Ray Tracing in Non-Euclidean Spaces

This dissertation describes a method for modeling, simulating and real-time rendering piecewise linear approximations of generic non-Euclidean 3D spaces. The 3D rendering pipeline most commonly used, where one multiplies each vertex coordinate by a 4x4 matrix to project it on the screen does not work for all cases where the space does not obey Euclid’s postulates (non-Euclidean space). Furthermore, while other non-Euclidean rendering tools only work for a limited type of spaces, our approach allows us to model, simulate, and render any isometrically embeddable non-Euclidean space and eventual objects lying therein. We envision at least two main applications for our approach. The first for helping mathematicians get a better understanding of what arbitrary spaces look like (e.g., hyperconical space, hyper-spherical space, and so forth). The second for assisting physicists to visualize and simulate the effects of bent space (e.g., black holes, wormholes, Alcubierre drive, and so forth) on light, and on physical objectsEsta dissertação descreve um método para modelar, simular e renderizar aproximações lineares de espaços não Euclideanos de forma genérica e em tempo real. A técnica de renderização 3D mais comum, que multiplica a matriz de projeção 4 x 4 por cada vértice para determinar as coordenadas do respetivo pixel no ecrã, nem sempre funciona quando o espaço não obedece a um postulado de Euclides (espaço não-Euclideano). Além disso, enquanto outras ferramentas para renderizar espaços não-Euclideanos só funcionam para certos tipos de espaços, a nossa técnica permite modelar, simular e renderizar qualquer espaço não-Euclideano embebível isometricamente, bem como eventuais objetos nele existentes. Antevemos pelo menos dois usos para a nossa técnica. A primeira para ajudar matemáticos a compreender melhor o aspeto de espaços arbitrários (e.g., espaço hiper-cónico, espaço hiper-esférico, etc.). A segunda para ajudar físicos a visualizar e simular os efeitos de espaço curvo (e.g., buracos negros, buracos de minhoca, deformações Alcubierra drive, etc.) em luz e objetos físicos circundantes

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Segmentation d'images IRM du cerveau pour la construction d'un modèle anatomique destiné à la simulation bio-mécanique

Comment obtenir des données anatomiques pendant une neurochirurgie ? a été ce qui a guidé le travail développé dans le cadre de cette thèse. Les IRM sont actuellement utilisées en amont de l'opération pour fournir cette information, que ce soit pour le diagnostique ou pour définir le plan de traitement. De même, ces images pre-opératoires peuvent aussi être utilisées pendant l'opération, pour pallier la difficulté et le coût des images per-opératoires. Pour les rendre utilisables en salle d'opération, un recalage doit être effectué avec la position du patient. Cependant, le cerveau subit des déformations pendant la chirurgie, phénomène appelé Brain Shift, ce qui altère la qualité du recalage. Pour corriger cela, d'autres données per-opératoires peuvent être acquises, comme la localisation de la surface corticale, ou encore des images US localisées en 3D. Ce nouveau recalage permet de compenser ce problème, mais en partie seulement. Ainsi, des modèles mécaniques ont été développés, entre autres pour apporter des solutions à l'amélioration de ce recalage. Ils permettent ainsi d'estimer les déformations du cerveau. De nombreuses méthodes existent pour implémenter ces modèles, selon différentes lois de comportement et différents paramètres physiologiques. Dans tous les cas, cela requiert un modèle anatomique patient-spécifique. Actuellement, ce modèle est obtenu par contourage manuel, ou quelquefois semi-manuel. Le but de ce travail de thèse est donc de proposer une méthode automatique pour obtenir un modèle du cerveau adapté sur l'anatomie du patient, et utilisable pour une simulation mécanique. La méthode implémentée se base sur les modèles déformables pour segmenter les structures anatomiques les plus pertinentes dans une modélisation bio-mécanique. En effet, les membranes internes du cerveau sont intégrées: falx cerebri and tentorium cerebelli. Et bien qu'il ait été démontré que ces structures jouent un rôle primordial, peu d'études les prennent en compte. Par ailleurs, la segmentation résultante de notre travail est validée par comparaison avec des données disponibles en ligne. De plus, nous construisons un modèle 3D, dont les déformations seront simulées en utilisant une méthode de résolution par Éléments Finis. Ainsi, nous vérifions par des expériences l'importance des membranes, ainsi que celle des paramètres physiologiques.The general problem that motivates the work developed in this thesis is: how to obtain anatomical information during a neurosurgery?. Magnetic Resonance (MR) images are usually acquired before the surgery to provide anatomical information for diagnosis and planning. Also, the same images are commonly used during the surgery, because to acquire MRI images in the operating room is complex and expensive. To make these images useful inside the operating room, a registration between them and the patient's position has to be processed. The problem is that the brain suffers deformations during the surgery, in a process called brain shift, degrading the quality of registration. To correct this, intra-operative information may be used, for example, the position of the brain surface or US images localized in 3D. The new registration will compensate this problem, but only to a certain extent. Mechanical models of the brain have been developed as a solution to improve this registration. They allow to estimate brain deformation under certain boundary conditions. In the literature, there are a variety of methods for implementing these models, different equation laws used for continuum mechanic, and different reported mechanical properties of the tissues. However, a patient specific anatomical model is always required. Currently, most mechanical models obtain the associated anatomical model by manual or semi-manual segmentation. The aim of this thesis is to propose and implement an automatic method to obtain a model of the brain fitted to the patient's anatomy and suitable for mechanical modeling. The implemented method uses deformable model techniques to segment the most relevant anatomical structures for mechanical modeling. Indeed, the internal membranes of the brain are included: falx cerebri and tentorium cerebelli. Even though the importance of these structures is stated in the literature, only a few of publications include them in the model. The segmentation obtained by our method is assessed using the most used online databases. In addition, a 3D model is constructed to validate the usability of the anatomical model in a Finite Element Method (FEM). And the importance of the internal membranes and the variation of the mechanical parameters is studied.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

Medial Axis Approximation and Regularization

Medial axis is a classical shape descriptor. Among many good properties, medial axis is thin, centered in the shape, and topology preserving. Therefore, it is constantly sought after by researchers and practitioners in their respective domains. However, two barriers remain that hinder wide adoption of medial axis. First, exact computation of medial axis is very difficult. Hence, in practice medial axis is approximated discretely. Though abundant approximation methods exist, they are either limited in scalability, insufficient in theoretical soundness, or susceptible to numerical issues. Second, medial axis is easily disturbed by small noises on its defining shape. A majority of current works define a significance measure to prune noises on medial axis. Among them, local measures are widely available due to their efficiency, but can be either too aggressive or conservative. While global measures outperform local ones in differentiating noises from features, they are rarely well-defined or efficient to compute. In this dissertation, we attempt to address these issues with sound, robust and efficient solutions. In Chapter 2, we propose a novel medial axis approximation called voxel core. We show voxel core is topologically and geometrically convergent to the true medial axis. We then describe a straightforward implementation as a result of our simple definition. In a variety of experiments, our method is shown to be efficient and robust in delivering topological promises on a wide range of shapes. In Chapter 3, we present Erosion Thickness (ET) to regularize instability. ET is the first global measure in 3D that is well-defined and efficient to compute. To demonstrate its usefulness, we utilize ET to generate a family of shape revealing and topology preserving skeletons. Finally, we point out future directions, and potential applications of our works in real world problems

Dynamic Multivariate Simplex Splines For Volume Representation And Modeling

Volume representation and modeling of heterogeneous objects acquired from real world are very challenging research tasks and playing fundamental roles in many potential applications, e.g., volume reconstruction, volume simulation and volume registration. In order to accurately and efficiently represent and model the real-world objects, this dissertation proposes an integrated computational framework based on dynamic multivariate simplex splines (DMSS) that can greatly improve the accuracy and efficacy of modeling and simulation of heterogenous objects. The framework can not only reconstruct with high accuracy geometric, material, and other quantities associated with heterogeneous real-world models, but also simulate the complicated dynamics precisely by tightly coupling these physical properties into simulation. The integration of geometric modeling and material modeling is the key to the success of representation and modeling of real-world objects. The proposed framework has been successfully applied to multiple research areas, such as volume reconstruction and visualization, nonrigid volume registration, and physically based modeling and simulation

Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs

We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation

Integrated biomechanical model of cells embedded in extracellular matrix

Nature encourages diversity in life forms (morphologies). The study of morphogenesis deals with understanding those processes that arise during the embryonic development of an organism. These processes control the organized spatial distribution of cells, which in turn gives rise to the characteristic form for the organism. Morphogenesis is a multi-scale modeling problem that can be studied at the molecular, cellular, and tissue levels. Here, we study the problem of morphogenesis at the cellular level by introducing an integrated biomechanical model of cells embedded in the extracellular matrix. The fundamental aspects of mechanobiology essential for studying morphogenesis at the cellular level are the cytoskeleton, extracellular matrix (ECM), and cell adhesion. Cells are modeled using tensegrity architecture. Our simulations demonstrate cellular events, such as differentiation, migration, and division using an extended tensegrity architecture that supports dynamic polymerization of the micro-filaments of the cell. Thus, our simulations add further support to the cellular tensegrity model. Viscoelastic behavior of extracellular matrix is modeled by extending one-dimensional mechanical models (by Maxwell and by Voigt) to three dimensions using finite element methods. The cell adhesion is modeled as a general Velcro-type model. We integrated the mechanics and dynamics of cell, ECM, and cell adhesion with a geometric model to create an integrated biomechanical model. In addition, the thesis discusses various computational issues, including generating the finite element mesh, mesh refinement, re-meshing, and solution mapping. As is known from a molecular level perspective, the genetic regulatory network of the organism controls this spatial distribution of cells along with some environmental factors modulating the process. The integrated biomechanical model presented here, besides generating interesting morphologies, can serve as a mesoscopic-scale platform upon which future work can correlate with the underlying genetic network

Flow visualization with quantified spatial and temporal errors using edge maps

pre-printRobust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures

Field D* pathfinding in weighted simplicial complexes

Includes abstract.Includes bibliographical references.The development of algorithms to efficiently determine an optimal path through a complex environment is a continuing area of research within Computer Science. When such environments can be represented as a graph, established graph search algorithms, such as Dijkstra’s shortest path and A*, can be used. However, many environments are constructed from a set of regions that do not conform to a discrete graph. The Weighted Region Problem was proposed to address the problem of finding the shortest path through a set of such regions, weighted with values representing the cost of traversing the region. Robust solutions to this problem are computationally expensive since finding shortest paths across a region requires expensive minimisation. Sampling approaches construct graphs by introducing extra points on region edges and connecting them with edges criss-crossing the region. Dijkstra or A* are then applied to compute shortest paths. The connectivity of these graphs is high and such techniques are thus not particularly well suited to environments where the weights and representation frequently change. The Field D* algorithm, by contrast, computes the shortest path across a grid of weighted square cells and has replanning capabilites that cater for environmental changes. However, representing an environment as a weighted grid (an image) is not space-efficient since high resolution is required to produce accurate paths through areas containing features sensitive to noise. In this work, we extend Field D* to weighted simplicial complexes – specifically – triangulations in 2D and tetrahedral meshes in 3D