final spline surface built from it. To solve this problem, we propose a new automatic algorithm, called periodic global parameterization. The basic idea is to find a “geometry-meaningful” parameterization guided by a pair of orthogonal anisotropic direction fields. Then, the iso-value lines of this parameterization will be extracted to define an initial control mesh, that satisfies the two criteria of a good control mesh. With the initial control mesh, we explain how to construct a T-spline approximation of the initial triangulated surface. We show several examples of the triangular mesh to T-spline conversion. The results show that thanks to the anisotropic control meshes, the final spline surfaces generated have much less oscillations as compared to results of previous methods, that do not take into account of the anisotropy

Le raffinement de nuage de points et la reconstruction de surface sont deux problèmes fondamentaux dans le traitement de la géométrie. La plupart des méthodes existantes ont été ciblées sur les données de capteur de distance et se sont avérées être mal adaptées aux données multi-vues. Dans cette thèse, deux nouvelles méthodes sont proposées respectivement pour les deux problèmes avec une attention particulière aux données multi-vues. La première méthode permet de lisser les nuages de points provenant de la reconstruction multi-vue sans endommager les données. Le problème est formulé comme une optimisation non-linéaire sous contrainte et ensuite résolu par une série de problèmes d’optimisation sans contrainte au moyen d’une méthode de barrière. La seconde méthode effectue une triangulation du nuage de points d’entrée pour générer un maillage en utilisant une stratégie de l’avancement du front pilotée par un critère de l’empilement compact de sphères. L’algorithme est simple et permet de produire efficacement des maillages de haute qualité. Les expérimentations sur des données synthétiques et du monde réel démontrent la robustesse et l’efficacité des méthodes proposées. Notre méthodes sont adaptées aux applications qui nécessitent des informations de position précises et cohérentes telles que la photogrammétrie et le suivi des objets en vision par ordinateur / Point cloud refinement and surface reconstruction are two fundamental problems in geometry processing. Most of the existing methods have been targeted at range sensor data and turned out be ill-adapted to multi-view data. In this thesis, two novel methods are proposed respectively for the two problems with special attention to multi-view data. The first method smooths point clouds originating from multi-view reconstruction without impairing the data. The problem is formulated as a nonlinear constrained optimization and addressed as a series of unconstrained optimization problems by means of a barrier method. The second method triangulates point clouds into meshes using an advancing front strategy directed by a sphere packing criterion. The method is algorithmically simple and can produce high-quality meshes efficiently. The experiments on synthetic and real-world data have been conducted as well, which demonstrates the robustness and the efficiency of the methods. The developed methods are suitable for applications which require accurate and consistent position information such photogrammetry and tracking in computer vision