Computations of Delaunay and Higher Order Triangulations, with Applications to Splines

Digital data that consist of discrete points are frequently captured and processed by scientific and engineering applications. Due to the rapid advance of new data gathering technologies, data set sizes are increasing, and the data distributions are becoming more irregular. These trends call for new computational tools that are both efficient enough to handle large data sets and flexible enough to accommodate irregularity. A mathematical foundation that is well-suited for developing such tools is triangulation, which can be defined for discrete point sets with little assumption about their distribution. The potential benefits from using triangulation are not fully exploited. The challenges fundamentally stem from the complexity of the triangulation structure, which generally takes more space to represent than the input points. This complexity makes developing a triangulation program a delicate task, particularly when it is important that the program runs fast and robustly over large data. This thesis addresses these challenges in two parts. The first part concentrates on techniques designed for efficiently and robustly computing Delaunay triangulations of three kinds of practical data: the terrain data from LIDAR sensors commonly found in GIS, the atom coordinate data used for biological applications, and the time varying volume data generated from from scientific simulations. The second part addresses the problem of defining spline spaces over triangulations in two dimensions. It does so by generalizing Delaunay configurations, defined as follows. For a given point set P in two dimensions, a Delaunay configuration is a pair of subsets (T, I) from P, where T, called the boundary set, is a triplet and I, called the interior set, is the set of points that fall in the circumcircle through T. The size of the interior set is the degree of the configuration. As recently discovered by Neamtu (2004), for a chosen point set, the set of all degree k Delaunay configurations can be associated with a set of degree k plus 1 splines that form the basis of a spline space. In particular, for the trivial case of k equals 0, the spline space coincides with the PL interpolation functions over the Delaunay triangulation. Neamtu’s definition of the spline space relies only on a few structural properties of the Delaunay configurations. This raises the question whether there exist other sets of configurations with identical structural properties. If there are, then these sets of configurations—let us call them generalized configurations from hereon—can be substituted for Delaunay configurations in Neamtu’s definition of spline space thereby yielding a family of splines over the same point set

Boolean operations on 3D selective Nef complexes : data structure, algorithms, optimized implementation, experiments and applications

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. Consequently, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. The generality of Nef complexes is essential for some applications. In this thesis, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysisNef-Polyeder sind d-dimensionale Punktmengen, die durch eine endliche Anzahl boolescher Operationen über Halbräumen generiert werden. Sie sind abgeschlossen hinsichtlich boolescher und topologischer Operationen. Als Konsequenz daraus können sie nicht-mannigfaltige Situationen, offene und geschlossene Mengen und gemischt-dimensionale Komplexe darstellen. Die Allgemeinheit von Nef-Komplexen ist unentbehrlich für einige Anwendungen. In dieser Doktorarbeit stellen wir eine neue Datenstruktur vor, die eine Randdarstellung von dreidimensionalen Nef-polyedern und Algorithmen für boolesche Operationen realisiert. Wir benutzen exakte Arithmetik um die bekannten Probleme mit Gleitkommaarithmetik und Degeneriertheiten zu vermeiden. Außerdem präsentieren wir wichtige Optimierungen der Algorithmen und bewerten die optimierte Implementierung an Hand umfassender Experimente. Weitere Experimente belegen die theoretische Laufzeitanalyse und vergleichen unsere Implementation mit dem kommerziellen CAD kernel ACIS. ACIS is meistens bis zu sechs mal schneller, aber es gibt auch Beispiele bei denen ACIS scheitert. Nef-Polyeder können bei einer Vielzahl von Anwendungen eingesetzt werden. Wir präsentieren einfache Implementationen zweier Anwendungen - von der visuellen Hülle und von der Minkowski-Summe zwei abgeschlossener Nef-Polyeder

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

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

Algorithms for the reconstruction, analysis, repairing and enhancement of 3D urban models from multiple data sources

Over the last few years, there has been a notorious growth in the field of digitization of 3D buildings and urban environments. The substantial improvement of both scanning hardware and reconstruction algorithms has led to the development of representations of buildings and cities that can be remotely transmitted and inspected in real-time. Among the applications that implement these technologies are several GPS navigators and virtual globes such as Google Earth or the tools provided by the Institut Cartogràfic i Geològic de Catalunya. In particular, in this thesis, we conceptualize cities as a collection of individual buildings. Hence, we focus on the individual processing of one structure at a time, rather than on the larger-scale processing of urban environments. Nowadays, there is a wide diversity of digitization technologies, and the choice of the appropriate one is key for each particular application. Roughly, these techniques can be grouped around three main families: - Time-of-flight (terrestrial and aerial LiDAR). - Photogrammetry (street-level, satellite, and aerial imagery). - Human-edited vector data (cadastre and other map sources). Each of these has its advantages in terms of covered area, data quality, economic cost, and processing effort. Plane and car-mounted LiDAR devices are optimal for sweeping huge areas, but acquiring and calibrating such devices is not a trivial task. Moreover, the capturing process is done by scan lines, which need to be registered using GPS and inertial data. As an alternative, terrestrial LiDAR devices are more accessible but cover smaller areas, and their sampling strategy usually produces massive point clouds with over-represented plain regions. A more inexpensive option is street-level imagery. A dense set of images captured with a commodity camera can be fed to state-of-the-art multi-view stereo algorithms to produce realistic-enough reconstructions. One other advantage of this approach is capturing high-quality color data, whereas the geometric information is usually lacking. In this thesis, we analyze in-depth some of the shortcomings of these data-acquisition methods and propose new ways to overcome them. Mainly, we focus on the technologies that allow high-quality digitization of individual buildings. These are terrestrial LiDAR for geometric information and street-level imagery for color information. Our main goal is the processing and completion of detailed 3D urban representations. For this, we will work with multiple data sources and combine them when possible to produce models that can be inspected in real-time. Our research has focused on the following contributions: - Effective and feature-preserving simplification of massive point clouds. - Developing normal estimation algorithms explicitly designed for LiDAR data. - Low-stretch panoramic representation for point clouds. - Semantic analysis of street-level imagery for improved multi-view stereo reconstruction. - Color improvement through heuristic techniques and the registration of LiDAR and imagery data. - Efficient and faithful visualization of massive point clouds using image-based techniques.Durant els darrers anys, hi ha hagut un creixement notori en el camp de la digitalització d'edificis en 3D i entorns urbans. La millora substancial tant del maquinari d'escaneig com dels algorismes de reconstrucció ha portat al desenvolupament de representacions d'edificis i ciutats que es poden transmetre i inspeccionar remotament en temps real. Entre les aplicacions que implementen aquestes tecnologies hi ha diversos navegadors GPS i globus virtuals com Google Earth o les eines proporcionades per l'Institut Cartogràfic i Geològic de Catalunya. En particular, en aquesta tesi, conceptualitzem les ciutats com una col·lecció d'edificis individuals. Per tant, ens centrem en el processament individual d'una estructura a la vegada, en lloc del processament a gran escala d'entorns urbans. Avui en dia, hi ha una àmplia diversitat de tecnologies de digitalització i la selecció de l'adequada és clau per a cada aplicació particular. Aproximadament, aquestes tècniques es poden agrupar en tres famílies principals: - Temps de vol (LiDAR terrestre i aeri). - Fotogrametria (imatges a escala de carrer, de satèl·lit i aèries). - Dades vectorials editades per humans (cadastre i altres fonts de mapes). Cadascun d'ells presenta els seus avantatges en termes d'àrea coberta, qualitat de les dades, cost econòmic i esforç de processament. Els dispositius LiDAR muntats en avió i en cotxe són òptims per escombrar àrees enormes, però adquirir i calibrar aquests dispositius no és una tasca trivial. A més, el procés de captura es realitza mitjançant línies d'escaneig, que cal registrar mitjançant GPS i dades inercials. Com a alternativa, els dispositius terrestres de LiDAR són més accessibles, però cobreixen àrees més petites, i la seva estratègia de mostreig sol produir núvols de punts massius amb regions planes sobrerepresentades. Una opció més barata són les imatges a escala de carrer. Es pot fer servir un conjunt dens d'imatges capturades amb una càmera de qualitat mitjana per obtenir reconstruccions prou realistes mitjançant algorismes estèreo d'última generació per produir. Un altre avantatge d'aquest mètode és la captura de dades de color d'alta qualitat. Tanmateix, la informació geomètrica resultant sol ser de baixa qualitat. En aquesta tesi, analitzem en profunditat algunes de les mancances d'aquests mètodes d'adquisició de dades i proposem noves maneres de superar-les. Principalment, ens centrem en les tecnologies que permeten una digitalització d'alta qualitat d'edificis individuals. Es tracta de LiDAR terrestre per obtenir informació geomètrica i imatges a escala de carrer per obtenir informació sobre colors. El nostre objectiu principal és el processament i la millora de representacions urbanes 3D amb molt detall. Per a això, treballarem amb diverses fonts de dades i les combinarem quan sigui possible per produir models que es puguin inspeccionar en temps real. La nostra investigació s'ha centrat en les següents contribucions: - Simplificació eficaç de núvols de punts massius, preservant detalls d'alta resolució. - Desenvolupament d'algoritmes d'estimació normal dissenyats explícitament per a dades LiDAR. - Representació panoràmica de baixa distorsió per a núvols de punts. - Anàlisi semàntica d'imatges a escala de carrer per millorar la reconstrucció estèreo de façanes. - Millora del color mitjançant tècniques heurístiques i el registre de dades LiDAR i imatge. - Visualització eficient i fidel de núvols de punts massius mitjançant tècniques basades en imatges

Visual Techniques for Geological Fieldwork Using Mobile Devices

Visual techniques in general and 3D visualisation in particular have seen considerable adoption within the last 30 years in the geosciences and geology. Techniques such as volume visualisation, for analysing subsurface processes, and photo-coloured LiDAR point-based rendering, to digitally explore rock exposures at the earth’s surface, were applied within geology as one of the first adopting branches of science. A large amount of digital, geological surface- and volume data is nowadays available to desktop-based workflows for geological applications such as hydrocarbon reservoir exploration, groundwater modelling, CO2 sequestration and, in the future, geothermal energy planning. On the other hand, the analysis and data collection during fieldwork has yet to embrace this ”digital revolution”: sedimentary logs, geological maps and stratigraphic sketches are still captured in each geologist’s individual fieldbook, and physical rocks samples are still transported to the lab for subsequent analysis. Is this still necessary, or are there extended digital means of data collection and exploration in the field ? Are modern digital interpretation techniques accurate and intuitive enough to relevantly support fieldwork in geology and other geoscience disciplines ? This dissertation aims to address these questions and, by doing so, close the technological gap between geological fieldwork and office workflows in geology. The emergence of mobile devices and their vast array of physical sensors, combined with touch-based user interfaces, high-resolution screens and digital cameras provide a possible digital platform that can be used by field geologists. Their ubiquitous availability increases the chances to adopt digital workflows in the field without additional, expensive equipment. The use of 3D data on mobile devices in the field is furthered by the availability of 3D digital outcrop models and the increasing ease of their acquisition. This dissertation assesses the prospects of adopting 3D visual techniques and mobile devices within field geology. The research of this dissertation uses previously acquired and processed digital outcrop models in the form of textured surfaces from optical remote sensing and photogrammetry. The scientific papers in this thesis present visual techniques and algorithms to map outcrop photographs in the field directly onto the surface models. Automatic mapping allows the projection of photo interpretations of stratigraphy and sedimentary facies on the 3D textured surface while providing the domain expert with simple-touse, intuitive tools for the photo interpretation itself. The developed visual approach, combining insight from all across the computer sciences dealing with visual information, merits into the mobile device Geological Registration and Interpretation Toolset (GRIT) app, which is assessed on an outcrop analogue study of the Saltwick Formation exposed at Whitby, North Yorkshire, UK. Although being applicable to a diversity of study scenarios within petroleum geology and the geosciences, the particular target application of the visual techniques is to easily provide field-based outcrop interpretations for subsequent construction of training images for multiple point statistics reservoir modelling, as envisaged within the VOM2MPS project. Despite the success and applicability of the visual approach, numerous drawbacks and probable future extensions are discussed in the thesis based on the conducted studies. Apart from elaborating on more obvious limitations originating from the use of mobile devices and their limited computing capabilities and sensor accuracies, a major contribution of this thesis is the careful analysis of conceptual drawbacks of established procedures in modelling, representing, constructing and disseminating the available surface geometry. A more mathematically-accurate geometric description of the underlying algebraic surfaces yields improvements and future applications unaddressed within the literature of geology and the computational geosciences to this date. Also, future extensions to the visual techniques proposed in this thesis allow for expanded analysis, 3D exploration and improved geological subsurface modelling in general.publishedVersio

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

Sistemas automáticos de informação e segurança para apoio na condução de veículos

Doutoramento em Engenharia MecânicaO objeto principal desta tese é o estudo de algoritmos de processamento e representação automáticos de dados, em particular de informação obtida por sensores montados a bordo de veículos (2D e 3D), com aplicação em contexto de sistemas de apoio à condução. O trabalho foca alguns dos problemas que, quer os sistemas de condução automática (AD), quer os sistemas avançados de apoio à condução (ADAS), enfrentam hoje em dia. O documento é composto por duas partes. A primeira descreve o projeto, construção e desenvolvimento de três protótipos robóticos, incluindo pormenores associados aos sensores montados a bordo dos robôs, algoritmos e arquitecturas de software. Estes robôs foram utilizados como plataformas de ensaios para testar e validar as técnicas propostas. Para além disso, participaram em várias competições de condução autónoma tendo obtido muito bons resultados. A segunda parte deste documento apresenta vários algoritmos empregues na geração de representações intermédias de dados sensoriais. Estes podem ser utilizados para melhorar técnicas já existentes de reconhecimento de padrões, deteção ou navegação, e por este meio contribuir para futuras aplicações no âmbito dos AD ou ADAS. Dado que os veículos autónomos contêm uma grande quantidade de sensores de diferentes naturezas, representações intermédias são particularmente adequadas, pois podem lidar com problemas relacionados com as diversas naturezas dos dados (2D, 3D, fotométrica, etc.), com o carácter assíncrono dos dados (multiplos sensores a enviar dados a diferentes frequências), ou com o alinhamento dos dados (problemas de calibração, diferentes sensores a disponibilizar diferentes medições para um mesmo objeto). Neste âmbito, são propostas novas técnicas para a computação de uma representação multi-câmara multi-modal de transformação de perspectiva inversa, para a execução de correcção de côr entre imagens de forma a obter mosaicos de qualidade, ou para a geração de uma representação de cena baseada em primitivas poligonais, capaz de lidar com grandes quantidades de dados 3D e 2D, tendo inclusivamente a capacidade de refinar a representação à medida que novos dados sensoriais são recebidos.The main object of this thesis is the study of algorithms for automatic information processing and representation, in particular information provided by onboard sensors (2D and 3D), to be used in the context of driving assistance. The work focuses on some of the problems facing todays Autonomous Driving (AD) systems and Advanced Drivers Assistance Systems (ADAS). The document is composed of two parts. The first part describes the design, construction and development of three robotic prototypes, including remarks about onboard sensors, algorithms and software architectures. These robots were used as test beds for testing and validating the developed techniques; additionally, they have participated in several autonomous driving competitions with very good results. The second part of this document presents several algorithms for generating intermediate representations of the raw sensor data. They can be used to enhance existing pattern recognition, detection or navigation techniques, and may thus benefit future AD or ADAS applications. Since vehicles often contain a large amount of sensors of different natures, intermediate representations are particularly advantageous; they can be used for tackling problems related with the diverse nature of the data (2D, 3D, photometric, etc.), with the asynchrony of the data (multiple sensors streaming data at different frequencies), or with the alignment of the data (calibration issues, different sensors providing different measurements of the same object). Within this scope, novel techniques are proposed for computing a multicamera multi-modal inverse perspective mapping representation, executing color correction between images for obtaining quality mosaics, or to produce a scene representation based on polygonal primitives that can cope with very large amounts of 3D and 2D data, including the ability of refining the representation as new information is continuously received