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We introduce the restricted constrained Delaunay triangulation (restricted CDT), a generalization of both the restricted Delaunay triangulation and the constrained Delaunay triangulation. The restricted CDT is a triangulation of a surface whose edges include a set of user-specified constraining segments. We define the restricted CDT to be the dual of a restricted Voronoi diagram defined on a surface that we have extended by topological surgery. We prove several properties of restricted CDTs, including sampling conditions under which the restricted CDT contains every constraining segment and is homeomorphic to the underlying surface., LIPIcs, Vol. 189, 37th International Symposium on Computational Geometry (SoCG 2021), pages 49:1-49:16

Depending on the specific method used to build a 3D structural model, and on the exact purpose of this model, its mesh must be adapted so that it enforces criteria on element types, maximum number of elements, and mesh quality. Meshing methods developed for applications others than geomodeling forbid any modification of the input model, that may be desirable in geomodeling to better control the number of elements in the final mesh and their quality. The objective of this thesis is to develop meshing methods that fulfill this requirement to better manage the geometrical complexity of B-Rep geological structural models. An analysis of the sources of geometrical complexity in those models is first proposed. The introduced measures are a first step toward the definition of tools allowing objective comparisons of structural models and permit to characterize the model zones that are more complicated to mesh. We then introduce two original meshing methods based on Voronoi diagrams: the first for surface remeshing, the second for hybrid gridding. The key ideas of these methods are identical: (1) the use of a centroidal Voronoi optimization to have a globally controlled number of elements of good quality, and (2) combinatorial considerations to locally build the final mesh while sometimes modifying the initial model. The surface remeshing method is automatic and permits to simplify a model at a given resolution. The gridding method generates a hybrid volumetric mesh. Prisms and pyramids fill the very thin layers of the model while the remaining regions are filled with tetrahedra.; Selon la méthode utilisée pour construire un modèle structural en trois dimensions et selon l'application à laquelle il est destiné, son maillage, en d'autres termes sa représentation informatique, doit être adapté afin de respecter des critères de type, de nombre et de qualité de ses éléments. Les méthodes de maillage développées dans d'autres domaines que la géomodélisation ne permettent pas de modifier le modèle d'entrée. Ceci est souhaitable en géomodélisation afin de mieux contrôler le nombre d'éléments du maillage et leur qualité. L'objectif de cette thèse est de développer des méthodes de maillage permettant de remplir ces objectifs afin de gérer la complexité géométrique des modèles structuraux définis par frontières. Premièrement, une analyse des sources de complexité géométrique dans ces modèles est proposée. Les mesures développées constituent une première étape dans la définition d'outils permettant la comparaison objective de différents modèles et aident à caractériser précisément les zones plus compliquées à mailler dans un modèle. Ensuite, des méthodes originales de remaillage surfacique et de maillage volumique fondées sur l'utilisation des diagrammes de Voronoï sont proposées. Les fondements de ces deux méthodes sont identiques : (1) une optimisation de type Voronoï barycentrique est utilisée pour globalement obtenir un nombre contrôlé d'éléments de bonne qualité et (2) des considérations combinatoires permettant de construire localement le maillage final, éventuellement en modifiant le modèle initial. La méthode de remaillage surfacique est automatique et permet de simplifier un modèle à une résolution donnée. L'originalité de la méthode de maillage volumique est que les éléments générés sont de types différents. Des prismes et pyramides sont utilisés pour remplir les zones très fines du modèle, tandis que le reste du modèle est rempli avec des tétraèdres.

International audience; In this paper we propose a method to generate mixed-element meshes (tetrahedra, triangular prisms, square pyramids) for B-Rep models. The vertices, edges, facets, and cells of the final volumetric mesh are determined from the combinatorial analysis of the intersections between the model components and the Voronoi diagram of sites distributed to sample the model. Inside the volumetric regions, Delaunay tetrahedra dual of the Voronoi diagram are built. Where the intersections of the Voronoi cells with the model surfaces have a unique connected component, tetrahedra are modified to fit the input triangulated surfaces. Where these intersections are more complicated, a correspondence between the elements of the Voronoi diagram and the elements of the mixed- element mesh is used to build the final volumetric mesh. The method which was motivated by meshing challenges encountered in geological modeling is demonstrated on several 3D synthetic models of subsurface rock volumes.

Most of the current surface reconstruction algorithms target high quality data and can produce some intractable results when used with point clouds acquired through profitable 3D acquisition methods. Our first contribution is a surface reconstruction algorithm from stereo vision data that copes with the data's fuzziness using information from both the acquired 3D point cloud and the calibrated images. After pre-processing the point cloud, the algorithm builds, using the calibrated images, a 3D triangular soup consistent with the surface of the scene through a combination of visibility and photo-consistency constraints. A mesh is then computed from the triangle soup using a combination of restricted Delaunay triangulation and Delaunay refinement methods. Our second contribution is an algorithm that builds, given a 3D point cloud sampled on a surface, an approximating surface mesh with an accurate representation of surface sharp edges, providing an enhanced trade-off between accuracy and mesh complexity. We first extract from the point cloud an approximation of the sharp edges of the underlying surface. Then, a feature preserving variant of a Delaunay refinement process generates a mesh combining a faithful representation of the extracted sharp edges with an implicit surface obtained from the point cloud. The method is shown to be flexible, robust to noise and tunable to adapt to the scale of the targeted mesh and to a user defined sizing field. We demonstrate the effectiveness of both contributions on a variety of scenes and models acquired with different hardware and show results that compare favourably, in terms of accuracy, with the current state of the art.; La majorité des algorithmes de reconstruction de surface sont optimisés pour s'appliquer à des données de haute qualité. Les résultats obtenus peuvent alors être inutilisables si les données proviennent de solutions d'acquisition bon marché. Notre première contribution est un algorithme de reconstruction de surfaces à partir de données de stéréo vision. Il combine les informations liées aux points 3D avec les images calibrées afin de combler l'imprécision des données. L'algorithme construit une soupe de triangles 3D à l'aide des images calibrées et à l'issue d'une phase de prétraitement du nuage de points. Pour épouser au mieux la surface de la scène, on contraint cette soupe de triangle 3D à respecter des critères de visibilité et de photo-consistance. On calcule ensuite un maillage à partir de la soupe de triangles à l'aide d'une technique de reconstruction qui combine les triangulations de Delaunay contraintes et le raffinement de Delaunay. Notre seconde contribution est un algorithme qui construit, à partir d'un nuage de points 3D échantillonnés sur une surface, un maillage de surface qui représente fidèlement les arêtes vives. Cet algorithme génère un bon compromis entre précision et complexité du maillage. Dans un premier temps, on extrait une approximation des arêtes vives de la surface sous-jacente à partir du nuage de points. Dans un deuxième temps, on utilise une variante du raffinement de Delaunay pour générer un maillage qui combine les arêtes vives extraites avec une surface implicite obtenue à partir du nuage de points. Notre méthode se révèle flexible, robuste au bruit; cette méthode peut prendre en compte la résolution du maillage ciblé et un champ de taille défini par l'utilisateur. Nos deux contributions génèrent des résultats efficaces sur une variété de scènes et de modèles. Notre méthode améliore l'état de l'art en termes de précision.

Special issue for ACCV09; International audience; This article describes an original method to reconstruct a 3D scene from a sequence of images. Our approach uses both the dense 3D point cloud extracted by multi-view stereovision and the calibrated images. It combines depth-maps construction in the image planes with surface reconstruction through restricted Delaunay triangulation. The method may handle very large scale outdoor scenes. Its accuracy has been tested on numerous outdoor scenes including the dense multi-view benchmark proposed by Strecha et al. Our results show that the proposed method compares favorably with the current state of the art.

International audience; Extracting a computer model of a real scene from a sequence of views, is one of the most challenging and fundamental problems in computer vision. Stereo vision algorithms allow us to extract from the images a sparse 3D point cloud on the scene surfaces. However, computing an accurate mesh of the scene based on such poor quality data points (noise, sparsity) is very difficult. Here we describe a simple yet original approach that uses both the stereo vision extracted point cloud and the calibrated images. Our method is a three-stage process in which the first stage merges, filters and smoothes the input 3D points. The second stage builds for each calibrated image a triangular depth-map and fuses the set of depth-maps into a triangle soup that minimize violations of size and visibility constraints. Finally, a mesh is computed from the triangle soup using a reconstruction method that combines restricted Delaunay triangulation and Delaunay refinement.

It is a well-known fact that, under mild sampling conditions, the restricted Delaunay triangulation provides good topological approximations of 1- and 2-manifolds. We show that this is not the case for higher-dimensional manifolds, even under stronger sampling conditions. Specifically, it is not true that, for any compact closed submanifold M of R^n, and any sufficiently dense uniform sampling L of M, the Delaunay triangulation of L restricted to M is homeomorphic to M, or even homotopy equivalent to it. Besides, it is not true either that, for any sufficiently dense set W of witnesses, the witness complex of L relative to M contains or is contained in the restricted Delaunay triangulation of L.