A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Le contrôle des inflexions et des extremums de courbure portés par les courbes et les surfaces B-Splines

RÉSUMÉ Le contrôle des propriétés différentielles des courbes et des surfaces B-splines est un enjeu important, en particulier pour le domaine de la conception géométrique assistée par ordinateur. Un enjeu qui sollicite autant les méthodes analytiques que numériques dans le but de permettre au concepteur de manipuler les formes avec une aisance toujours croissante. Ce texte explore les possibilités offertes lorsqu'on combine des méthodes numériques de pointe aux travaux de grands géomètres du 19e siècle. Ainsi, de nouveaux algorithmes pour l’optimisation sous contraintes des B-splines ont été développés. Ensuite, ces algorithmes ont été combinés à la théorie des groupes de transformations comme elle a été développée à l’origine par des pionniers comme Sophus Lie, Gaston Darboux et Felix Klein. Ceci permet d’ouvrir des portes vers de nouveaux horizons. Il devient possible de générer de larges espaces de formes sur lesquels on contrôle les propriétés différentielles. Il devient également possible d’éliminer des oscillations de façon sélective ou de manipuler les formes sans introduire d’oscillations indésirables. Avant de progresser vers cet objectif ambitieux, il faut d’abord être en mesure de bien comprendre et de bien visualiser ces propriétés différentielles que l’on souhaite contrôler. L’histoire de la géométrie différentielle classique des courbes et des surfaces est très riche. Cette histoire est revisitée avec une perspective nouvelle. Soit la perspective du contrôle des inflexions et des extremums de courbure. Ceci permet de faire émerger des liens importants entre la géométrie différentielle, la théorie des singularités, les groupes de transformations et l’optique géométrique. Ensuite viennent les algorithmes d’optimisation des B-splines sous contraintes. Les variables indépendantes sont les positions des points de contrôle de la B-spline alors que les contraintes portent sur la position des points de contrôle d’une fonction qui représente les propriétés différentielles de la B-spline. Les algorithmes sont d’abord développés pour les fonctions B-splines à une et deux variables. Une fois ces algorithmes développés, plusieurs possibilités nouvelles s’offrent à nous. Il devient possible, par exemple d’obtenir la courbe qui s’approche le plus d’une autre courbe quelconque sous la contrainte de posséder certaines propriétés différentielles. De cette manière, il devient possible de travailler avec un plus grand nombre de points de contrôle et ainsi dans un espace de forme plus riche sans avoir à se soucier d’oscillations arbitraires. Ceci permet en particulier d’éliminer de façon sélective des oscillations indésirables sur des profils aérodynamiques.----------ABSTRACT Control of B-spline differential properties is an important stake, especially for the field of computer-aided geometric design. An issue that calls for analytical and numerical skills to allow the designer to manipulate shapes in an increasingly efficient way. This text explores possibilities offered by combining new numerical methods with works of 19th century great geometers. Thus, new algorithms for constrained optimization of B-splines are selected and then grafted to the group theory of transformations as it was originally developed by pioneers such as Sophus Lie, Gaston Darboux and Felix Klein. This opens doors to new horizons. It becomes possible to generate large spaces of shapes with a control over their differential properties. This also gives us a selective eraser of curvature extrema and the option to manipulate shapes without introducing undesirables oscillations. Before progressing towards this ambitious goal, one must first be able to understand and visualize these differential properties that one wishes to control. This rich history of the classical differential geometry is revisited with a new perspective. This new perspective is the one of the control of inflections and extrema of curvature. A perspective that allows to establish important links between differential geometry, the theory of singularities, groups of transformations and geometric optics. Next comes the B-splines optimization algorithms with constraints. The independent variables are the B-spline position of the control points, while the constraints are applied to the control points of a function which represents the differential properties of the B-spline. The algorithms are first developed for B-spline functions. Once these algorithms have been developed, several new possibilities open up to us. It becomes possible, for example, to find the closest curve to another one under specified differential properties. This particular algorithm is introduced as an extension to the standard B-spline least squares method to approximate a series of points. The extension consists in adding constraints to produce curve segments with monotonously increasing or decreasing curvature. The interior point method is used to solve the constrained optimization problem. The method requires gradients and those are provided by symbolic B-spline operators. Therefore, the algorithm relies on the arithmetic, differential and variation diminishing properties of the Bsplines to apply the constraints. Thereby, it becomes possible to work with a greater number of control points and thus in a richer shape space without having to manage undesired oscillations

Squelettes pour la reconstruction 3D : de l'estimation de la projection du squelette dans une image 2D à la triangulation du squelette en 3D

La reconstruction 3D consiste à acquérir des images d’un objet, et à s’en servir pour en estimer un modèle 3D. Dans ce manuscrit, nous développons une méthode de reconstruction basée sur la modélisation par squelette. Cette méthode a l’avantage de renvoyer un modèle 3D qui est un objet virtuel complet (i.e. fermé) et aisément éditable, grâce à la structure du squelette. Enfin, l’objet acquis n’a pas besoin d’être texturé, et entre 3 et 5 images sont suffisantes pour la reconstruction. Dans une première partie, nous étudions les aspects 2D de l’étude. En effet, l’estimation d’un squelette 3D nécessite d’étudier la formation de la silhouette de l’objet à partir de son squelette, et donc les propriétés de sa projection perspective, appelée squelette perspectif. Cette étude est suivie par notre première contribution : un algorithme d’estimation de la projection perspective d’un squelette 3D curviligne, constitué d’un ensemble de courbes. Cet algorithme a toutefois tendance, comme beaucoup d’algorithmes estimant un squelette, à générer des branches peu informatives, notamment sur une image rastérisée. Notre seconde contribution est donc un algorithme d’estimation de squelette 2D, capable de prendre en compte la discrétisation du contour de la forme 2D, et d’éviter ces branches peu informatives. Cet algorithme, d’abord conçu pour estimer un squelette classique, est ensuite généralisé à l’estimation d’un squelette perspectif. Dans une seconde partie, nous estimons le squelette 3D d’un objet à partir de ses projections. Tout d’abord, nous supposons que le squelette de l’objet 3D à reconstruire est curviligne. Ainsi, chaque squelette perspectif estimé correspond à la projection du squelette 3D de l’objet, sous différents points de vue. La topologie du squelette étant affectée par la projection, nous proposons notre troisième contribution, l’estimation de la topologie du squelette 3D à partir de l’ensemble de ses projections. Une fois celle-ci estimée, la projection d’une branche 3D du squelette est identifiée sur chaque image, i.e. sur chacun des squelettes perspectifs. Avec cette identification, nous pouvons trianguler les branches du squelette 3D, ce qui constitue notre quatrième contribution : nous sommes donc en mesure d’estimer un squelette curviligne associé à un ensemble d’images d’un objet. Toutefois, les squelettes 3D ne sont pas tous constitués d’un ensemble de courbes : certains d’entre eux possèdent aussi des parties surfaciques. Notre dernière contribution, pour reconstruire des squelettes 3D surfaciques, est une nouvelle approche pour l’estimation d’un squelette 3D à partir d’images : son principe est de faire grandir le squelette 3D, sous les contraintes données par les images de l’objet

Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon

Transform domain texture synthesis on surfaces

In the recent past application areas such as virtual reality experiences, digital cinema and computer gamings have resulted in a renewed interest in advanced research topics in computer graphics. Although many research challenges in computer graphics have been met due to worldwide efforts, many more are yet to be met. Two key challenges which still remain open research problems are, the lack of perfect realism in animated/virtually-created objects when represented in graphical format and the need for the transmissiim/storage/exchange of a massive amount of information in between remote locations, when 3D computer generated objects are used in remote visualisations. These challenges call for further research to be focused in the above directions. Though a significant amount of ideas have been proposed by the international research community in their effort to meet the above challenges, the ideas still suffer from excessive complexity related issues resulting in high processing times and their practical inapplicability when bandwidth constraint transmission mediums are used or when the storage space or computational power of the display device is limited. In the proposed work we investigate the appropriate use of geometric representations of 3D structure (e.g. Bezier surface, NURBS, polygons) and multi-resolution, progressive representation of texture on such surfaces. This joint approach to texture synthesis has not been considered before and has significant potential in resolving current challenges in virtual realism, digital cinema and computer gaming industry. The main focus of the novel approaches that are proposed in this thesis is performing photo-realistic texture synthesis on surfaces. We have provided experimental results and detailed analysis to prove that the proposed algorithms allow fast, progressive building of texture on arbitrarily shaped 3D surfaces. In particular we investigate the above ideas in association with Bezier patch representation of 3D objects, an approach which has not been considered so far by any published world wide research effort, yet has flexibility of utmost practical importance. Further we have discussed the novel application domains that can be served by the inclusion of additional functionality within the proposed algorithms.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Calculation of Realistic Charged-Particle Transfer Maps

The study and computation of nonlinear charged-particle transfer maps is fundamental to understanding single-particle beam dynamics in accelerator devices. Transfer maps for individual elements of the beamline can in general depend sensitively on nonlinear fringe-field and high-multipole effects. The inclusion of these effects requires a detailed and realistic model of the interior and fringe magnetic fields, including knowledge of high spatial derivatives. Current methods for computing such maps often rely on idealized models of beamline elements. This Dissertation describes the development and implementation of a collection of techniques for computing realistic (as opposed to idealized) charged-particle transfer maps for general beamline elements, together with corresponding estimates of numerical error. Each of these techniques makes use of 3-dimensional measured or numerical field data on a grid as provided, for example, by various 3-dimensional finite element field codes. The required high derivatives of the corresponding vector potential A, required to compute transfer maps, cannot be reliably computed directly from this data by numerical differentiation due to numerical noise whose effect becomes progressively worse with the order of derivative desired. The effect of this noise, and its amplification by numerical differentiation, can be overcome by fitting on a bounding surface far from the axis and then interpolating inward using the Maxwell equations. The key ingredients are the use of surface data and the smoothing property of the inverse Laplacian operator. We explore the advantages of map computation using realistic field data on surfaces of various geometry. Maps obtained using these techniques can then be used to compute realistically all derived linear and nonlinear properties of both single pass and circular machines. Although the methods of this Dissertation have been applied primarily to magnetic beamline elements, they can also be applied to electric and radio-frequency beamline elements

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop