Modelado jerárquico de objetos 3D con superficies de subdivisión

Las SSs (Superficies de Subdivisión) son un potente paradigma de modelado de objetos 3D (tridimensionales) que establece un puente entre los dos enfoques tradicionales a la aproximación de superficies, basados en mallas poligonales y de parches alabeados, que conllevan problemas uno y otro. Los esquemas de subdivisión permiten definir una superficie suave (a tramos), como las más frecuentes en la práctica, como el límite de un proceso recursivo de refinamiento de una malla de control burda, que puede ser descrita muy compactamente. Además, la recursividad inherente a las SSs establece naturalmente una relación de anidamiento piramidal entre las mallas / NDs (Niveles de Detalle) generadas/os sucesivamente, por lo que las SSs se prestan extraordinariamente al AMRO (Análisis Multiresolución mediante Ondículas) de superficies, que tiene aplicaciones prácticas inmediatas e interesantísimas, como la codificación y la edición jerárquicas de modelos 3D. Empezamos describiendo los vínculos entre las tres áreas que han servido de base a nuestro trabajo (SSs, extracción automática de NDs y AMRO) para explicar como encajan estas tres piezas del puzzle del modelado jerárquico de objetos de 3D con SSs. El AMRO consiste en descomponer una función en una versión burda suya y un conjunto de refinamientos aditivos anidados jerárquicamente llamados "coeficientes ondiculares". La teoría clásica de ondículas estudia las señales clásicas nD: las definidas sobre dominios paramétricos homeomorfos a R" o (0,1)n como el audio (n=1), las imágenes (n=2) o el vídeo (n=3). En topologías menos triviales, como las variedades 2D) (superficies en el espacio 3D), el AMRO no es tan obvio, pero sigue siendo posible si se enfoca desde la perspectiva de las SSs. Basta con partir de una malla burda que aproxime a un bajo ND la superficie considerada, subdividirla recursivamente y, al hacerlo, ir añadiendo los coeficientes ondiculares, que son los detalles 3D necesarios para obtener aproximaciones más y más finas a la superficie original. Pasamos después a las aplicaciones prácticas que constituyen nuestros principal desarrollo original y, en particular, presentamos una técnica de codificación jerárquica de modelos 3D basada en SSs, que actúa sobre los detalles 3D mencionados: los expresa en un referencial normal loscal; los organiza según una estructura jerárquica basada en facetas; los cuantifica dedicando menos bits a sus componentes tangenciales, menos energéticas, y los "escalariza"; y los codifica dinalmente gracias a una técnica similar al SPIHT (Set Partitioning In Hierarchical Tress) de Said y Pearlman. El resultado es un código completamente embebido y al menos dos veces más compacto, para superficies mayormente suaves, que los obtenidos con técnicas de codificación progresiva de mallas 3D publicadas previamente, en las que además los NDs no están anidados piramidalmente. Finalmente, describimos varios métodos auxiliares que hemos desarrollado, mejorando técnicas previas y creando otras propias, ya que una solución completa al modelado de objetos 3D con SSs requiere resolver otros dos problemas. El primero es la extracción de una malla base (triangular, en nuestro caso) de la superficie original, habitualmente dada por una malla triangular fina con conectividad arbitraria. El segundo es la generación de un remallado recursivo con conectividad de subdivisión de la malla original/objetivo mediante un refinamiento recursivo de la malla base, calculando así los detalles 3D necesarios para corregir las posiciones predichas por la subdivisión para nuevos vértices

Automatic model construction with Gaussian processes

This thesis develops a method for automatically constructing, visualizing and describing a large class of models, useful for forecasting and finding structure in domains such as time series, geological formations, and physical dynamics. These models, based on Gaussian processes, can capture many types of statistical structure, such as periodicity, changepoints, additivity, and symmetries. Such structure can be encoded through kernels, which have historically been hand-chosen by experts. We show how to automate this task, creating a system that explores an open-ended space of models and reports the structures discovered. To automatically construct Gaussian process models, we search over sums and products of kernels, maximizing the approximate marginal likelihood. We show how any model in this class can be automatically decomposed into qualitatively different parts, and how each component can be visualized and described through text. We combine these results into a procedure that, given a dataset, automatically constructs a model along with a detailed report containing plots and generated text that illustrate the structure discovered in the data. The introductory chapters contain a tutorial showing how to express many types of structure through kernels, and how adding and multiplying different kernels combines their properties. Examples also show how symmetric kernels can produce priors over topological manifolds such as cylinders, toruses, and Möbius strips, as well as their higher-dimensional generalizations. This thesis also explores several extensions to Gaussian process models. First, building on existing work that relates Gaussian processes and neural nets, we analyze natural extensions of these models to deep kernels and deep Gaussian processes. Second, we examine additive Gaussian processes, showing their relation to the regularization method of dropout. Third, we combine Gaussian processes with the Dirichlet process to produce the warped mixture model: a Bayesian clustering model having nonparametric cluster shapes, and a corresponding latent space in which each cluster has an interpretable parametric form.This work was supported by the National Sciences and Engineering Research Council of Canada, the Cambridge Commonwealth Trust, Pembroke College, a grant from the Engineering and Physical Sciences Research Council, and a grant from Google

Advanced General Relativity Notes

These lecture notes are intended as a guide to Graduate level readers that are already familiar with basic General Relativity. They present in a concise way some advanced concepts and problems encountered in the study of gravitation. In these notes are covered: Alternates forms of the Schwarzschild Black Hole solution, including the classic Kruskal extension; An account of the building of Conformal, Carter-Penrose, diagrams; A discussion of Birkhoff Theorem; A discussion of tools for Geodesics and congruences, including Energy Conditions; A discussion of Horizons and an approach to some of the singularity theorems; An exploration of the Kerr Black Hole solution properties, including the Penrose Process and Black Hole Thermodynamics; A discussion of the Eckart and Israel-Stewart Relativistic Thermodynamics; A discussion of Tetrads in Relativity, in Einstein-Cartan theory and in Newman-Penrose formalism; An explicitation of calculations on Geodesics approach from Hamilton-Jacobi Formalism; A derivation from Least action of the equation of Motion of a top in Relativity, the M.P.D. equationsComment: 164pp, 40figs. Lecture notes for Graduates with GR1 knowledge. Feedback appreciate

Controle híbrido para estabilização de pose usando quaternions duais

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2018.Motivado tanto pelas vantagens da representação em dual quatérnios duais e por problemas relativos à obstrução topológica de se ter um equilíbrio assintótico globalmente estável, esse trabalho visa usar o formalismo de quaternion dual e as ferramentas de sistemas dinâmicos híbridos para tratar o problema de estabilização de pose de corpos rígidos. O grupo de Lie dos quatérnios duais proporciona um modo eficiente de representar a cinemática linear e rotacional de um corpo rígido sem singularidades. Algumas estratégias híbridas são propostas para lidar com o problema de “chattering” presente em todos os controladores por realimentação descontínuos enquanto ao mesmo tempo garantindo atratividade global da pose de estabilização do corpo rígido.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF).Motivated both by the advantages of the dual quaternion representation and by the problems concerning the topological obstruction to global asymptotic stability, this work addresses the rigid body pose stabilization problem using dual quaternion formalism and dynamic hybrid systems tools. The Lie group of unit dual quaternions provides a computationally efficient way to represent coupled linear and rotational kinematics without singularities. Some hybrid control strategies are proposed to overcome the chattering problem present in all discontinuous-based feedback controllers while at same time also guaranteeing global attractivity of the stabilization pose of the rigid body

Multi-Modal Partial Surface Matching for Intra-Operative Registration

An important task for computer-assisted surgical interventions is the alignment of pre- and intra-operative spaces allowing the transfer of pre-operative information to the current patient situation, known as intra-operative registration. Registration is usually performed by using markers or image-based techniques. Another approach is the intra-operative acquisition of organ surfaces by 3D range scanners, which are then matched to pre-operatively generated surfaces. However, this approach is not trivial, as methods for intra-operative surface matching must be able to deal with noise, distortions, deformations, and the availability of only partially overlapping, nearly flat surfaces. For these reasons, surface matching for intra-operative registration has so far only been used to account for displacements that occur in local scales, while the actual alignment is still performed manually. The main contributions of this thesis are two different approaches for automatic surface matching in intra-operative environments. The focus here is the registration of surfaces acquired by different modalities, dealing with the aforementioned issues and without relying on unique landmarks. For the first approach, surfaces are converted to graph representations and correspondences between them are identified by means of graph matching. Graphs are obtained automatically by segmenting the surfaces into regions with similar properties. As the graph matching problem is known to be NP-hard, it was solved by iteratively computing node similarity scores, and converting it to a linear assignment problem. In the second approach, correspondences are identified by the selection of two spatial configurations of landmarks that can be better fitted to each other, according to an error metric. This error metric does not only incorporate a fitting error, but also a new measure for spatial configuration reliability. The optimization problem is solved by means of a greedy algorithm. Evaluation of the two approaches was performed with several experiments, simulating intra-operative conditions. While the graph matching approach proved to be robust for the registration of small partial data, the point-based approach proved to be more reliable for noisy surfaces. Apart from being a significant contribution to the field of feature-less partial surface matching, this work represents a great effort towards the achievement of a fully automatic, marker-less, registration system for computer-assisted surgery guidance

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

A Geometric Approach to Converting CAD Models to CAM Models: an Application on Aeronautical Structure Parts

"RÉSUMÉ:" La conversion d'un modèle de CAO en un modèle de FAO est la première étape de fabrication intégrée par ordinateur. Les principaux problèmes qui concernent la conversion sont les suivants: définir des volumes de matériau amovible géométriquement, vérifier les accessibilités aux volumes ainsi obtenus, associer les opérations d'usinage avec ces volumes individuellement, sélectionner les outils de coupe, mettre en séquençage les opérations d'usinage et assigner une machine pour exécuter le processus. La détermination des volumes individuels de matériel amovible est le premier problème de la conversion. Dans les dernières décennies, de nombreuses approches ont été développées avec d'énormes efforts, mais aucune étude à ce jour a examiné de manière exhaustive les approches pour générer des volumes de matériau amovible pour traiter des pièces complexes, telles que celles qu’on rencontre dans en aéronautique dans la partie structurelle. Dans la perspective de définir les volumes du matériau amovible, les méthodes existantes se limitent aux fonctions prismatiques. L'objectif principal de cette recherche était de développer des approches systématiques, pour générer automatiquement l'ensemble des volumes de matières amovibles selon les modèles 3D d’une pièce aéronautique structurelle. Il faut alors partir du brut (un morceau de matière première) et usiner toutes les surfaces requises. Grâce à l'outil mathématique disponible des opérations booléennes, il est possible de séparer des géométries volumiques très complexes en volumes plus petits relativement simples. La décomposition du volume delta présente des avantages dans la création des volumes amovibles. Dans cette recherche, les approches de décomposition de volume ont été développées dans le but que chaque volume de matériau puisse être usiné en une seule opération d'usinage. Des arêtes concaves impliquent éventuellement des opérations d'usinage différentes. La détection du bord concave est la première étape de la décomposition de volume intérieur. Dans cette étude, une approche mathématique a été développée afin de vérifier la concavité d'une arête dans la limite d'un modèle solide 3D et une approche de détection des bords concaves est proposée. Générer des faces de séparation est une étape clé pour définir un volume décomposé. Selon la complexité de l'élément de construction, les algorithmes sont conçus pour créer différents types de décompositions de faces correspondant à des formes locales de la pièce à décomposer.----------"ABSTRACT:" Conversion of a CAD model to a CAM model is the initial step of computer integrated manufacturing. Main issues concerning the conversion are as follows: defining volumes of removable material geometrically, verifying accessibilities to so obtained volumes, associating machining operations with these volumes individually, selecting cutting tools, sequencing machining operations, and assign a machine to perform the process. Determination of individual volumes of removable material is the first issue of the conversion. In the past decades many approaches have been developed by enormous efforts but no study up to date has comprehensively discussed approaches to generate volumes of removable material for producing a complex aeronautical structural part. In the perspective of volumetric definition of removable material, existing methods are limited to prismatic features. The main objective of this research was to develop systematic approaches to generating automatically the complete set of volumes of removable material according to the 3D models of both an aeronautical structural part to be produced and the stock (a piece of raw material) to be machined. Due to powerful mathematical tool of Boolean operations available for separating very complex volumetric geometries into relatively simple smaller volumes, delta volume decomposition has advantages in generating removable volumes. In this research volume decomposition approaches were developed for the purpose that every volume of material can be machined in one machining operation. Concave edges imply possible requirement of different machining operations. Detecting concave edge is the premier step of interior volume decomposition. In this study a mathematical approach was developed to verify the concavity of an edge in the boundary of a 3D solid model. Approaches to detecting concave edges were proposed. Generating splitting faces is the key step to define a decomposed volume. According to the complexity of the structural component, algorithms are developed to create different kinds of splitting faces corresponding to local shapes of the part to perform decompositions. Face union is a powerful tool to separate volumes bounded by faces of complex geometries. This research proposed recursive procedures of decomposition. Using the proposed approaches the 3D design model of an aeronautic structural component is converted into volumes of removable material (named sub delta volume and denoted SDV in this research) by means of delta volume decomposition