Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

A GPU accelerated algorithm for 3D Delaunay triangulation

Abstract: We propose the first algorithm to compute the 3D Delaunay triangulation (DT) on the GPU. Our algorithm uses massively parallel point insertion followed by bilateral flipping, a powerful local operation in computational geometry. Although a flipping algorithm is very amenable to parallel processing and has been employed to construct the 2D DT and the 3D convex hull on the GPU, to our knowledge there is no such successful attempt for constructing the 3D DT. This is because in 3D when many points are inserted in parallel, flipping gets stuck long before reaching the DT, and thus any further correction to obtain the DT is costly. In contrast, we show that by alternating between parallel point insertion and flipping, together with picking an appropriate point insertion order, one can still obtain a triangulation very close to Delaunay. We further propose an adaptive star splaying approach to subsequently transform this result into the 3D DT efficiently. In addition, we introduce several GPU speedup techniques for our implementation, which are also useful for general computational geometry algorithms. On the whole, our hybrid approach, with the GPU accelerating the main work of constructing a near-Delaunay structure and the CPU transforming that into the 3D DT, outperforms all existing sequential CPU algorithms by up to an order of magnitude, in both synthetic and real-world inputs. We also adapt our approach to the 2D DT problem and obtain similar speedup over the best sequential CPU algorithms, and up to 2 times over previous GPU algorithms.

Structural Analysis of the SDSS Cosmic Web I.Nonlinear Density Field Reconstructions

Abstract: We investigate the ability of three reconstruction techniques to analyze and investigate weblike features and geometries in a discrete distribution of objects. The three methods are the linear Delaunay Tessellation Field Estimator (DTFE), its higher order equivalent Natural Neighbour Field Estimator (NNFE) and a version of Kriging interpolation adapted to the specific circumstances encountered in galaxy redshift surveys, the Natural Lognormal Kriging technique. DTFE and NNFE are based on the local geometry defined by the Voronoi and Delaunay tessellations of the galaxy distribution. The three reconstruction methods are analysed and compared using mock magnitude-limited and volume-limited SDSS redshift surveys, obtained on the basis of the Millennium simulation. We investigate error trends, biases and the topological structure of the resulting fields, concentrating on the void population identified by the Watershed Void Finder. Environmental effects are addressed by evaluating the density fields on a range of Gaussian filter scales. Comparison with the void population in the original simulation yields the fraction of false void mergers and false void splits. In most tests DTFE, NNFE and Kriging have largely similar density and topology error behaviour. Cosmetically, higher order NNFE and Kriging methods produce more visually appealing reconstructions. Quantitatively, however, DTFE performs better, even while computationally far less demanding. A successful recovery of the void population on small scales appears to be difficult, while the void recovery rate improves significantly on scales > 3 h-1Mpc. A study of small scale voids and the void galaxy population should therefore be restricted to the local Universe, out to at most 100 h-1Mpc.

Reconstruction using witness complexes

Abstract: We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew's surface meshing algorithm, with one notable difference: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.

Pavement Crack Classification via Spatial Distribution Features

Abstract: Pavement crack types provide important information for making pavement maintenance strategies. This paper proposes an automatic pavement crack classification approach, exploiting the spatial distribution features (i.e., direction feature and density feature) of the cracks under a neural network model. In this approach, a direction coding (D-Coding) algorithm is presented to encode the crack subsections and extract the direction features, and a Delaunay Triangulation technique is employed to analyze the crack region structure and extract the density features. As regarding skeletonized crack sections rather than crack pixels, the spatial distribution features hold considerable feature significance for each type of cracks. Empirical study indicates a classification precision of over 98% of the proposed approach.

ADMESH: An advanced, automatic unstructured mesh generator for shallow water models

Abstract: In this paper, we present the development and application of a two-dimensional, automatic unstructured mesh generator for shallow water models called Admesh. Starting with only target minimum and maximum element sizes and points defining the boundary and bathymetry/ topography of the domain, the goal of the mesh generator is to automatically produce a high-quality mesh from this minimal set of input. From the geometry provided, properties such as local features, curvature of the boundary, bathymetric/topographic gradients, and approximate flow characteristics can be extracted, which are then used to determine local element sizes. The result is a high-quality mesh, with the correct amount of refinement where it is needed to resolve all the geometry and flow characteristics of the domain. Techniques incorporated include the use of the so-called signed distance function, which is used to determine critical geometric properties, the approximation of piecewise linear coastline data by smooth cubic splines, a so-called mesh function used to determine element sizes and control the size ratio of neighboring elements, and a spring-based force equilibrium approach used to improve the element quality of an initial mesh obtained from a simple Delaunay triangulation. Several meshes of shallow water domains created by the new mesh generator are presented.