Ridges and umbilics of a sampled smooth surface: a complete picture gearing toward topological coherence

Consider a smooth surface, and at each point which is not an umbilic, respectively paint in blue (red) anything related to the maximum (minimum) principal curvature. Given such a surface, a blue (red) ridge is a curve on the surface such that at each of its points, the principal blue (red) curvature has an extremum along its blue (red) curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. Surprisingly, no method developed so far to report ridges from a mesh approximating a smooth surface comes with a careful analysis, which entails that one does not know whether the ridges are reported in a coherent fashion. This paper aims at bridging this gap with the following contributions. First, a careful analysis of the Acute rule - an orientation procedure used in most algorithms - is presented. Second, given a triangulation $T$ approximating a smooth generic surface $S$, we present sufficient conditions on $T$ together with a certified algorithm reporting ridges in a topologically coherent fashion. Third, we develop an algorithm and a filtering procedure aiming at reporting the most salient features of a coarse mesh $T$

Differential topology and geometry of smooth embedded surfaces: selected topics

The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere. This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis. Finally, properties of the medial axis are used to present sufficient conditions ensuring that two embedded surfaces are ambient isotopic. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be useful for those interested in certified approximations of smooth surfaces

Jet_fitting_3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting

Surfaces of R^3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties. This paper presents a generic C++ software package solving these problems. On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. On the implementation side and following state-of-the-art coding rules in Computational Geometry, genericity of the package is achieved thanks to three template classes accounting for (a) the type of the input points (b) the internal geometric computations and (c) the linear algebra operations. An instantiation within the Computational Geometry Algorithms Library (CGAL, version 3.3) and using CLAPACK is also provided

Smooth surfaces, umbilics, lines of curvatures, foliations, ridges and the medial axis: a concise overview

The understanding of surfaces embedded in R^3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As an example and assuming this makes sense for the applications encompassed, we are not aware as of today of any algorithm able to report ---under reasonable assumptions--- a topologically correct medial axis or foliation from a sampled surface. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of global objects related to curvature properties of a smooth generic surface. Gathering from differential topology and singularity theory sources, our presentation focuses on the geometric intuition rather than the technicalities. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere. This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be helpful for those interested in the manipulation of surfaces without using global parametrizations, and also for those aiming at producing globally coherent approximations of surfaces

Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting

International audienceSurfaces of R3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. %% One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties. This paper presents a generic C++ software package solving these problems. %% On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. %% On the implementation side and following state-of-the-art coding rules in Computational Geometry, genericity of the package is achieved thanks to four template classes accounting for (a) the type of the input points (b) the internal geometric computations (c) a convertion mechanism between these two geometries and (d) the linear algebra operations. An instantiation within the Computational Geometry Algorithms Library (CGAL, version 3.3) and using LAPACK is also provided

Les enjeux de la conception en phase d'esquisse pour les systèmes du génie électrique : illustration sur le cas des systèmes énergétiques pour les bâtiments

International audienceCe papier propose de donner un statut, et un outil, aux phases de conception initiale, dans lesquelles les caractéristiques principales du système énergétique utilisant l'énergie électrique doivent être esquissées. Nous proposons ainsi, d'utiliser un type de modèle adapté (modèle macroscopique et statique basé sur des bilans énergétiques et financiers). Ces modèles sont réalisés à l'échelle des composants du système et ils comportent des équations et des contraintes, qui sont ensuite composées à l'échelle du système, via des ports énergétiques et financiers. Il en résulte un modèle qui peut dès lors, être utilisé par des techniques d'optimisation. Cela permet de fournir un dimensionnement global et simultané des caractéristiques du système (taille et dimension des systèmes énergétiques) tout en esquissant une stratégie de gestion optimale sur des cycles de fonctionnement type. La fonction objectif est un coût global du système intégrant les coûts d'investissement, de maintenance et d'exploitation. Cette approche est illustrée sur un exemple de gare, représentatif à la fois des enjeux de la filière bâtiment, mais aussi représentatif de cette problématique de conception générique des systèmes hybrides comportant des sources, des charges et des moyens de stockage, auquel le génie électrique est de plus en plus confronté via des application dans les transports terrestres, aéronautiques, le smart-building, le smart-grid, ..

Les enjeux de la conception système en phase d'esquisse pour les systèmes du génie électrique : illustration sur le cas des systèmes énergétiques pour les bâtiments

Ce papier propose de donner un statut, et un outil, aux phases de conception initiale, dans lesquelles les caractéristiques principales du système énergétique utilisant l'énergie électrique doivent être esquissées. Nous proposons ainsi, d'utiliser un type de modèle adapté (modèle macroscopique et statique basé sur des bilans énergétiques et financiers). Ces modèles sont réalisés à l'échelle des composants du système et ils comportent des équations et des contraintes, qui sont ensuite composées à l'échelle du système, via des ports énergétiques et financiers. Il en résulte un modèle qui peut dès lors, être utilisé par des techniques d'optimisation. Cela permet de fournir un dimensionnement global et simultané des caractéristiques du système (taille et dimension des systèmes énergétiques) tout en esquissant une stratégie de gestion optimale sur des cycles de fonctionnement type. La fonction objectif est un coût global du système intégrant les coûts d'investissement, de maintenance et d'exploitation. Cette approche est illustrée sur un exemple de gare, représentatif à la fois des enjeux de la filière bâtiment, mais aussi représentatif de cette problématique de conception générique des systèmes hybrides comportant des sources, des charges et des moyens de stockage, auquel le génie électrique est de plus en plus confronté via des application dans les transports terrestres, aéronautiques, le smart-building, le smart-grid, ..

Impact of Complex-Logic Cell Layout on the Single-Event Transient Sensitivity

International audienceThe design methodology based on standard cells is widely used in a broad range of VLSI applications. Further, several optimization algorithms can be employed to address different constraints such as power consumption or reliability. This work evaluates the implications of the usage of complex-logic cells from a 45 nm Standard-Cell library to the Single-Event Transient sensitivity under heavy ions. Results show that even though a reduction in the layout area is obtained when adopting complex-logic gates, a slight reduction in the total sensitive area of the circuit is observed. Moreover, the effectiveness of logical masking can be suppressed, leading to a higher SET cross-section

VirtualEnaction: A Platform for Systemic Neuroscience Simulation.

International audienceConsidering the experimental study of systemic models of the brain as a whole (in contrast to models of one brain area or aspect), there is a real need for tools designed to realistically simulate these models and to experiment them. We explain here why a robotic setup is not necessarily the best choice, and what are the general requirements for such a bench-marking platform. A step further, we describe an effective solution, freely available on line and already in use to validate functional models of the brain. This solution is a digital platform where the brainy-bot implementing the model to study is embedded in a simplified but realistic controlled environment. From visual, tactile and olfactory input, to body, arm and eye motor command, in addition to vital somesthetic cues, complex survival behaviors can be experimented. The platform is also complemented with algorithmic high-level cognitive modules, making the job of building biologically plausible bots easier.Dans le domaine de l'étude expérimentale des modèles systémiques du cerveau pris dans son ensemble (par opposition aux modèles de la zone du cerveau ou une image), il y a un réel besoin d'outils conçus pour simuler de manière réaliste ces modèles et les expérimenter. Nous expliquons ici pourquoi une installation robotique n'est pas nécessairement le meilleur choix, et quelles sont les exigences générales d'une telle plateforme en terme de benchmarking. Nous décrivons alors une solution efficace, disponible gratuitement en ligne et déjà utilisées pour valider les modèles fonctionnels du cerveau. Cette solution est une plate-forme numérique où un "brainy-bot" implémente le modèle étudié et permet son intégration dans un environnement de survie contrôlé, simplifié, mais réaliste. Des entrées visuelles, tactiles et olfactive, un corps très simplifié, un bras et une commandande mootrice des yeux, en plus de la somesthésie des variavles vitales sont disponibles. De ce fait, des comportements de survie complexes peuvent être expérimentées. La plate-forme est également complétée par des modules algorithmiques de simulation de fonctions cognitives de haut niveau, facilitant le travail de construction de comportement biologiquement plausibles