Guillaume Coiffier

Neural implicit surfaces are a powerful tool for encoding a surface as the zero level set of a neural function. Trained using gradient‐descent based optimizers, these methods however suffer from a low‐frequency bias that prevents them to fit fine details of the surface. Solutions to break this bias exist but often damage the regularity of the long range implicit function, making geometrical queries more difficult. In this work, we combine a 1‐Lipschitz neural function with a detail field made of compactly supported radial basis functions. While the neural function handles long‐range queries with low surface accuracy, the detail field is fitted to perturb the neural function so that its zero level set precisely goes through sampled points of the surface and locally accounts for small‐scale details. As it is compactly supported, regions where the 1‐Lipschitz property breaks are finely controlled. We therefore achieve the best of both worlds by greatly increasing the representation power of the implicit function near the zero level set while maintaining robustness far from the zero level set.

Algorithmes de paramétrisation globale pour maillages quadrilatéraux: Les maillages quadrangulés sont une structure de données centrale au domaine du traitement automatique de la géométrie, trouvant des applications en infographie comme en simulation numérique. De nos jours, les techniques de génération automatique calculant les maillages quads de meilleure qualité se basent sur le calcul d'une représentation à plat de la surface à mailler. Pour que des quadrilatères puissent être extraits, cette paramétrisation se doit d'être "sans couture", c'est-à-dire de respecter un ensemble de contraintes sur son bord et ses découpes. Cela demande une quantification en nombre entier de certains degrés de liberté. Ces contraintes sont généralement imposées petit à petit dans un pipeline d'opérations désormais bien étudié, consistant en un calcul de champ de repères lisse, définissant un champ de singularités au défaut d'angle multiple de "pi/2", une phase d'intégration pour obtenir une paramétrisation aux coutures sans rotation, suivie d'une phase de quantification déterminant les degrés de liberté en translation. Cette thèse s'intéresse à l'amélioration des différentes étapes du pipeline de génération de maillages quadrilatéraux. En nous appuyant sur des notions de géométrie différentielle, nous proposons des formulations du problème évitant les écueils de l'approche actuelle. Premièrement, nous abandonnons la résolution de problèmes en nombre entier pour certaines étapes (connue pour être difficiles à résoudre) pour la remplacer par la minimisation de fonctions objectif continues (bien que non convexe). Deuxièmement, nous fusionnons certaines étapes du pipeline en une seule optimisation déterminant en un seul coup les degrés de liberté correspondants. Cela permet plus de versatilité et de contrôle utilisateur sur le maillage quad final, et évite les cas d'échecs classiques causés par l'approche gloutonne du pipeline actuel. Ces formulations théoriques du problème de paramétrisation sans couture s'accompagnent d'implémentations pratiques dans lesquelles nous démontrons la viabilité de nos approches sur une grande variété de modèles CAO. Finalement, notre travail est théoriquement généralisable au problème plus difficile du maillage hexaédrique, là où les algorithmes de paramétrisation actuels sont soit uniquement valables pour les surfaces, soit échouent à produire des résultats de façon robuste.

Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods exhibit great visual fidelity and quality near the surface, yet their properties tend to degrade with distance, making geometrical queries hard to perform without the help of complex range analysis techniques. Based on recent advancements in Lipschitz neural networks, we introduce a new method for approximating the signed distance function of a given object. As our neural function is made 1‐Lipschitz by construction, it cannot overestimate the distance, which guarantees robustness even far from the surface. Moreover, the 1‐Lipschitz constraint allows us to use a different loss function, called the hinge‐Kantorovitch‐Rubinstein loss, which pushes the gradient as close to unit‐norm as possible, thus reducing computation costs in iterative queries. As this loss function only needs a rough estimate of occupancy to be optimized, this means that the true distance function need not to be known. We are therefore able to compute neural implicit representations of even bad quality geometry such as noisy point clouds or triangle soups. We demonstrate that our methods is able to approximate the distance function of any closed or open surfaces or curves in the plane or in space, while still allowing sphere tracing or closest point projections to be performed robustly.

This article introduces a new representation of surface global parametrization based on Cartan’s method of moving frames . We show that a system of structure equations , characterizing the local coordinates changes with respect to a local frame system, completely characterizes the set of possible cone parametrizations. The discretization of this system provably provides necessary and sufficient conditions for the existence of a valid mapping. We are able to derive a versatile algorithm for surface parametrization, allowing feature constraints and singularities. As the first structure equation is independent of the global coordinate system, we do not require prior knowledge of cuts or cone positions. So, a single non-linear least-square problem is enough to place quantized cones while minimizing a given distortion energy. We are therefore able to take full advantage of the link between the parametrization geometry and the topology of its cone metric to solve challenging constrained parametrization problems.

Deep Neural Networks are state-of-the-art in a large number of challenges in machine learning. However, to reach the best performance they require a huge pool of parameters. Indeed, typical deep convolutional architectures present an increasing number of feature maps as we go deeper in the network, whereas spatial resolution of inputs is decreased through downsampling operations. This means that most of the parameters lay in the final layers, while a large portion of the computations are performed by a small fraction of the total parameters in the first layers. In an effort to use every parameter of a network at its maximum, we propose a new convolutional neural network architecture, called ThriftyNet. In ThriftyNet, only one convolutional layer is defined and used recursively, leading to a maximal parameter factorization. In complement, normalization, non-linearities, downsamplings and shortcut ensure sufficient expressivity of the model. ThriftyNet achieves competitive performance on a tiny parameters budget, exceeding 91% accuracy on CIFAR-10 with less than 40 k parameters in total, 74.3% on CIFAR-100 with less than 600 k parameters, and 67.1% On ImageNet ILSVRC 2012 with no more than 4.15 M parameters. However, the proposed method typically requires more computations than existing counterparts.