On the number of higher order Delaunay triangulations

"Vegeu el resum a l'inici del document del fitxer adjunt"

One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

Feature-sensitive and Adaptive Image Triangulation: A Super-pixel-based Scheme for Image Segmentation and Mesh Generation

With increasing utilization of various imaging techniques (such as CT, MRI and PET) in medical fields, it is often in great need to computationally extract the boundaries of objects of interest, a process commonly known as image segmentation. While numerous approaches have been proposed in literature on automatic/semi-automatic image segmentation, most of these approaches are based on image pixels. The number of pixels in an image can be huge, especially for 3D imaging volumes, which renders the pixel-based image segmentation process inevitably slow. On the other hand, 3D mesh generation from imaging data has become important not only for visualization and quantification but more critically for finite element based numerical simulation. Traditionally image-based mesh generation follows such a procedure as: (1) image boundary segmentation, (2) surface mesh generation from segmented boundaries, and (3) volumetric (e.g., tetrahedral) mesh generation from surface meshes. These three majors steps have been commonly treated as separate algorithms/steps and hence image information, once segmented, is not considered any more in mesh generation. In this thesis, we investigate a super-pixel based scheme that integrates both image segmentation and mesh generation into a single method, making mesh generation truly an image-incorporated approach. Our method, called image content-aware mesh generation, consists of several main steps. First, we generate a set of feature-sensitive, and adaptively distributed points from 2D grayscale images or 3D volumes. A novel image edge enhancement method via randomized shortest paths is introduced to be an optional choice to generate the features’ boundary map in mesh node generation step. Second, a Delaunay-triangulation generator (2D) or tetrahedral mesh generator (3D) is then utilized to generate a 2D triangulation or 3D tetrahedral mesh. The generated triangulation (or tetrahedralization) provides an adaptive partitioning of a given image (or volume). Each cluster of pixels within a triangle (or voxels within a tetrahedron) is called a super-pixel, which forms one of the nodes of a graph and adjacent super-pixels give an edge of the graph. A graph-cut method is then applied to the graph to define the boundary between two subsets of the graph, resulting in good boundary segmentations with high quality meshes. Thanks to the significantly reduced number of elements (super-pixels) as compared to that of pixels in an image, the super-pixel based segmentation method has tremendously improved the segmentation speed, making it feasible for real-time feature detection. In addition, the incorporation of image segmentation into mesh generation makes the generated mesh well adapted to image features, a desired property known as feature-preserving mesh generation

Algorithms for Triangles, Cones & Peaks

Three different geometric objects are at the center of this dissertation: triangles, cones and peaks. In computational geometry, triangles are the most basic shape for planar subdivisions. Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc. We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set. Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of $k$ cones drawn around it. They are used to aid the construction of Euclidean minimum spanning trees or in wireless networks for topology control and routing. We present the first implementation of an optimal $\mathcal{O}(n \log n)$-time sweepline algorithm to construct Yao graphs. One metric to quantify the importance of a mountain peak is its isolation. Isolation measures the distance between a peak and the closest point of higher elevation. Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms. We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

Real-time Photorealistic Visualisation of Large-scaleMultiresolution Terrain Models

Height field terrain rendering is an important aspect of GIS, outdoor virtual reality applicationssuch as flight simulation, 3-D games, etc. A polygonal model of very large terrain data requiresa large number of triangles. So, even most high-performance graphics workstations have greatdifficulty to display even moderately sized height fields at interactive frame rates. To bringphotorealism in visualisation, it is required to drape corresponding high-resolution satellite oraerial phototexture over 3-D digital terrain and also to place multiple collections of point-location-based static objects such as buildings, trees, etc and to overlay polyline vector objects suchas roads on top of the terrain surface. It further complicates the requirement of interactive framerates while navigation over the terrain. This paper describes a novel approach for objects andterrain visualisation by combination of two algorithms, one for terrain data and the other forobjects. The terrain rendering is accomplished by an efficient dynamic multiresolution view-dependent level-of-detail mesh simplification algorithm. It is augmented with out-of-corevisualisation of large-height geometry and phototexture terrain data populated with 3-D/2-Dstatic objects as well as vector overlays without extensive memory load. The proposedmethodology provides interactive frame rates on a general-purpose desktop PC with OpenGL-enabled graphics hardware. The software TREND has been successfully tested on different real-world height maps and satellite phototextures of sizes up to 16K*16K coupled with thousandsof static objects and polyline vector overlays

An Urban-Conscious Rapid Wind Downscaling Model for Early Design Stages

Assessments of urban contexts using existing microclimate models mostly fall short, when considering topographies along with complex layouts of buildings and streets, regardless of their significant influences on building performances and outdoor environments. The challenge exists mainly due to modelâ??s inherent complexities and the associated high computational costs. This becomes especially challenging at early design stages when time, expertise, and computational resources are limited, even though the opportunities for performance enhancement are greater than at later stages. This dissertation develops a wind downscaling model that can rapidly assess urban contexts to relate climate data in a large spatial resolution for a smaller-scale site. Surrounding slopes and terrains, up to a few kilometers in diameter, are considered to predict wind pressure on the volumetric boundary of a neighborhood and local wind speed. The new model strives for prediction accuracy and computational efficiency by employing the capacities of a computational fluid dynamics (CFD) simulation and of an existing mathematical method. The proposed model is composed of three parts: pressure database, speed database, and interpolation. The databases store wind data for existing urban contexts that are generated with CFD simulations. Using the databases, the interpolation approximates the pressure outcomes for a new urban context; thus, real-time CFD runs can be avoided for the model users. Independent development of data for pressure and speed facilitates the flexibility and expandability of the model. The proposed model showed an acceptable prediction accuracy, with average errors of less than 10%, compared to the full-scale CFD simulation for the same territorial scope. An exceptional computational efficiency is also shown, with a runtime in 0.308 seconds, which is 16568 times faster than the CFD simulation. This rate allows creation of a yearlong prediction in a few tens of minutes with a personal desktop computer. For non-experts, the pertinence of the model is enhanced with a limited number of parameters, making it easily adaptable during early design stages of buildings and urban design scales. Geometric sensitivities are embedded for incremental study, which is crucial to finding optimal solutions, toward more efficient, yet healthier, urban environments