Showing papers on "Voronoi diagram published in 2003"
01 Jan 2003
TL;DR: A unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes, using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method is proposed.
Abstract: This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.
2,003 citations
TL;DR: The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the L/sub p/ and chamfer metrics.
Abstract: A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a k-dimensional binary image in time linear in the total number of voxels N. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the L/sub p/ and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O(N) time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O(N/p) time with p processors.
907 citations
TL;DR: In this article, adaptive spatial binning of Integral-Field Spectroscopic (IFS) data to reach a chosen constant signal-to-noise ratio per bin is presented.
Abstract: We present new techniques to perform adaptive spatial binning of Integral-Field Spectroscopic (IFS) data to reach a chosen constant signal-to-noise ratio per bin. These methods are required for the proper analysis of IFS observations, but can also be used for standard photometric imagery or any other two-dimensional data. Various schemes are tested and compared by binning and extracting the stellar kinematics of the Sa galaxy NGC2273 from spectra obtained with the panoramic IFS SAURON.
770 citations
01 Sep 2003
TL;DR: In this paper, a set of points that are evenly spaced across the surface is automatically found, and then a geodesic Delaunay triangulation of the points is created, using a Voronoi diagram construction based on Fast Marching.
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.
183 citations
08 Dec 2003
TL;DR: This paper proposes a novel approach to tracking moving objects and their identity using noisy, sparse information collected by id-sensors such as infrared and ultrasound badge systems and demonstrates that EM-based learning of behavior patterns increases the tracking performance and provides valuable information for high-level behavior recognition.
Abstract: Tracking the activity of people in indoor environments has gained considerable attention in the robotics community over the last years. Most of the existing approaches are based on sensors, which allow to accurately determining the locations of people but do not provide means to distinguish between different persons. In this paper we propose a novel approach to tracking moving objects and their identity using noisy, sparse information collected by id-sensors such as infrared and ultrasound badge systems. The key idea of our approach is to use particle filters to estimate the locations of people on the Voronoi graph of the environment. By restricting particles to a graph, we make use of the inherent structure of indoor environments. The approach has two key advantages. First, it is by far more efficient and robust than unconstrained particle filters. Second, the Voronoi graph provides a natural discretization of human motion, which allows us to apply unsupervised learning techniques to derive typical motion patterns of the people in the environment. Experiments using a robot to collect ground-truth data indicate the superior performance of Voronoi tracking. Furthermore, we demonstrate that EM-based learning of behavior patterns increases the tracking performance and provides valuable information for high-level behavior recognition.
163 citations
08 Jun 2003
TL;DR: An algorithm is developed for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20A, as measured from the skewed perspective of any point in the triangle.
Abstract: We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteed-quality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionality---most notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20A, as measured from the skewed perspective of any point in the triangle.
163 citations
TL;DR: Algorithms for the construction of the constrained CVDT from an initial Delaunay tetrahedral mesh of a three-dimensional domain are discussed, establishing an appropriate relationship between the density function and the specified sizing field and applying the Lloyd's iteration.
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
153 citations
TL;DR: A nondiscrete approximation straight from the Voronoi diagram with a guarantee of convergence of the medial axis of a surface in 3D is presented and its convergence analysis is presented.
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.
137 citations
22 Oct 2003
TL;DR: This paper presents a signed distance transform algorithm using graphics hardware, which computes the scalar valued function of the Euclidean distance to a given manifold of co-dimension one, if the manifold is closed and orientable.
Abstract: This paper presents a signed distance transform algorithm using graphics hardware, which computes the scalar valued function of the Euclidean distance to a given manifold of co-dimension one. If the manifold is closed and orientable, the distance has a negative sign on one side of the manifold and a positive sign on the other. Triangle meshes are considered for the representation of a two-dimensional manifold and the distance function is sampled on a regular Cartesian grid. In order to achieve linear complexity in the number of grid points, to each primitive we assign a simple polyhedron enclosing its Voronoi cell. Voronoi cells are known to contain exactly all points that lay closest to its corresponding primitive. Thus, the distance to the primitive only has to be computed for grid points inside its polyhedron. Although Voronoi cells partition space, the polyhedrons enclosing these cells do overlap. In regions where these overlaps occur, the minimum of all computed distances is assigned to a grid point. In order to speed up computations, points inside each polyhedron are determined by scan conversion of grid slices using graphics hardware. For this task, a fragment program is used to perform the nonlinear interpolation and minimization of distance values.
135 citations
08 Jun 2003
TL;DR: Under a mild uniform sampling condition, it is shown that the complexity of the 3D Delaunay triangulation of the points is O(N log N).
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).
118 citations
TL;DR: A generic automatic approach for the parcellation of the cortical surface into labeled gyri defined from a set of pairs of sulci selected by the user, illustrated with 12 different hemispheres.
TL;DR: This paper presents simple point insertion and deletion operations in Voronoi diagrams and Delaunay triangulations which may be useful for a wide variety of applications, either where interactivity is important, or where local modification of the topology is preferable to global rebuilding.
TL;DR: In this article, a set of stereoscopic imaging techniques is proposed for the measurement of rapidly flowing dispersions of opaque particles observed near a transparent wall using projective geometry and the Voronoi diagram.
Abstract: A set of stereoscopic imaging techniques is proposed for the measurement of rapidly flowing dispersions of opaque particles observed near a transparent wall. The methods exploit projective geometry and the Voronoi diagram. They rely on purely geometrical principles to reconstruct 3D particle positions, concentrations, and velocities. The methods are able to handle position and motion ambiguities, as well as particle-occlusion effects, difficulties that are common in the case of dense dispersions of many identical particles. Fluidization cell experiments allow validation of the concentration estimates. A mature debris-flow experimental run is then chosen to test the particle-tracking algorithm. The Voronoi stereo methods are found to perform well in both cases, and to present significant advantages over monocular imaging measurements.
TL;DR: Results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects.
Abstract: This paper presents a novel processing scheme for the automatic and robust computation of a medial shape model, which represents an object population with shape variability The sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique These problems are approached with the computation of a model with common medial branching topology and grid sampling This model is then used for a medial shape description of individual objects via a constrained model fit
The process starts from parametric 3D boundary representations with existing point-to-point homology between objects The Voronoi skeleton of each sampled object boundary is partitioned into non-branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion Using the surface homology, medial sheets are combined to form a common medial branching topology Finally, the medial sheets are sampled and represented as meshes of medial primitives
Results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects
16 Jun 2003
TL;DR: A Voronoi-based skeletal mesh is extracted from a given original mesh, modified by free-form deformations, and a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh.
Abstract: In this paper, we propose a new scheme for free-form skeleton-driven global mesh deformations. First a Voronoi-based skeletal mesh is extracted from a given original mesh. Next the skeletal mesh is modified by free-form deformations. Then a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh. We develop a mesh fairing procedure allowing us to avoid possible global and local self-intersections of the reconstructed mesh. Finally, using a displaced subdivision surface representation [18] improves the speed and robustness of our approach.
TL;DR: The paper describes an algorithm for generating an approximation of the medial axis transform (MAT) for planar objects with free form boundaries by a tracing technique that marches along the object boundary rather than the bisectors of the boundary entities.
Abstract: The paper describes an algorithm for generating an approximation of the medial axis transform (MAT) for planar objects with free form boundaries. The algorithm generates the MAT by a tracing technique that marches along the object boundary rather than the bisectors of the boundary entities. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction or branch points and the search for these branch points is more efficient than while tracing the bisectors. The algorithm works for multiply connected objects as well. Results of implementation are provided.
TL;DR: In this paper, a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids is proposed, which can be viewed as a point collocation technique, and the satisfaction of the discrete maximum principle (stability) is established.
Abstract: Voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two-fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well-known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive-definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: A finite volume scheme for the discretization of partial differential equations (PDEs) on the sphere is developed and the notion of constrained centroidal Voronoi tessellations (CCVTs) of the sphere are introduced.
TL;DR: The proposed methodology develops a methodology for parameter structure identification in ground water modeling, in which the true transmissivity field is characterized by either a continuous distribution or a distribution that can be characterized by zones.
Abstract: This research develops a methodology for parameter structure identification in ground water modeling. For a given set of observations, parameter structure identification seeks to identify the parameter dimension, its corresponding parameter pattern and values. Voronoi tessellation is used to parameterize the unknown distributed parameter into a number of zones. Accordingly, the parameter structure identification problem is equivalent to finding the number and locations as well as the values of the basis points associated with the Voronoi tessellation. A genetic algorithm (GA) is allied with a grid search method and a quasi-Newton algorithm to solve the inverse problem. GA is first used to search for the near-optimal parameter pattern and values. Next, a grid search method and a quasi-Newton algorithm iteratively improve the GA's estimates. Sensitivities of state variables to parameters are calculated by the sensitivity-equation method. MODFLOW and MT3DMS are employed to solve the coupled flow and transport model as well as the derived sensitivity equations. The optimal parameter dimension is determined using criteria based on parameter uncertainty and parameter structure discrimination. Numerical experiments are conducted to demonstrate the proposed methodology, in which the true transmissivity field is characterized by either a continuous distribution or a distribution that can be characterized by zones. We conclude that the optimized transmissivity zones capture the trend and distribution of the true transmissivity field.
TL;DR: The problem of dynamic maintenance of a Voronoi diagram for a set of spheres moving independently in d-dimensional space is addressed and the method to study the structure of a system of polydisperse spheres in a three-dimensional Euclidean space is discussed.
TL;DR: In this article, an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell of a two-dimensional Poisson-Voronoi tessellation is given.
Abstract: In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.
Book•
01 Jan 2003TL;DR: Mesh algorithms for computational geometry: preliminaries the convex hull smallest enclosing figures nearest point problem line segments and simple polygons intersection of convex sets diameter iso-oriented rectangles and polygons voronoi diagram.
Abstract: Part 1 Overview: models of computation forms of input problems data movement operations sample algorithms further remarks. Part 2 Fundamental mesh algorithms: definitions lower bounds primitive mesh algorithms matrix algorithms algorithms involving ordered data further remarks. Part 3 Mesh algorithms for images and graphs: fundamental graph algorithms connected components internal distances convexity external distances further remarks. Part 4 Mesh algorithms for computational geometry: preliminaries the convex hull smallest enclosing figures nearest point problem line segments and simple polygons intersection of convex sets diameter iso-oriented rectangles and polygons voronoi diagram further remarks. Part 5 Tree-like pyramid algorithms: definitions lower bounds fundamental algorithms image algorithms further remarks. Part 6 Hybrid pyramid algorithms: graphs as unordered edges graphs as adjacency matrices digitized pictures convexity data movement operations optimality further remarks.
TL;DR: In this article, the authors quantified flow-induced void and crack microstructures by locating all particle centroids with quantitative image processing, by performing Voronoi volume tessellation with computational geometry and by analyzing the particle number density fluctuations as a function of averaging volume.
Abstract: Unconstrained uniaxial compression (or squeeze flow) of high volume fraction gels of fluorescent silica particles of diameter 832 nm results in the formation of voids (at φ=0.26) and cracks (at φ=0.40) that are of scale 10–100 μm. This evidence of inhomogeneous material deformation was obtained by direction visualization of three-dimensional structure by confocal laser scanning microscopy. Flow-induced void and crack microstructures are quantified by locating all particle centroids with quantitative image processing, by performing Voronoi volume tessellation with computational geometry and by analyzing the particle number density fluctuations as a function of averaging volume. Average short-range real space structural measures, such as the pair correlation function, are little changed by the flow. However, the probability distribution of excess normalized Voronoi polyhedra volume is profoundly extended by squeeze flow, particularly at large polyhedra volumes. Comparison of the Voronoi polyhedra volume distributions and particle bond distributions indicates that: (1) in the low φ gel, large flow-induced voids are formed by the reorganization of the existing quiescent voids without significant effect on the local structure; and (2) in the high φ gel, cracks are formed by reorganization of the local structure itself. Analysis of the number density fluctuations shows that the gels respond to applied squeeze flow deformation with structural distortion on the length scale of 5–10 particle diameters.Unconstrained uniaxial compression (or squeeze flow) of high volume fraction gels of fluorescent silica particles of diameter 832 nm results in the formation of voids (at φ=0.26) and cracks (at φ=0.40) that are of scale 10–100 μm. This evidence of inhomogeneous material deformation was obtained by direction visualization of three-dimensional structure by confocal laser scanning microscopy. Flow-induced void and crack microstructures are quantified by locating all particle centroids with quantitative image processing, by performing Voronoi volume tessellation with computational geometry and by analyzing the particle number density fluctuations as a function of averaging volume. Average short-range real space structural measures, such as the pair correlation function, are little changed by the flow. However, the probability distribution of excess normalized Voronoi polyhedra volume is profoundly extended by squeeze flow, particularly at large polyhedra volumes. Comparison of the Voronoi polyhedra volume dis...
12 Jan 2003
TL;DR: The equivalence relationship between additively weighted Voronoi cells and convex hulls of spheres permits us to compute a single additively Weight Vor onoi cel1 in dimension d in worst case optimal time.
Abstract: In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in dimension d; (2) the convex hull of a set of d-dimensional spheres. In particular, given a set of n spheres in dimension d, we show that the worst case complexity of both a single additively weighted Voronoi cell and the convex hull of the set of spheres is Θ(n[d/2]). The equivalence between additively weighted Voronoi cells and convex hulls of spheres permits us to compute a single additively weighted Voronoi cel1 in dimension d in worst case optimal timeO(n log n+n[d/2]).
01 Jan 2003
11 May 2003
TL;DR: Improved algorithms, with a preprocessing time of O(n log n), to compute a maximum breach/support path P in optimal (|P|) time or the maximum Breach/support value in O(1) time are presented.
Abstract: This paper discusses the computation of optimal coverage paths in an ad-hoc network consisting of n sensors. Improved algorithms, with a preprocessing time of O(n log n), to compute a maximum breach/support path P in optimal (|P|) time or the maximum breach/support value in O(1) time are presented. Algorithms for computing a shortest path that has maximum breach/support are also provided. Experimental results for breach paths show that the shortest path length is on the average 30% less and is not much worse that the ideal straight line path. For applications that require redundancy (i.e., detection by multiple sensors), a generalization of Voronoi diagrams allows us to compute maximum breach paths where breach is defined as the distance to the kth nearest sensor in the field. Extensive experimental results are provided.
TL;DR: In this article, the authors give an explicit expression for the distribution of the number of sides of the typical cell of a two-dimensional Poisson-Voronoi tessellation.
Abstract: In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.
TL;DR: In this paper, a simple algorithm for computing the Time Voronoi Diagram, that is, the Voronoa Diagram of a set of points using the time distance is given.
12 Jan 2003
TL;DR: A simple method is proposed that allows to remove any vertex even when the points are in very degenerate configurations in a three-dimensional 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.
TL;DR: An algorithm for material interface reconstruction for data sets where fractional material information is given as a percentage for each element of the underlying grid and can treat data sets containing any number of materials.
Abstract: The paper presents an algorithm for material interface reconstruction for data sets where fractional material information is given as a percentage for each element of the underlying grid. The reconstruction problem is transformed to a problem that analyzes a dual grid, where each vertex in the dual grid has an associated barycentric coordinate tuple that represents the fraction of each material present. Material boundaries are constructed by analyzing the barycentric coordinate tuples of a tetrahedron in material space and calculating intersections with Voronoi cells that represent the regions where one material dominates. These intersections are used to calculate intersections in the Euclidean coordinates of the tetrahedron. By triangulating these intersection points, one creates the material boundary. The algorithm can treat data sets containing any number of materials. The algorithm can also create nonmanifold boundary surfaces if necessary. By clipping the generated material boundaries against the original cells, one can examine the error in the algorithm. Error analysis shows that the algorithm preserves volume fractions within an error range of 0.5 percent per material.