Two-dimensional Delaunay Triangulation Research Articles

Variable resolution Poisson-disk sampling for meshing discrete fracture networks

We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to generate point distributions for variable resolution Delaunay triangular and tetrahedral meshes in two and three-dimensions, respectively. nMAPS consists of two principal stages. In the first stage, an initial point distribution is produced using a cell-based rejection algorithm. In the second stage, holes in the sample are detected using an efficient background grid and filled in to obtain a near-maximal covering. Extensive testing shows that nMAPS generates a variable resolution mesh in linear run time with the number of accepted points. We demonstrate nMAPS capabilities by meshing three-dimensional discrete fracture networks (DFN) and the surrounding volume. The discretized boundaries of the fractures, which are represented as planar polygons, are used as the seed of 2D-nMAPS to produce a conforming Delaunay triangulation. The combined mesh of the DFN is used as the seed for 3D-nMAPS, which produces conforming Delaunay tetrahedra surrounding the network. Under a set of conditions that naturally arise in maximal Poisson-disk samples and are satisfied by nMAPS, the two-dimensional Delaunay triangulations are guaranteed to only have well-behaved triangular faces. While nMAPS does not provide triangulation quality bounds in more than two dimensions, we found that low-quality tetrahedra in 3D are infrequent, can be readily detected and removed, and a high quality balanced mesh is produced.

Bridge-surface panoramic-image generation for automated bridge-inspection using deepmatching

Visual inspection is important for the efficient maintenance of bridge structures and has recently been supplemented with the use of image-processing techniques that can localize and quantify damages using images captured from bridges. A series of overlapping bridge images can be combined for constructing a panoramic bridge-surface image in which the locations and sizes of the damages can be noted. Despite the excellent performance of image-processing techniques, generating panoramic images from a series of bridge-surface images is challenging as bridge-surface images may not possess distinct patterns or patterns that can act as reference feature points for stitching adjacent images. To address this issue, this paper presents a general method for stitching bridge-surface images using Deepmatching, which determines a pixel-wise correspondence between an image pair in comparison with conventional feature-wise matching methods. To employ Deepmatching for panoramic-image generation, (1) image matching pair search using 2D Delaunay triangulation, (2) parametric model for optimal image stitching were developed, and (3) field validation was conducted in this study. First, possible image matching pairs are organized using the two-dimensional Delaunay triangulation, and then Deepmatching is used to determine the matching points between possible image pairs. The developed parametric model refines the valid image matching pair, which is used for obtaining optimal global homographies for panoramic-image generation. For the validation of the proposed method, a lab-scale experiment on a flat concrete wall and a field experiment on a concrete bridge were conducted. The experimental validation demonstrates that the proposed method successfully identifies dense matching points between image pairs and generates a panoramic image while minimizing the occurrence of ghosting and drift.

Critical behavior of SIS model on two-dimensional quasiperiodic tilings

We investigated the critical behavior of SIS (susceptible-infected-susceptible) model on Penrose and Ammann–Beenker quasiperiodic tilings by means of numerical simulations and finite size scaling technique. We used the reactivation dynamics, which consists of inserting a spontaneous infected particle without contact in the system when infection dies out, to avoid the dynamics being trapped in the absorbing state. We obtained the mean infection density, its fluctuation and 5-order Binder ratio, in order to determine the universality class of the system. We showed that the system still obeys two-dimensional directed percolation universality class, in according to Harris-Barghathi-Vojta criterion, which states quasiperiodic order is irrelevant for this system and do not induce any change in the universality class. Our results are in agreement with a previous investigation of the contact process model on two-dimensional Delaunay triangulations, which still obeys directed percolation universality class according to the same criterion.

Triangulations from topologically correct digital Voronoi diagrams

We prove that the dual of the digital Voronoi diagram constructed by flooding the plane from the data points gives a geometrically and topologically correct dual triangulation. This provides the proof of correctness for recently developed GPU algorithms that outperform traditional CPU algorithms for constructing two-dimensional Delaunay triangulations.

A New Reverse Engineering Method for the 3D Surface Reconstruction without Contact

In this paper, a new reverse engineering method is proposed for 3d surface reconstruction without contact. This method is based on the combination of stereovision and structure light methods. The originality of the proposed technique is that the pattern is a two-dimensional Delaunay triangulation. A key characterizing the vertices is computed for each vertex. The pattern matching between two images is reduced to the comparison of two meshes. The three dimensional triangulation of the surface can then be obtained by using the geometric properties of the acquisition devices.

Density-independent model of self-propelled particles

We examine a density-independent modification of the Vicsek model in which a particle interacts with neighbors defined by Delaunay triangulation. To feasibly simulate the model, an algorithm for repairing the triangulation over time was developed. This algorithm may also be applied to any time varying two-dimensional Delaunay triangulation. This model exhibits a continuous phase transition with noise, and a distinct set of critical exponents were measured which satisfy a hyperscaling relationship. The critical exponents are found to vary between a low and high velocity regime, but they are robust under the inclusion of a repulsive interaction. We present evidence that the correlation length approximately scales with the size of the system even in the ordered phase.

Triangulation of cubic panorama for view synthesis

An unstructured triangulation approach, new to our knowledge, is proposed to apply triangular meshes for representing and rendering a scene on a cubic panorama (CP). It sophisticatedly converts a complicated three-dimensional triangulation into a simple three-step triangulation. First, a two-dimensional Delaunay triangulation is individually carried out on each face. Second, an improved polygonal triangulation is implemented in the intermediate regions of each of two faces. Third, a cobweblike triangulation is designed for the remaining intermediate regions after unfolding four faces to the top/bottom face. Since the last two steps well solve the boundary problem arising from cube edges, the triangulation with irregular-distribution feature points is implemented in a CP as a whole. The triangular meshes can be warped from multiple reference CPs onto an arbitrary viewpoint by face-to-face homography transformations. The experiments indicate that the proposed triangulation approach provides a good modeling for the scene with photorealistic rendered CPs.

Vertex removal in two-dimensional Delaunay triangulation: Speed-up by low degrees optimization

The theoretical complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O ( d ) , O ( d log d ) , or O ( d 2 ) . In fact, the asymptotic complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3 ⩽ d ⩽ 7 , it improves the global behavior of the removal for random points by more than 45%.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nic...

A probabilistic result on multi-dimensional Delaunay triangulations, and its application to the 2D case

This paper exploits the notion of “unfinished site”, introduced by Katajainen and Koppinen (1998) in the analysis of a two-dimensional Delaunay triangulation algorithm, based on a regular grid. We generalize the notion and its properties to any dimension k⩾2: in the case of uniform distributions, the expected number of unfinished sites in a k-rectangle is O(N 1−1/k) . This implies, under some specific assumptions, the linearity of a class of divide-and-conquer schemes based on balanced k-d trees. This general result is then applied to the analysis of a new algorithm for constructing Delaunay triangulations in the plane. According to Su and Drysdale (1995, 1997), the best known algorithms for this problem run in linear expected time, thanks in particular to the use of bucketing techniques to partition the domain. In our algorithm, the partitioning is based on a 2-d tree instead, the construction of which takes Θ(N logN) time, and we show that the rest of the algorithm runs in linear expected time. This “preprocessing” allows the algorithm to adapt efficiently to irregular distributions, as the domain is partitioned using point coordinates, as opposed to a fixed, regular basis (buckets or grid). We checked that even for the largest data sets that could fit in internal memory (over 10 million points), constructing the 2-d tree takes noticeably less CPU time than triangulating the data. With this in mind, our algorithm is only slightly slower than the reputedly best algorithms on uniform distributions, and is even the most efficient for data sets of up to several millions of points distributed in clusters.

Finite Element Approximation of the Diffusion Operator on Tetrahedra

Linear Galerkin finite element discretizations of the Laplace operator produce nonpositive stiffness coefficients for internal element edges of two-dimensional Delaunay triangulations. This property, also called the positive transmissibility (PT) condition, is a prerequisite for the existence of an M-matrix and ensures that nonphysical local extrema are not present in the solution. For tetrahedral elements, it has already been shown that the linear Galerkin approach does not in general satisfy the PT condition. We propose a modification of the three-dimensional Galerkin scheme that, if a Delaunay triangulation is used, satisfies the PT condition for internal edges and, if further conditions on the boundary are specified, yields an M-matrix.The proposed approach can also be extended to the general diffusion operator with nonconstant or anisotropic coefficients.

A fast algorithm for generating constrained delaunay triangulations

A fast algorithm for generating constrained two-dimensional Delaunay triangulations is described. The scheme permits certain edges to be specified in the final triangulation, such as those that correspond to domain boundaries or natural interfaces, and is suitable for mesh generation and contour plotting applications. Detailed timing statistics indicate that its CPU time requirement is roughly proportional to the number of points in the data set. Subject to the conditions imposed by the edge constraints, the Delaunay scheme automatically avoids the formation of long thin triangles and thus gives high quality grids. A major advantage of the method is that it does not require extra points to be added to the data set in order to ensure that the specified edges are present.

An implementation of Watson's algorithm for computing 2-dimensional delaunay triangulations

A FORTRAN 77 implementation of Watson's algorithm for computing two-dimensional Delaunay triangulations is described. The algorithm is shown to have an asymptotic time complexity bound which is better than O(N 1.5) by applying it to collections of N points generated randomly within the unit square. The computer code obeys strict FORTRAN 77 syntax. Excluding the memory needed to store the co-ordinates of the points, it requires slightly greater than 9N integer words of memory to assemble and store the Delaunay triangulation.