Showing papers on "Delaunay triangulation published in 2013"

TetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 --- User's Manual)

Abstract: TetGen is a program to generate tetrahedral meshes from 3d polyhedral domains. Its goal is to generate good quality tetrahedral meshes suitable for various applications in scientific computing. It can be used as either a standalone program or a library component integrated into other software. The purpose of this document is to give a brief explanation of the kind of tetrahedralizations and meshing problems handled by TetGen and to give fairly detailed documentation about the usage of the program. Readers will learn how to create tetrahedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained. keywords: tetrahedral mesh generation, Delaunay tetrahedralization, weighted Delaunay triangulation, constrained Delaunay tetrahedralization, mesh quality, mesh refinement, mesh adaption, mesh coarsening AMS Classification: 65M50, 65N50

Line Assisted Light Field Triangulation and Stereo Matching

Abstract: Light fields are image-based representations that use densely sampled rays as a scene description. In this paper, we explore geometric structures of 3D lines in ray space for improving light field triangulation and stereo matching. The triangulation problem aims to fill in the ray space with continuous and non-overlapping simplices anchored at sampled points (rays). Such a triangulation provides a piecewise-linear interpolant useful for light field super-resolution. We show that the light field space is largely bilinear due to 3D line segments in the scene, and direct triangulation of these bilinear subspaces leads to large errors. We instead present a simple but effective algorithm to first map bilinear subspaces to line constraints and then apply Constrained Delaunay Triangulation (CDT). Based on our analysis, we further develop a novel line-assisted graph-cut (LAGC) algorithm that effectively encodes 3D line constraints into light field stereo matching. Experiments on synthetic and real data show that both our triangulation and LAGC algorithms outperform state-of-the-art solutions in accuracy and visual quality.

Surface Reconstruction through Point Set Structuring

Abstract: We present a method for reconstructing surfaces from point sets. The main novelty lies in a structure-preserving approach where the input point set is first consolidated by structuring and resampling the planar components, before reconstructing the surface from both the consolidated components and the unstructured points. The final surface is obtained through solving a graph-cut problem formulated on the 3D Delaunay triangulation of the structured point set where the tetrahedra are labeled as inside or outside cells. Structuring facilitates the surface reconstruction as the point set is substantially reduced and the points are enriched with structural meaning related to adjacency between primitives. Our approach departs from the common dichotomy between smooth/piecewise-smooth and primitive-based representations by gracefully combining canonical parts from detected primitives and free-form parts of the inferred shape. Our experiments on a variety of inputs illustrate the potential of our approach in terms of robustness, flexibility and efficiency.

Variational Anisotropic Surface Meshing with Voronoi Parallel Linear Enumeration

Abstract: This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvature-adapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d or larger). In this high dimensional space, the mesh is optimized by computing a Centroidal Voronoi Tessellation (CVT), i.e. the minimizer of a C 2 objective function that depends on the coordinates at the vertices (quantization noise power). Optimizing this objective function requires to compute the intersection between the (higher dimensional) Voronoi cells and the surface (Restricted Voronoi Diagram). The method overcomes the d-factorial cost of computing a Voronoi diagram of dimension d by directly computing the restricted Voronoi cells with a new algorithm that can be easily parallelized (Vorpaline: Voronoi Parallel Linear Enumeration). The method is demonstrated with several examples comprising CAD and scanned meshes.

Efficient computation of clipped Voronoi diagram for mesh generation

Abstract: The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partially outside of the domain, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm to compute the clipped Voronoi diagram for a set of sites with respect to a compact 2D region or a 3D volume. We also apply the proposed method to optimal mesh generation based on the centroidal Voronoi tessellation.

The Stretch Factor of the Delaunay Triangulation Is Less than 1.998

Abstract: Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The stretch factor (also known as dilation or spanning ratio) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long-standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation is less than $\rho = 1.998$, significantly improving the current best upper bound of 2.42 by Keil and Gutwin [``The Delaunay triangulation closely approximates the complete Euclidean graph,” in Proceedings of the 1st Workshop on Algorithms and Data Structures (WADS), 1989, pp. 47--56]. Our bound of 1.998 also improves the upper bound of the best stretch factor that can be achieved by a plane spanner of a Euclidean graph (the current best upper bound is 2). Our result has a direc...

Particle-based anisotropic surface meshing

Abstract: This paper introduces a particle-based approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metric-adapted mesh. The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one, called "embedding space". The vertices of the mesh are generated by uniformly sampling the surface in this higher dimensional embedding space, and the sampling is further regularized by optimizing an energy function with a quasi-Newton algorithm. All the computations can be re-expressed in terms of the dot product in the embedding space, and the Jacobian matrices of the mappings that connect different spaces. This transform makes it unnecessary to explicitly represent the coordinates in the embedding space, and also provides all necessary expressions of energy and forces for efficient computations. Through energy optimization, it naturally leads to the desired anisotropic particle distributions in the original space. The triangles are then generated by computing the Restricted Anisotropic Voronoi Diagram and its dual Delaunay triangulation. We compare our results qualitatively and quantitatively with the state-of-the-art in anisotropic surface meshing on several examples, using the standard measurement criteria.

Map Generalization with a Triangulated Data Structure

Abstract: Automation of map generalization requires facilities to monitor the spatial relationships and interactions among multiple map objects An experimental map generalization system has been developed which addresses this issue by representing spatial objects within a simplicial data structure (SDS) based on constrained Delaunay triangulation of the source data Geometric generalization operators that have been implemented include object exaggeration, collapse, amalgamation, boundary reduction and displacement The generalization operators exploit a set of primitive SDS functions to determine topological and proximal relationships, measure map objects, apply transformations, and detect and resolve spatial conflicts Proximal search functions are used for efficient analysis of the structure and dimensions of the intervening spaces between map objects Because geometric generalization takes place within a fully triangulated representation of the map surface, the presence of overlap conflicts, resulting from indi

Generating haptic texture models from unconstrained tool-surface interactions

Abstract: If you pick up a tool and drag its tip across a table, a rock, or a swatch of fabric, you are able to feel variations in the textures even though you are not directly touching them. These vibrations are characteristic of the material and the motions made when interacting with the surface. This paper presents a new method for creating haptic texture models from data recorded during natural and unconstrained motions using a new haptic recording device. The recorded vibration data is parsed into short segments that represent the feel of the surface at the associated tool force and speed. We create a low-order auto-regressive (AR) model for each data segment and construct a Delaunay triangulation of models in force-speed space for each surface. During texture rendering, we stably interpolate between these models using barycentric coordinates and drive the interpolated model with white noise to output synthetic vibrations. Our methods were validated through application to data recorded by eight human subjects and the experimenter interacting with six textures. We present a new spectral metric for determining perceptual match of the models in order to evaluate the effectiveness and consistency of the segmenting and modeling approach. Multidimensional scaling (MDS) on the pairwise differences in the synthesized vibrations shows that the 54 created texture models cluster by texture in a two-dimensional perceptual space.

A frontal Delaunay quad mesh generator using the L∞norm

Abstract: A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well known algorithm of the graph theory, namely the Blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time. Then, the triangulation itself is taylored with the aim of producing right triangles in the domain. This is done using the infinity norm to compute distances in the meshing process. The alignement of the triangles is controlled by a cross field that is defined on the domain. Meshes constructed this way have their points aligned with the cross field direction and their triangles are almost right everywhere. Then, recombination with our Blossom-based approach yields quadrilateral meshes of excellent quality.

The natural radial element method

Abstract: SUMMARY In this work an innovative numerical approach is proposed, which combines the simplicity of low-order finite elements connectivity with the geometric flexibility of meshless methods. The natural neighbour concept is applied to enforce the nodal connectivity. Resorting to the Delaunay triangulation a background integration mesh is constructed, completely dependent on the nodal mesh. The nodal connectivity is imposed through nodal sets with reduce size, reducing significantly the test function construction cost. The interpolations functions, constructed using Euclidian norms, are easily obtained. To prove the good behaviour of the proposed interpolation function several data-fitting examples and first-order partial differential equations are solved. The proposed numerical method is also extended to the elastostatic analysis, where classic solid mechanics benchmark examples are solved. Copyright © 2013 John Wiley & Sons, Ltd.

Head-related transfer function interpolation in azimuth, elevation, and distance.

Abstract: Although distance-dependent head-related transfer function (HRTF) databases provide interesting possibilities, e.g., for rendering virtual sounds in the near-field, there is a lack of algorithms and tools to make use of them. Here, a framework is proposed for interpolating HRTF measurements in 3-D (i.e., azimuth, elevation, and distance) using tetrahedral interpolation with barycentric weights. For interpolation, a tetrahedral mesh is generated via Delaunay triangulation and searched via an adjacency walk, making the framework robust with respect to irregularly positioned HRTF measurements and computationally efficient. An objective evaluation of the proposed framework indicates good accordance between measured and interpolated near-field HRTFs.

An intelligent fused approach for face recognition

Abstract: Abstract Face detection plays important roles in many applications such as human–computer interaction, security and surveillance, face recognition, etc. This article presents an intelligent enhanced fused approach for face recognition based on the Voronoi diagram (VD) and wavelet moment invariants. Discrete wavelet transform and moment invariants are used for feature extraction of the facial face. Finally, VD and the dual tessellation (Delaunay triangulation, DT) are used to locate and detect original face images. Face recognition results based on this new fusion are promising in the state of the art.

Incremental Surface Extraction from Sparse Structure-from-Motion Point Clouds

Abstract: In this paper we propose a new method to incrementally extract a surface from a consecutively growing Structure-from-Motion (SfM) point cloud in real-time. Our method is based on a Delaunay triangulation (DT) on the 3D points. The core idea is to robustly label all tetrahedra into freeand occupied space using a random field formulation and to extract the surface as the interface between differently labeled tetrahedra. For this reason, we propose a new energy function that achieves the same accuracy as state-of-the-art methods but reduces the computational effort significantly. Furthermore, our new formulation allows us to extract the surface in an incremental manner, i. e. whenever the point cloud is updated we adapt our energy function. Instead of minimizing the updated energy with a standard graph cut, we employ the dynamic graph cut of Kohli et al. [1] which enables efficient minimization of a series of similar random fields by re-using the previous solution. In such a way we are able to extract the surface from an increasingly growing point cloud nearly independent of the overall scene size. Energy Function for Surface Extraction Our method formulates surface extraction as a binary labeling problem, with the goal of assigning each tetrahedron either a free or occupied label. For this reason, we model the probabilities that a tetrahedron is free- or occupied space analyzing the set of rays that connect all 3D points to image features. Following the idea of the truncated signed distance function (TSDF), which is known from voxel-based surface reconstructions, a tetrahedron in front of a 3D point X has a high probability to be free space, whereas a tetrahedron behind X is presumably occupied space. We further assume that it is very unlikely that neighboring tetrahedra obtain different labels, except for pairs of tetrahedra that have a ray through the face connecting both. Such a labeling problem can be elegantly formulated as a pairwise random field and since our priors are submodular, we can efficiently find a global optimal labeling solution e. g. using graph cuts. In contrast to existing methods like [2], our energy depends only on the visibility information that is directly connected to the four 3D points that span the tetrahedraVi. Hence a modification of the tetrahedral structure by inserting new points has only limited effect on the energy function. This property enables us to easily adopt the energy function to a modified tetrahedral structure. Incremental Surface Extraction To enable efficient incremental surface reconstruction, our method has to consecutively integrate new scene information (3D points as well as visibility information) in the energy function and to minimize the modified energy efficiently. Integrating new visibility information, i. e. adding rays for newly available 3D points, affects only those terms of the energy function that relate

A robust 2D point-sequence curve offset algorithm with multiple islands for contour-parallel tool path

Abstract: An offset algorithm is important to the contour-parallel tool path generation process. Usually, it is necessary to offset with islands. In this paper a new offset algorithm for a 2D point-sequence curve (PS-curve) with multiple islands is presented. The algorithm consists of three sub-processes, the islands bridging process, the raw offset curve generation and the global invalid loops removal. The input of the algorithm is a set of PS-curves, in which one of them is the outer profile and the others are islands. The bridging process bridges all the islands to the outer profile with the Delaunay triangulation method, forming a single linked PS-curve. With the fact that local problems are caused by intersections of adjacent bisectors, the concept of stuck circle is proposed. Based on stuck circle, local problems are fixed by updating the original profile with the proposed basic rule and append rule, so that a raw offset curve can be generated. The last process first reports all the self-intersections on the raw offset PS-curve, and then a procedure called tree analysis puts all the self-intersections into a tree. All the points between the nodes in even depth and its immediate children are collected using the collecting rule. The collected points form the valid loops, which is the output of the proposed algorithm. Each sub-process can be complete in near linear time, so the whole algorithm has a near linear time complexity. This can be proved by the examples tested in the paper.

Graph Induced Complex on Point Data

Abstract: The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is huge--particularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the {\em graph induced complex} that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.

Forward modeling of gravity data using finite-volume and finite-element methods on unstructured grids

Abstract: Minimum-structure inversion is one of the most effective tools for the inversion of gravity data. However, the standard Gauss-Newton algorithms that are commonly used for the minimization procedure and that employ forward solvers based on analytic formulas require large memory storage for the formation and inversion of the involved matrices. An alternative to the analytical solvers are numerical ones that result in sparse matrices. This sparsity suits gradient-based minimization methods that avoid the explicit formation of the inversion matrices and that solve the system of equations using memory-efficient iterative techniques. We have developed several numerical schemes for the forward modeling of gravity data using the finite-element and finite-volume methods on unstructured grids. In the finite-volume method, a Delaunay tetrahedral grid and its dual Voronoi grid are used to find the primary solution (i.e., gravitational potential) at the centers and vertices of the tetrahedra, respectively (cel...

Sparse Non-Negative Stencils for Anisotropic Diffusion

Abstract: We introduce a new discretization scheme for Anisotropic Diffusion, AD-LBR, on two and three dimensional cartesian grids. The main features of this scheme is that it is non-negative, and has a stencil cardinality bounded by 6 in 2D, by 14 in 3D, despite allowing diffusion tensors of arbitrary anisotropy. Our scheme also has good spectral properties, which permits larger time steps and avoids e.g. chessboard artifacts. AD-LBR relies on Lattice Basis Reduction, a tool from discrete mathematics which has recently shown its relevance for the discretization on grids of strongly anisotropic Partial Differential Equations. We prove that AD-LBR is in 2D asymptotically equivalent to a finite element discretization on an anisotropic Delaunay triangulation, a procedure more involved and computationally expensive. Our scheme thus benefits from the theoretical guarantees of this procedure, for a fraction of its cost. Numerical experiments in 2D and 3D illustrate our results.

A Node Deployment Algorithm Based on Van Der Waals Force in Wireless Sensor Networks

Abstract: The effectiveness of wireless sensor networks (WSN) depends on the regional coverage provided by node deployment, which is one of the key topics in WSN. Virtual force-based algorithms (VFA) are popular approaches for this problem. In VFA, all nodes are seen as points subject to repulsive and attractive force exerted among them and can move according to the calculated force. In this paper, a sensor deployment algorithm for mobile WSN based on van der Waals force is proposed. Friction force is introduced into the equation of force, the relationship of adjacency of nodes is defined by Delaunay triangulation, and the force calculated produce acceleration for nodes to move. An evaluation metric called pair correlation function is introduced here to evaluate the uniformity of the node distribution. Simulation results and comparisons have showed that the proposed approach has higher coverage rate, more uniformity in configuration, and moderate convergence time compared to some other virtual force algorithms.

Computing 2D Constrained Delaunay Triangulation Using the GPU

Abstract: We propose the first graphics processing unit (GPU) solution to compute the 2D constrained Delaunay triangulation (CDT) of a planar straight line graph (PSLG) consisting of points and edges. There are many existing CPU algorithms to solve the CDT problem in computational geometry, yet there has been no prior approach to solve this problem efficiently using the parallel computing power of the GPU. For the special case of the CDT problem where the PSLG consists of just points, which is simply the normal Delaunay triangulation (DT) problem, a hybrid approach using the GPU together with the CPU to partially speed up the computation has already been presented in the literature. Our work, on the other hand, accelerates the entire computation on the GPU. Our implementation using the CUDA programming model on NVIDIA GPUs is numerically robust, and runs up to an order of magnitude faster than the best sequential implementations on the CPU. This result is reflected in our experiment with both randomly generated PSLGs and real-world GIS data having millions of points and edges.

Visual-Based Human Crowds Behavior Analysis Based on Graph Modeling and Matching

Abstract: Modeling human crowds is an important issue for video surveillance and is a challenging task due to their unpredictable behavior. In this paper, the position of an isolated region that comprises an individual person or a set of occluded persons is detected by background subtraction. Each isolated region is considered a vertex and a human crowd is thus modeled by a graph. To construct a graph, Delaunay triangulation is used to systematically connect vertices and therefore the problem of event detection of human crowds is formulated as measuring the topology variation of consecutive graphs in temporal order. To effectively model the topology variations, local characteristics, such as triangle deformations and eigenvalue-based subgraph analysis, and global features, such as moments, are used and are finally combined as an indicator to detect if any anomalies of human crowd(s) present in the scene. Experimental results obtained by using extensive dataset show that our system is effective in detecting anomalous events for uncontrolled environment of surveillance videos.

How the Spatial Position of Individuals Affects Their Influence on Swarms: A Numerical Comparison of Two Popular Swarm Dynamics Models

Abstract: Schools of fish and flocks of birds are examples of self-organized animal groups that arise through social interactions among individuals. We numerically study two individual-based models, which recent empirical studies have suggested to explain self-organized group animal behavior: (i) a zone-based model where the group communication topology is determined by finite interacting zones of repulsion, attraction, and orientation among individuals; and (ii) a model where the communication topology is described by Delaunay triangulation, which is defined by each individual's Voronoi neighbors. The models include a tunable parameter that controls an individual's relative weighting of attraction and alignment. We perform computational experiments to investigate how effectively simulated groups transfer information in the form of velocity when an individual is perturbed. A cross-correlation function is used to measure the sensitivity of groups to sudden perturbations in the heading of individual members. The results show how relative weighting of attraction and alignment, location of the perturbed individual, population size, and the communication topology affect group structure and response to perturbation. We find that in the Delaunay-based model an individual who is perturbed is capable of triggering a cascade of responses, ultimately leading to the group changing direction. This phenomenon has been seen in self-organized animal groups in both experiments and nature.

Technical note: Delaunay Hodge star

Abstract: We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.

Indexing and retrieving in fingerprint databases under structural distortions

Abstract: This paper presents a new algorithm for fingerprint indexing, which is based on minutia triplets, and it is very tolerant to missing and spurious minutiae. In this sense, a novel representation for fingerprints is proposed by defining a triangle set based on extensions of Delaunay triangulations. Moreover, a set of robust features is used to build indices. Finally, a recovery method based on calculating the recommendation score is introduced, using a new similarity function between geometric transformations. Our proposal was tested on well known databases, showing that it outperforms most of the already reported methods, especially under conditions of distortions.

Parallel algorithms for planar and spherical Delaunay construction with an application to centroidal Voronoi tessellations

Abstract: . A new algorithm, featuring overlapping domain decompositions, for the parallel construction of Delaunay and Voronoi tessellations is developed. Overlapping allows for the seamless stitching of the partial pieces of the global Delaunay tessellations constructed by individual processors. The algorithm is then modified, by the addition of stereographic projections, to handle the parallel construction of spherical Delaunay and Voronoi tessellations. The algorithms are then embedded into algorithms for the parallel construction of planar and spherical centroidal Voronoi tessellations that require multiple constructions of Delaunay tessellations. This combination of overlapping domain decompositions with stereographic projections provides a unique algorithm for the construction of spherical meshes that can be used in climate simulations. Computational tests are used to demonstrate the efficiency and scalability of the algorithms for spherical Delaunay and centroidal Voronoi tessellations. Compared to serial versions of the algorithm and to STRIPACK-based approaches, the new parallel algorithm results in speedups for the construction of spherical centroidal Voronoi tessellations and spherical Delaunay triangulations.