Improved Incremental Randomized Delaunay Triangulation

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the levels below. Point location is done by marching in a triangulation to determine the nearest neighbor of the query at that level, then the march restarts from that neighbor at the level below. Using a small sample (3%) allows a small memory occupation; the march and the use of the nearest neighbor to change levels quickly locate the query.Comment: 19 pages, 7 figures Proc. 14th Annu. ACM Sympos. Comput. Geom., 106--115, 199

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

Octilinear Force-Directed Layout with Mental Map Preservation for Schematic Diagrams

We present an algorithm for automatically laying out metro map style schematics using a force-directed approach, where we use a localized version of the standard spring embedder forces combined with an octilinear magnetic force. The two types of forces used during layout are naturally conflicting, and the existing method of simply combining these to generate a resultant force does not give satisfactory results. Hence we vary the forces, emphasizing the standard forces in the beginning to produce a well distributed graph, with the octilinear forces becoming prevalent at the end of the layout, to ensure that the key requirement of line angles at intervals of 45? is obtained. Our method is considerably faster than the more commonly used search-based approaches, and we believe the results are superior to the previous force-directed approach. We have further developed this technique to address the issues of dynamic schematic layout. We use a Delaunay triangulation to construct a schematic “frame”, which is used to retain relative node positions and permits full control of the level of mental map preservation. This technique is the first to combine mental map preservation techniques with the additional layout criteria of schematic diagrams. To conclude, we present the results of a study to investigate the relationship between the level of mental map preservation and the user response time and accuracy

Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element (FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients

Efficient moving point handling for incremental 3D manifold reconstruction

As incremental Structure from Motion algorithms become effective, a good sparse point cloud representing the map of the scene becomes available frame-by-frame. From the 3D Delaunay triangulation of these points, state-of-the-art algorithms build a manifold rough model of the scene. These algorithms integrate incrementally new points to the 3D reconstruction only if their position estimate does not change. Indeed, whenever a point moves in a 3D Delaunay triangulation, for instance because its estimation gets refined, a set of tetrahedra have to be removed and replaced with new ones to maintain the Delaunay property; the management of the manifold reconstruction becomes thus complex and it entails a potentially big overhead. In this paper we investigate different approaches and we propose an efficient policy to deal with moving points in the manifold estimation process. We tested our approach with four sequences of the KITTI dataset and we show the effectiveness of our proposal in comparison with state-of-the-art approaches.Comment: Accepted in International Conference on Image Analysis and Processing (ICIAP 2015

Localization in Unstructured Environments: Towards Autonomous Robots in Forests with Delaunay Triangulation

Autonomous harvesting and transportation is a long-term goal of the forest industry. One of the main challenges is the accurate localization of both vehicles and trees in a forest. Forests are unstructured environments where it is difficult to find a group of significant landmarks for current fast feature-based place recognition algorithms. This paper proposes a novel approach where local observations are matched to a general tree map using the Delaunay triangularization as the representation format. Instead of point cloud based matching methods, we utilize a topology-based method. First, tree trunk positions are registered at a prior run done by a forest harvester. Second, the resulting map is Delaunay triangularized. Third, a local submap of the autonomous robot is registered, triangularized and matched using triangular similarity maximization to estimate the position of the robot. We test our method on a dataset accumulated from a forestry site at Lieksa, Finland. A total length of 2100\,m of harvester path was recorded by an industrial harvester with a 3D laser scanner and a geolocation unit fixed to the frame. Our experiments show a 12\,cm s.t.d. in the location accuracy and with real-time data processing for speeds not exceeding 0.5\,m/s. The accuracy and speed limit is realistic during forest operations

A Variational r-Adaption and Shape-Optimization Method for Finite-Deformation Elasticity

This paper is concerned with the formulation of a variational r-adaption method for finite-deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configuration forces for isoparametric elements and nonlinear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi-infinite crack in linear and nonlinear elastic bodies; and the optimization of the shape of elastic inclusions

Advancing In Situ Modeling of ICMEs: New Techniques for New Observations

It is generally known that multi-spacecraft observations of interplanetary coronal mass ejections (ICMEs) more clearly reveal their three-dimensional structure than do observations made by a single spacecraft. The launch of the STEREO twin observatories in October 2006 has greatly increased the number of multipoint studies of ICMEs in the literature, but this field is still in its infancy. To date, most studies continue to use on flux rope models that rely on single track observations through a vast, multi-faceted structure, which oversimplifies the problem and often hinders interpretation of the large-scale geometry, especially for cases in which one spacecraft observes a flux rope, while another does not. In order to tackle these complex problems, new modeling techniques are required. We describe these new techniques and analyze two ICMEs observed at the twin STEREO spacecraft on 22-23 May 2007, when the spacecraft were separated by ~8 degrees. We find a combination of non-force-free flux rope multi-spacecraft modeling, together with a new non-flux rope ICME plasma flow deflection model, better constrains the large-scale structure of these ICMEs. We also introduce a new spatial mapping technique that allows us to put multispacecraft observations and the new ICME model results in context with the convecting solar wind. What is distinctly different about this analysis is that it reveals aspects of ICME geometry and dynamics in a far more visually intuitive way than previously accomplished. In the case of the 22-23 May ICMEs, the analysis facilitates a more physical understanding of ICME large-scale structure, the location and geometry of flux rope sub-structures within these ICMEs, and their dynamic interaction with the ambient solar wind