Showing papers on "Delaunay triangulation published in 2015"

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

Abstract: TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

Copy–Move Forgery Detection by Matching Triangles of Keypoints

Abstract: Copy–move forgery is one of the most common types of tampering for digital images Detection methods generally use block-matching approaches, which first divide the image into overlapping blocks and then extract and compare features to find similar ones, or point-based approaches, in which relevant keypoints are extracted and matched to each other to find similar areas In this paper, we present a very novel hybrid approach, which compares triangles rather than blocks, or single points Interest points are extracted from the image, and objects are modeled as a set of connected triangles built onto these points Triangles are matched according to their shapes (inner angles), their content (color information), and the local feature vectors extracted onto the vertices of the triangles Our methods are designed to be robust to geometric transformations Results are compared with a state-of-the-art block matching method and a point-based method Furthermore, our data set is available for use by academic researchers

Offshore wind farm electrical cable layout optimization

Abstract: This article explores an automated approach for the efficient placement of substations and the design of an inter-array electrical collection network for an offshore wind farm through the minimization of the cost. To accomplish this, the problem is represented as a number of sub-problems that are solved in series using a combination of heuristic algorithms. The overall problem is first solved by clustering the turbines to generate valid substation positions. From this, a navigational mesh pathfinding algorithm based on Delaunay triangulation is applied to identify valid cable paths, which are then used in a mixed-integer linear programming problem to solve for a constrained capacitated minimum spanning tree considering all realistic constraints. The final tree that is produced represents the solution to the inter-array cable problem. This method is applied to a planned wind farm to illustrate the suitability of the approach and the resulting layout that is generated.

Finite element mesh generation

Abstract: Introduction Finite element method What is finite element mesh generation? Why finite element mesh generation? Problem definition, scope and philosophy, science or art? General strategies, robustness, difficulties and methodologies Mathematics Historical development So far achieved and what lies ahead Topics discussed in the chapters Fundamentals Introduction Notations, symbols and abbreviations Terminologies and data structures Geometrical operations and formulas Topological operations and algorithms Sorting Background grid Mesh generation on planar domain Introduction Structured mesh on planar domain Unstructured mesh on planar domain Meshing by quadtree decomposition Delaunay triangulation (DT) Advancing front approach Meshing by a combined scheme of DT and ADF approach Enhanced quadtree meshing Quadrilateral mesh Mesh generation over curved surfaces Introduction Parametric mapping method Mesh generation by packing ellipses Direct mesh generation on surface Mesh generation by surface intersection Quadrilateral surface mesh Mesh generation in three dimensions Introduction Delaunay triangulation (3D) Boundary recovery for 3D DT Boundary protection in DT Generation of tetrahedral mesh by ADF approach Delaunay-ADF meshing Generation of tetrahedral mesh by sphere packing Generation of hexahedral mesh Mesh optimisation Introduction Shape measure and quality coefficient Optimisation by shifting of nodes Optimisation by topological operations Mesh generation by parallel processing Introduction Fundamentals and strategies Parallel Delaunay triangulation in 2D Parallel Delaunay triangulation in 3D Partition of discretised surface for parallel processing Auxiliary meshing techniques Surface verification and preparation Multi-grid insertion of non-uniform point distributions (2D) Multi-grid insertion of non-uniform point distributions (3D) Mesh generation and adaptation by edge refinement Meshing volume bounded by analytical curved surfaces Merging of tetrahedral meshes Merging of hexahedral meshes Curvilinear finite element mesh Adaptive refinement analysis References Appendix Index

CGALmesh: A Generic Framework for Delaunay Mesh Generation

Abstract: CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes—surface triangular meshes or volume tetrahedral meshes—from input surfaces, 3D domains, and 3D multidomains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality and on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.

Time–Frequency Filtering Based on Spectrogram Zeros

Abstract: For a proper choice of the analysis window, a short-time Fourier transform is known to be completely characterized by its zeros, which coincide with those of the associated spectrogram. A simplified representation of the time-frequency structure of a signal can therefore be given by the Delaunay triangulation attached to spectrogram zeros. In the case of multicomponent nonstationary signals embedded in white Gaussian noise, it turns out that each time–frequency domain attached to a given component can be viewed as the union of adjacent Delaunay triangles whose edge length is an outlier as compared to the distribution in noise-only regions. Identifying such domains offers a new way of disentangling the different components in the time–frequency plane, as well as of reconstructing the corresponding waveforms.

A New Technique for Multi-Oriented Scene Text Line Detection and Tracking in Video

Abstract: Text detection and tracking in video is challenging due to contrast, resolution and background variations, and different orientations and text movements. In addition, the presence of both caption and scene texts in video aggravates the problem because these two text types differ in characteristics significantly . This paper proposes a new technique for detecting and tracking video texts of any orientation by using spatial and temporal information, respectively. The technique explores gradient directional symmetry at component level for smoothing edge components before text detection. Spatial information is preserved by forming Delaunay triangulation in a novel way at this level, which results in text candidates. Text characteristics are then proposed in a different way for eliminating false text candidates , which results in potential text candidates. Then grouping is proposed for combining potential text candidates regardless of orientation based on the nearest neighbor criterion. To tackle the problems of multi-font and multi-sized texts, we propose multi-scale integration by a pyramid structure, which helps in extracting full text lines. Then, the detected text lines are tracked in video by matching the subgraphs of triangulation. Experimental results for text detection and tracking on our video dataset, the benchmark video datasets, and the natural scene image benchmark datasets show that the proposed method is superior to the state-of-the-art methods in terms of recall, precision , and F-measure.

Efficient construction and simplification of Delaunay meshes

Abstract: Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a model-dependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK log K) time complexity. Delaunay meshes have many favorable geometric and numerical properties. For example, a DM has exactly the same geometry as the input mesh, and it can be encoded by any mesh data structure. Moreover, the empty geodesic circumcircle property implies that the commonly used cotangent Laplace-Beltrami operator has non-negative weights. Therefore, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. We observe that DMs can improve the accuracy of the heat method for computing geodesic distances. Also, popular parameterization techniques, such as discrete harmonic mapping, produce more stable results on the DMs than on the input meshes.

Complex Root Finding Algorithm Based on Delaunay Triangulation

Abstract: A simple and flexible algorithm for finding zeros of a complex function is presented. An arbitrary-shaped search region can be considered and a very wide class of functions can be analyzed, including those containing singular points or even branch cuts. The proposed technique is based on sampling the function at nodes of a regular or a self-adaptive mesh and on the analysis of the function sign changes. As a result, a set of candidate points is created, where the signs of the real and imaginary parts of the function change simultaneously. To verify and refine the results, an iterative algorithm is applied. The validity of the presented technique is supported by the results obtained in numerical tests involving three different types of functions.

Proximity-based grouping of buildings in urban blocks: a comparison of four algorithms

Abstract: Grouping of buildings based on proximity is a pre-processing step of urban pattern (structure) recognition for contextual cartographic generalization. This paper presents a comparison of grouping algorithms for polygonal buildings in urban blocks. Four clustering algorithms, Minimum Spanning Tree (MST), Density-Based Spatial Clustering Application with Noise (DBSCAN), CHAMELEON and Adaptive Spatial Clustering based on Delaunay Triangulation (ASCDT) are reviewed and analysed to detect building groups. The success of the algorithms is evaluated based on group distribution characteristics (i.e. distribution of the buildings in groups) with two methods: S_Dbw and newly proposed Cluster Assessment Circles. A proximity matrix of the nearest distances between the building polygons, and Delaunay triangulation of building vertices are created as an input for the algorithms. A topographic data-set at 1:25,000 scale is used for the experiments. Urban block polygons are created to constrain the clustering processes f...

Pressure-field extraction on unstructured flow data using a Voronoi tessellation-based networking algorithm: a proof-of-principle study

Abstract: A novel technique is described for pressure extraction from Lagrangian particle-tracking data. The technique uses a Poisson solver to extract the pressure field on a network of data nodes, which is constructed using the Voronoi tessellation and the Delaunay triangulation. The technique is demonstrated on two cases: synthetic Lagrangian data generated for the analytical case of Hill’s spherical vortex, and the flow in the wake behind a NACA 0012 which was impulsively accelerated to $$Re = 7{,}500$$ . The experimental data were collected using four-camera, three-dimensional particle-tracking velocimetry. For both the analytical case and the experimental case, the dependence of pressure-field error or sensitivity on the normalized spatial particle density was found to follow similar power-law relationships. It was shown that in order to resolve the salient flow structures from experimental data, the required particle density was an order of magnitude greater than for the analytical case. Furthermore, additional sub-structures continued to be identified in the experimental data as the particle density was increased. The increased density requirements of the experimental data were assumed to be due to a combination of phase-averaging error and the presence of turbulent coherent structures in the flow. Additionally, the computational requirements of the technique were assessed. It was found that in the current implementation, the computational requirements are slightly nonlinear with respect to the number of particles. However, the technique will remain feasible even as advancements in particle-tracking techniques in the future increase the density of Lagrangian data.

A Robust Delaunay Triangulation Matching for Multispectral/Multidate Remote Sensing Image Registration

Abstract: A novel dual-graph-based matching method is proposed in this letter particularly for the multispectral/multidate images with low overlapping areas, similar patterns, or large transformations. First, scale invariant feature transform based matching is improved by normalizing gradient orientations and maximizing the scale ratio similarity of all corresponding points. Next, Delaunay graphs are generated for outlier removal, and the candidate outliers are selected by comparing the distinction of Delaunay graph structures. In order to bring back the inliers removed in Delaunay triangulation matching iterations and to exclude the remaining outliers, the recovery strategy equipped with the dual graph of Delaunay is explored. Inliers located in the corresponding Voronoi cells are recovered to the residual sets. The experimental results demonstrate the accuracy and robustness of the proposed algorithm for various representative remote sensing images.

Anisotropic Delaunay Mesh Generation

Abstract: Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations are known to be well suited for interpolation of functions or solving PDEs. Assuming that the anisotropic shape requirements for mesh elements are given through a metric field varying over the domain, we propose a new approach to anisotropic mesh generation, relying on the notion of anisotropic Delaunay meshes. An anisotropic Delaunay mesh is defined as a mesh in which the star of each vertex $v$ consists of simplices that are Delaunay for the metric associated to vertex $v$. This definition works in any dimension and allows us to define a simple refinement algorithm. The algorithm takes as input a domain and a metric field and provides, after completion, an anisotropic mesh whose elements are sized and shaped according to the metric field.

Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

Abstract: We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ...

Discrete Conformal Deformation: Algorithm and Experiments

Abstract: In this paper, we introduce the definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, which is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle a surface with any topology, with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss--Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.

Anisotropic Delaunay Meshes of Surfaces

Abstract: Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface representation. Given a surface S endowed with a metric tensor field, we propose a new approach to generate an anisotropic mesh that approximates S with elements shaped according to the metric field. The algorithm relies on the well-established concepts of restricted Delaunay triangulation and Delaunay refinement and comes with theoretical guarantees. The star of each vertex in the output mesh is Delaunay for the metric attached to this vertex. Each facet has a good aspect ratio with respect to the metric specified at any of its vertices. The algorithm is easy to implement. It can mesh various types of surfaces like implicit surfaces, polyhedra, or isosurfaces in 3D images. It can handle complicated geometries and topologies, and very anisotropic metric fields.

A Non-parametric Approach to Shape Reconstruction from Planar Point Sets through Delaunay Filtering

Abstract: In this paper, we present a fully automatic Delaunay based sculpting algorithm for approximating the shape of a finite set of points S in R 2 . The algorithm generates a relaxed Gabriel graph ( R G G ) that consists of most of the Gabriel edges and a few non-Gabriel edges induced by the Delaunay triangulation. Holes are characterized through a structural pattern called as body-arm formed by the Delaunay triangles in the void regions. R G G is constructed through an iterative removal of Delaunay triangles subjected to circumcenter (of triangle) and topological regularity constraints in O ( n log n ) time using O ( n ) space. We introduce the notion of directed boundary samples which characterizes the two dimensional objects based on the alignment of their boundaries in the cavities. Theoretically, we justify our algorithm by showing that under given sampling conditions, the boundary of R G G captures the topological properties of objects having directed boundary samples. Unlike many other approaches, our algorithm does not require tuning of any external parameter to approximate the geometric shape of point set and hence human intervention is completely eliminated. Experimental evaluations of the proposed technique are done using L 2 error norm measure, which is the symmetric difference between the boundaries of reconstructed shape and the original shape. We demonstrate the efficacy of our automatic shape reconstruction technique by showing several examples and experiments with varying point set densities and distributions.

Incremental reconstruction of urban environments by Edge-Points Delaunay triangulation

Abstract: Urban reconstruction from a video captured by a surveying vehicle constitutes a core module of automated mapping. When computational power represents a limited resource and, a detailed map is not the primary goal, the reconstruction can be performed incrementally, from a monocular video, carving a 3D Delaunay triangulation of sparse points; this allows online incremental mapping for tasks such as traversability analysis or obstacle avoidance. To exploit the sharp edges of urban landscape, we propose to use a Delaunay triangulation of Edge-Points, which are the 3D points corresponding to image edges. These points constrain the edges of the 3D Delaunay triangulation to real-world edges. Besides the use of the Edge-Points, a second contribution of this paper is the Inverse Cone Heuristic that preemptively avoids the creation of artifacts in the reconstructed manifold surface. We force the reconstruction of a manifold surface since it makes it possible to apply computer graphics or photometric refinement algorithms to the output mesh. We evaluated our approach on four real sequences of the public available KITTI dataset by comparing the incremental reconstruction against Velodyne measurements.

An image-based technique for 3d building reconstruction using multi-view uav images

Abstract: Nowadays, with the development of the urban areas, the automatic reconstruction of the buildings, as an important objects of the city complex structures, became a challenging topic in computer vision and photogrammetric researches. In this paper, the capability of multi-view Unmanned Aerial Vehicles (UAVs) images is examined to provide a 3D model of complex building facades using an efficient image-based modelling workflow. The main steps of this work include: pose estimation, point cloud generation, and 3D modelling. After improving the initial values of interior and exterior parameters at first step, an efficient image matching technique such as Semi Global Matching (SGM) is applied on UAV images and a dense point cloud is generated. Then, a mesh model of points is calculated using Delaunay 2.5D triangulation and refined to obtain an accurate model of building. Finally, a texture is assigned to mesh in order to create a realistic 3D model. The resulting model has provided enough details of building based on visual assessment.

Reconstruction of water-tight surfaces through Delaunay sculpting

Abstract: Given a finite set of points S ? R 2 , we define a proximity graph called as shape-hull graph ( SHG ( S ) ) that contains all Gabriel edges and a few non-Gabriel edges of Delaunay triangulation of S . For any S , SHG ( S ) is topologically regular with its boundary (referred to as shape-hull ( SH )) homeomorphic to a simple closed curve. We introduce the concept of divergent concavity for simple, closed, planar curves based on the alignment of curves in concave portions and discuss various measures to characterize curves having divergent concavity. Under sufficiently dense sampling, we prove that SH ( S ) , where S is sampled from a divergent concave curve Σ D , represents a piece-wise linear approximation of Σ D . We extend this result to provide a sculpting algorithm for closed surface reconstruction from a set of raw samples. The surface is constructed through a repeated elimination of Delaunay tetrahedra subjected to circumcenter and topological constraints. Theoretically, we justify our algorithm by establishing a topological guarantee on the 3D shape-hull with the help of topological rules. We demonstrate the effectiveness of our approach with experimental results on models with sharp features and sparsely distributed point clouds. Compared to existing sculpting approaches for surface reconstruction that require either a parameter tuning or several stages, our approach is simple, non-parametric, single stage and reconstructs topologically correct piece-wise linear approximation for divergent concave surfaces. Delaunay-based surface reconstruction algorithm has been proposed.It is a non-parametric and single stage approach.Theoretical guarantee has been discussed.

Isotopic approximation within a tolerance volume

Abstract: We introduce in this paper an algorithm that generates from an input tolerance volume a surface triangle mesh guaranteed to be within the tolerance, intersection free and topologically correct. A pliant meshing algorithm is used to capture the topology and discover the anisotropy in the input tolerance volume in order to generate a concise output. We first refine a 3D Delaunay triangulation over the tolerance volume while maintaining a piecewise-linear function on this triangulation, until an isosurface of this function matches the topology sought after. We then embed the isosurface into the 3D triangulation via mutual tessellation, and simplify it while preserving the topology. Our approach extends to surfaces with boundaries and to non-manifold surfaces. We demonstrate the versatility and efficacy of our approach on a variety of data sets and tolerance volumes.