Showing papers on "Delaunay triangulation published in 2003"

Coverage in wireless ad hoc sensor networks

Abstract: Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking. One of the fundamental problems in sensor networks is the calculation of the coverage. In Meguerdichian et al. (2001), it is assumed that the sensor has uniform sensing ability. We provide efficient distributed algorithms to optimally solve the best-coverage problem raised in the above-mentioned article. In addition, we consider a more general sensing model: the sensing ability diminishes as the distance increases. As energy conservation is a major concern in wireless (or sensor) networks, we also consider how to find an optimum best-coverage-path with the least energy consumption and how to find an optimum best-coverage-path that travels a small distance. In addition, we justify the correctness of the method proposed above that uses the Delaunay triangulation to solve the best coverage problem and show that the search space of the best coverage problem can be confined to the relative neighborhood graph, which can be constructed locally.

Localized Delaunay triangulation with application in ad hoc wireless networks

Abstract: Several localized routing protocols guarantee the delivery of the packets when the underlying network topology is a planar graph. Typically, relative neighborhood graph (RING) or Gabriel graph (GG) is used as such planar structure. However, it is well-known that the spanning ratios of these two graphs are not bounded by any constant (even for uniform randomly distributed points). Bose et al. (1999) recently developed a localized routing protocol that guarantees that the distance traveled by the packets is within a constant factor of the minimum if Delaunay triangulation of all wireless nodes is used, in addition, to guarantee the delivery of the packets. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we model the network as a unit-disk graph (UDG), in which a link uv exists only if the distance /spl par/uv/spl par/ is at most the maximum transmission range. In this paper, we present a novel localized networking protocol that constructs a planar 2 5-spanner of UDG, called the localized Delaunay triangulation (LDEL), as network topology. It contains all edges that are both in the unit-disk graph and the Delaunay triangulation of all nodes. The total communication cost of our networking protocol is O(n log n) bits, which is within a constant factor of the optimum to construct any structure in a distributed manner. Our experiments show that the delivery rates of some of the existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed topologies. Our simulations also show that the traveled distance of the packets is significantly less when the FACE routing algorithm is applied on LDEL, rather than applied on GG.

Geodesic Remeshing Using Front Propagation

Abstract: In this paper, we present a method for remeshing triangulated manifolds by using geodesic path calculations and distance maps. Our work builds on the Fast Marching algorithm, which has been extended to arbitrary meshes by Sethian and Kimmel. First, a set of points that are evenly spaced across the surface is automatically found. A geodesic Delaunay triangulation of the set of points is then created, using a Voronoi diagram construction based on Fast Marching. At last, we use the distance information to find a simple parameterization of the manifold. Marching algorithm makes this method computationally inexpensive, and gives very good results. Examples are shown for synthetic and real surfaces.

0-Efficient Triangulations of 3-Manifolds

Abstract: 0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S3, ℝP3 or L(3, 1). Similarly, any triangulation of a compact, orientable, irreducible, ∂-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, ∂-irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then Mо admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3-manifold can be modified to a 0-efficient ideal triangulation. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S3 and the triangulation has precisely two vertices. 0-efficient triangulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each boundary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds. © 2003 Applied Probability Trust.

Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations

Abstract: The centroidal Voronoi tessellation based Delaunay triangulation (CVDT) provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high-quality mesh In this paper, we discuss algorithms for the construction of the constrained CVDT from an initial Delaunay tetrahedral mesh of a three-dimensional domain By establishing an appropriate relationship between the density function and the specified sizing field and applying the Lloyd's iteration, the constrained CVDT mesh is obtained as a natural global optimization of the initial mesh Simple local operations such as edges/faces flippings are also used to further improve the CVDT mesh Several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method Copyright © 2003 John Wiley & Sons, Ltd

Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee

Abstract: The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to improve their approximations. Voronoi diagrams turn out to be useful for this approximation. Although it is known that Voronoi vertices for a sample of points from a curve in 2D approximate its medial axis, a similar result does not hold in 3D. Recently, it has been discovered that only a subset of Voronoi vertices converge to the medial axis as sample density approaches infinity. However, most applications need a nondiscrete approximation as opposed to a discrete one. To date no known algorithm can compute this approximation straight from the Voronoi diagram with a guarantee of convergence. We present such an algorithm and its convergence analysis in this paper. One salient feature of the algorithm is that it is scale and density independent. Experimental results corroborate our theoretical claims.

Complexity of the delaunay triangulation of points on surfaces the smooth case

Abstract: It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).

Provably good surface sampling and approximation

Abstract: We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3-dimensional Delaunay triangulation of the points. The method is very simple and works in 3d-space without requiring to parameterize the surface. Taking advantage of recent results on the restricted Delaunay triangulation, we prove that the algorithm can generate good samples on S as well as triangulated surfaces that approximate S. More precisely, we show that the restricted Delaunay triangulation Del\S of the points has the same topology type as S, that the Hausdorff distance between Del\S and S can be made arbitrarily small, and that we can bound the aspect ratio of the facets of Del\S. The algorithm has been implemented and we report on experimental results that provide evidence that it is very effective in practice. We present results on implicit surfaces, on CSG models and on polyhedra. Although most of our theoretical results are given for smooth closed surfaces, the method is quite robust in handling smooth surfaces with boundaries, and even non-smooth surfaces.

Parallel Delaunay mesh generation kernel

Abstract: We present the results of an evaluation study on the re-structuring of a latency-bound mesh generation algorithm into a latency-tolerant parallel kernel. We use concurrency at a fine-grain level to tolerate long, variable, and unpredictable latencies of remote data gather operations required for parallel guaranteed quality Delaunay triangulations. Our performance data from a 16 node SP2 and 32 node Cluster of Sparc Workstations suggest that more than 90% of the latency from remote data gather operations can be masked effectively at the cost of increasing communication overhead between 2 and 20% of the total run time. Despite the increase in the communication overhead the latency-tolerant mesh generation kernel we present in this paper can generate tetrahedral meshes for parallel field solvers eight to nine times faster than the traditional approach. Copyright © 2003 John Wiley & Sons, Ltd.

Watermarking 2D vector maps in the mesh-spectral domain

Abstract: The paper proposes a digital watermarking algorithm for 2D vector digital maps The watermark is a robust, informed-detection watermark to be used to prevent such abuses as an intellectual property rights violation The algorithm proposed in the paper embeds watermarks in the frequency-domain representation of a 2D vector digital map Our method treats vertices in the map as a point set, and imposes connectivity among the points by using Delaunay triangulation The method then computes the mesh-spectral coefficients (Karni, 2000) from the mesh created Modifications of the coefficients according to the message bits, and inverse transforming the coefficients back into the coordinate domain produces the watermarked map Our evaluation experiments showed that the watermark produced by the method is resistant against additive random noise, similarity transformation, vertex insertion and removal It is also resistant, to some extent, against cropping Compared to our previous algorithm (Ohbuchi, 2002), the algorithm described in this paper showed significantly improved attack resiliency

A fast approach for accurate content-adaptive mesh generation

Abstract: Mesh modeling is an important problem with many applications in image processing. A key issue in mesh modeling is how to generate a mesh structure that well represents an image by adapting to its content. We propose a new approach to mesh generation, which is based on a theoretical result derived on the error bound of a mesh representation. In the proposed method, the classical Floyd-Steinberg error-diffusion algorithm is employed to place mesh nodes in the image domain so that their spatial density varies according to the local image content. Delaunay triangulation is next applied to connect the mesh nodes. The result of this approach is that fine mesh elements are placed automatically in regions of the image containing high-frequency features while coarse mesh elements are used to represent smooth areas. The proposed algorithm is noniterative, fast, and easy to implement. Numerical results demonstrate that, at very low computational cost, the proposed approach can produce mesh representations that are more accurate than those produced by several existing methods. Moreover, it is demonstrated that the proposed algorithm performs well with images of various kinds, even in the presence of noise.

A hybrid approach for computing visual hulls of complex objects

Abstract: This paper addresses the problem of computing visual hulls from image contours. We propose a new hybrid approach, which overcomes the precision-complexity trade-off inherent to voxel based approaches by taking advantage of surface based approaches. To this aim, we introduce a space discretization, which does not rely on a regular grid where most cells are ineffective, but rather on an irregular grid where sample points lie on the surface of the visual hull. Such a grid is composed of tetrahedral cells obtained by applying a Delaunay triangulation on the sample points. These cells are carved afterward according to image silhouette information. The proposed approach keeps the robustness of volumetric approaches while drastically improving their precision and reducing their time and space complexities. It thus allows modeling of objects with complex geometry, and it also makes real time feasible for precise models. Preliminary results with synthetic and real data are presented.

The flow complex: a data structure for geometric modeling

Abstract: Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

A geometric convection approach of 3-D reconstruction

Abstract: This paper introduces a fast and efficient algorithm for surface reconstruction. As many algorithms of this kind, it produces a piecewise linear approximation of a surface S from a finite, sufficiently dense, subset of its points. Originally, the starting point of this work does not come from the computational geometry field. It is inspired by an existing numerical scheme of surface convection developed by Zhao, Osher and Fedkiw. We have translated this scheme to make it depend on the geometry of the input data set only, and not on the precision of some grid around the surface. Our algorithm deforms a closed oriented pseudo-surface embedded in the 3D Delaunay triangulation of the sampled points, and the reconstructed surface consists of a set of oriented facets located in this 3D Delaunay triangulation. This paper provides an appropriate data structure to represent a pseudo-surface, together with operations that manage deformations and topological changes. The algorithm can handle surfaces with boundaries, surfaces of high genus and, unlike most of the other existing schemes, it does not involve a global heuristic. Its complexity is that of the 3D Delaunay triangulation of the points. We present some results of the method, which turns out to be efficient even on noisy input data.

Adaptive whole Earth tomography

Abstract: [1] An approach for seismic tomography is presented which allows the parameterization to be refined during the inversion. The objective is to use the data to refine the mesh and the velocity model together, and hence both are considered part of the solution. Some simple rules are used to identify the volumes of a three-dimensional model in need of refinement. The self-adaptive parameterization is applied to an initial mesh built from uniformly distributed spherical triangles and Delaunay tetrahedra. Application of the technique to a typical summary ray P-wave arrival time data set shows it to be both feasible and practical for large scale whole Earth tomography. A noticeable trend in the resulting models, as the parameterization is refined, is the thinning of the Farallon and Tethys subduction features imaged in the mid mantle, together with an increase in amplitude of the velocity perturbation.

?Ultimate? robustness in meshing an arbitrary polyhedron

Abstract: Given a boundary surface mesh (a set of triangular facets) of a polyhedron, the problem of deciding whether or not a triangulation exists is reported to be NP-hard. In this paper, an algorithm to triangulate a general polyhedron is presented which makes use of a classical Delaunay triangulation algorithm, a phase for recovering the missing boundary facets by means of facet partitioning, and a final phase that makes it possible to remove the additional points defined in the previous step. Following this phase, the resulting mesh conforms to the given boundary surface mesh. The proposed method results in a discussion of theoretical interest about existence and complexity issues. In practice, however, the method should provide what we call ‘ultimate’ robustness in mesh generation methods. Copyright © 2003 John Wiley & Sons, Ltd.

Updating and constructing constrained delaunay and constrained regular triangulations by flips

Abstract: I discuss algorithms based on bistellar flips for inserting and deleting constraining (d - 1)-facets in d-dimensional constrained Delaunay triangulations (CDTs) and weighted CDTs, also known as constrained regular triangulations. The facet insertion algorithm is likely to outperform other known algorithms on most inputs. The facet deletion algorithm is the first proposed for d > 2, short of recomputing the CDT from scratch. An incremental facet insertion algorithm that begins with an unconstrained Delaunay triangulation can construct the CDT of a ridge-protected piecewise linear complex with nv vertices in O(nv[d / 2] + 1 log nv) time. Hence, in odd dimensions, CDT construction by incremental facet insertion is within a factor of log nv of worst-case optimal. Perhaps the most important feature of these algorithms is that they are relatively easy to implement.

Density estimators in particle hydrodynamics DTFE versus regular SPH

Abstract: We present the results of a study comparing density maps reconstructed by the Delaunay Tessellation Field Estimator (DTFE) and by regular SPH kernel-based techniques The density maps are constructed from the outcome of an SPH particle hydrodynamics simulation of a multiphase interstellar medium The comparison between the two methods clearly demonstrates the superior performance of the DTFE with respect to conventional SPH methods, in particular at locations where SPH appears to fail Filamentary and sheetlike structures form telling examples The DTFE is a fully self-adaptive technique for reconstructing continuous density fields from discrete particle distributions, and is based upon the corresponding Delaunay tessellation Its principal asset is its complete independence of arbitrary smoothing functions and parameters specifying the properties of these As a result it manages to faithfully reproduce the anisotropies of the local particle distribution and through its adaptive and local nature proves to be optimally suited for uncovering the full structural richness in the density distribution Through the improvement in local density estimates, calculations invoking the DTFE will yield a much better representation of physical processes which depend on density This will be crucial in the case of feedback processes, which play a major role in galaxy and star formation The presented results form an encouraging step towards the application and insertion of the DTFE in astrophysical hydrocodes We describe an outline for the construction of a particle hydrodynamics code in which the DTFE replaces kernel-based methods Further discussion addresses the issue and possibilities for a moving grid-based hydrocode invoking the DTFE, and Delaunay tessellations, in an attempt to combine the virtues of the Eulerian and Lagrangian approaches

Perturbations and vertex removal in a 3D delaunay triangulation

Abstract: Though Delaunay triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional Delaunay triangulation is still a problem in practice.We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations. The solution is available in CGAL.

Approximating geometric bottleneck shortest paths

Abstract: In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: Given two points p and q of S and a real number L, compute (or approximate) a shortest path in the subgraph of the complete graph on S consisting of all edges whose length is less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on minimum spanning trees, spanners, the Delaunay triangulation, and planar separators.

Delaunay triangulation algorithms useful for multibeam echosounding

Abstract: The Delaunay triangulation is a widely appreciated and investigated mathematical model for topographic surface represen- tation. After a brief theoretical description, six possible basic algorithms to construct a Delaunay triangulation are analyzed and properties that can be exploited for multibeam echosounder data processing are investigated. Two concepts will be treated in more depth: the divide-and-conquer construction algorithm and the incremental method. The calculation speed of the divide-and-conquer method makes it an ideal candidate to construct the initial triangulation of multibeam data. Its runtime performance is compared to that of the incremental algorithm to demonstrate this. The algorithm's merge step appears to be useful also in replacing triangulated areas of existing triangula- tions by new data. The incremental algorithm does not seem an effective construction method but it can easily be adapted to accommodate insertion of individual vertices into an existing triangulation and as such it is useful for editing purposes.

Graded conforming Delaunay tetrahedralization with bounded radius-edge ratio

Abstract: We propose an algorithm to compute a conforming Delaunay mesh of a polyhedral domain. Arbitrarily small input angles are allowed. The output mesh is graded and has bounded radius-edge ratio everywhere.

Properties of the Delaunay Triangulation

Abstract: We defined several functionals on the set of all triangulations of the finite system of points in d-space achieving global minimum on the Delaunay triangulation (DT). We consider a so called "parabolic" functional and prove it attains its minimum on DT in all dimensions. As the second example we treat "mean radius" functional (mean of circumcircle radii of triangles) for planar triangulations. As the third example we treat a so called "harmonic" functional. For a triangle this functional equals the sum of squares of sides over area. Finally, we consider a discrete analog of the Dirichlet functional. DT is optimal for these functionals only in dimension two.

Topology design for free space optical networks

Abstract: In this paper, we consider the problem of designing a topology for deploying a free space optical (FSO) link based network. The problem is to create a topology with strong connectivity and short diameter with uniform degree bounds on each node. Two centralized approaches are presented. The first approach constructs a backbone network by Delaunay triangulation. The basic structure is then refined to meet the design objectives. The second approach called the closest neighbor (CN) algorithm constructs a degree constrained minimum weight spanning tree. The tree is developed into a network with good connectivity and small diameter by forming edges with the closest neighbors. We prove that the CN algorithm forms a connected network. Through simulation and analysis we also show that this approach results in high reliability and small diameter.

Efficient Exact Geometric Predicates for Delaunay Triangulations

Abstract: A time efficient implementation of the exact computation paradigm relies on arithmetic filters which are used to speed up the exact computation of easy instances of the geometric predicates. Depending of what is called ``easy instances'', we usually classify filters as static or dynamic and also some in between categories often called semi-static. In this paper, we propose, in the context of three dimensional Delaunay triangulations: --- automatic tools for the writing of static and semi-static filters, --- a new semi-static level of filtering called translation filter, --- detailed benchmarks of the success rates of these filters and comparison with rounded arithmetic, long integer arithmetic and filters provided in Shewchuk's predicates. Our method is general and can be applied to all geometric predicates on points that can be expressed as signs of polynomial expressions. This work is applied in the CGAL library.