Showing papers on "Voronoi diagram published in 1999"

Centroidal Voronoi Tessellations: Applications and Algorithms

Abstract: A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods for computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.

Geophysical inversion with a neighbourhood algorithm—I. Searching a parameter space

Abstract: SUMMARY This paper presents a new derivative-free search method for finding models of acceptable data fit in a multidimensional parameter space. It falls into the same class of method as simulated annealing and genetic algorithms, which are commonly used for global optimization problems. The objective here is to find an ensemble of models that preferentially sample the good data-fitting regions of parameter space, rather than seeking a single optimal model. (A related paper deals with the quantitative appraisal of the ensemble.) The new search algorithm makes use of the geometrical constructs known as Voronoi cells to derive the search in parameter space. These are nearest neighbour regions defined under a suitable distance norm. The algorithm is conceptually simple, requires just two ‘tuning parameters’, and makes use of only the rank of a data fit criterion rather than the numerical value. In this way all diYculties associated with the scaling of a data misfit function are avoided, and any combination of data fit criteria can be used. It is also shown how Voronoi cells can be used to enhance any existing direct search algorithm, by intermittently replacing the forward modelling calculations with nearest neighbour calculations. The new direct search algorithm is illustrated with an application to a synthetic problem involving the inversion of receiver functions for crustal seismic structure. This is known to be a non-linear problem, where linearized inversion techniques suVer from a strong dependence on the starting solution. It is shown that the new algorithm produces a sophisticated type of ‘self-adaptive’ search behaviour, which to our knowledge has not been demonstrated in any previous technique of this kind.

Surface Reconstruction by Voronoi Filtering

Abstract: We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs.

Fast computation of generalized Voronoi diagrams using graphics hardware

Abstract: We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation. Additional

Textons, contours and regions: cue integration in image segmentation

Abstract: The paper makes two contributions: it provides (1) an operational definition of textons, the putative elementary units of texture perception, and (2) an algorithm for partitioning the image into disjoint regions of coherent brightness and texture, where boundaries of regions are defined by peaks in contour orientation energy and differences in texton densities across the contour. B. Julesz (1981) introduced the term texton, analogous to a phoneme in speech recognition, but did not provide an operational definition for gray-level images. We re-invent textons as frequently co-occurring combinations of oriented linear filter outputs. These can be learned using a K-means approach. By mapping each pixel to its nearest texton, the image can be analyzed into texton channels, each of which is a point set where discrete techniques such as Voronoi diagrams become applicable. Local histograms of texton frequencies can be used with a /spl chi//sup 2/ test for significant differences to find texture boundaries. Natural images contain both textured and untextured regions, so we combine this cue with that of the presence of peaks of contour energy derived from outputs of odd- and even-symmetric oriented Gaussian derivative filters. Each of these cues has a domain of applicability, so to facilitate cue combination we introduce a gating operator based on a statistical test for isotropy of Delaunay neighbors. Having obtained a local measure of how likely two nearby pixels are to belong to the same region, we use the spectral graph theoretic framework of normalized cuts to find partitions of the image into regions of coherent texture and brightness. Experimental results on a wide range of images are shown.

Resampling of data between arbitrary grids using convolution interpolation

Abstract: For certain medical applications resampling of data is required. In magnetic resonance tomography (MRT) or computer tomography (CT), e.g., data may be sampled on nonrectilinear grids in the Fourier domain. For the image reconstruction a convolution-interpolation algorithm, often called gridding, can be applied for resampling of the data onto a rectilinear grid. Resampling of data from a rectilinear onto a nonrectilinear grid are needed, e.g., if projections of a given rectilinear data set are to be obtained. In this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another. The basic algorithm can be split into two steps. First, the data are resampled from the arbitrary input grid onto a rectilinear grid and second, the rectilinear data is resampled onto the arbitrary output grid. Furthermore, the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm. For fast, sampling-pattern-independent determination of the sampling density function the Voronoi diagram of the sample distribution is calculated. The volume of the Voronoi cell around each sample is used as a measure for the sampling density. It is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids. Furthermore, it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns. Examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral, radial, and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image.

Distance transformations: fast algorithms and applications to medical image processing

Abstract: Medical image processing is a demanding domain, both in terms of CPU and memory requirements. The volume of data to be processed is often large (a typical MRI dataset requires 10 MBytes) and many processing tools are only useful to the physician if they are available as real-time applications, i.e. if they run in a few seconds at most. Of course, a large part of these demands are - and will be - handled by the development of more powerful hardware. On the other hand, when faced with non-linear computational complexity, the development of improved algorithms is obviously the best solution. Distance transformations, a powerful image analysis tool used in a number of problems such as image registration, requires such improvements. A distance map is an image where the value of each pixel is the distance from this pixel to the nearest pixel belonging to a given set or object. A distance transformation (DT) is an algorithm that computes a distance map from a binary image representing this set of pixels. This definition is global in the sense that it requires finding the minimum on a set of distances computed between all image pixels and all object pixels. Therefore, a direct application of the definition usually leads to an unacceptable computational complexity. Numerous algorithms have been proposed to localize this definition of distance to the nearest pixel and allow a faster DT computation, but up to now, none of them combines both exactness and linear complexity. Numerous applications of distance transformations to image analysis and pattern recognition have been reported and those related to medical image processing are explored in what follows. Chapter 1 introduces a few basic concepts, a typical application of distance transformations in pattern recognition and the key challenges in producing a DT algorithm. Chapter 2 contains an exhaustive critical review of published algorithms. The strong and weak points of the most popular ones are discussed and the core principles for our original algorithms are derived. Chapters 3, 5, 6, 8 and 10 present original distance transformation algorithms. Each of those chapters is organized in a somewhat similar fashion. First we describe the algorithm. Then we evaluate its computational complexity and compare it to the state of the art. Chapter 4, 7, 9 and 11 each present an application to a particular problem in medical image processing, using the algorithm developed in the previous chapter. Ideally, the description of any medical image processing problem should include a medical justification of the need for an automated processing, a complete review of the state of the art in the field, a detailed description of the proposed processing method, and an evaluation of the accuracy of the results and their medical significance. Because of both time and space constraints in this thesis, such an exhaustive work will only be presented for the application in chapter 4, while the other applications will be described more briefly. Chapter 3 describes a new exact Euclidean distance transformation using ordered propagation. It is based on a variation of Ragnelmam's approximate Euclidean DT. We analyze the error patterns for approximate Euclidean DT using finite masks, and we derive a rule defining, for any pixel location, the size of the neighborhood that guarantees the exactness of the DT. This algorithm is particularly well-suited to implement mathematical morphology operations, which are examined in details. In Chapter 4, we apply the algorithm of chapter 3 to the segmentation of neuronal fibers from microscopic images of the sciatic nerve. In particular, it is used to determine the thickness of the myelin sheath surrounding the center of the fiber. This study was carried out in collaboration with the Neural Rehabilitation Engineering Laboratory, UCL. Chapter 5 proposes another exact Euclidean distance transformation, based on the explicit computation of the Voronoi division of the image. Possible error locations are detected at the corners of the Voronoi polygons and corrected if needed. This algorithm is shown to be the fastest exact EDT to date. It approaches the theoretical optimal complexity, a CPU time proportional to the number of pixels on which the distance is computed. Chapter 6 investigates how the algorithms of chapters 3 and 5 can be extended to 3 dimensional images. It shows the limitations of both approaches and proposes an hybrid algorithm mixing the method of chapter 5 and Saito's. In Chapter 7, the 3D Euclidean DT is applied to the registration of MR images of the brain where the matching criterion is the distance between the surfaces of similar objects (skin, cortex, ventricular system, ...) in both images. Examples are shown, from projects with the Neuro-physiology Laboratory, UCL, and with the Positron Tomography Laboratory, UCL. Chapter 8 discusses an extension of the distance transformation concept: geodesic distances on non-convex domains. Because geodesic distances are based on the notion of paths, a trade-off has to be introduced between the accuracy with which straight lines are represented and the way curves of the domain are followed. It is shown that, whatever the trade-off chosen, there is an efficient implementation of the geodesic DT by propagation. By back-tracking the geodesic distance propagation, one can find the shortest path between a target and a starting point. In chapter 9, this is used to plan the optimal path for the camera movements in virtual endoscopy, a work done in collaboration with the Surgical Planning Laboratory, Harvard Medical School, Boston. Chapter 10 extends the Euclidean distance transformation from finding the nearest object pixel to finding the k nearest object pixels. It is shown that this can be done with a complexity increasing linearly with k. In Chapter 11, the k-DT is used as a fast implementation of the k Nearest Neighbors (k-NN) classification between different tissue types in multi-modal MR imaging. This is illustrated through the classification of multiple sclerosis lesions from T1-T2 images, provided by the Radiology unit, St-Luc Hospital, UCL, via the Positron Tomography Laboratory, UCL. Finally, a general conclusion is drawn. It reviews the main contributions of the thesis, its applications and explores some new domains in which their applications could also be useful. Ultimately, the publications related to this thesis are briefly reviewed.

New algorithms in 3D image analysis and their application to the measurement of a spatialized pore size distribution in soils

Abstract: We introduce a skeletization method based on the Voronoi diagram to determine local pole sizes in any porous medium. Using the skeleton of the pore space in a 3D image of the porous medium, a pole size value is assigned to each voxel and a reconstructed image of a spatialized local pore size distribution is created. The reconstructed image provides a means for calculating the global volume versus size pore distribution. It is also used to carry out fluid invasion simulation which take into account the connectivity of and constrictions in the pore network. As an example we simulate mercury intrusion in a 3D soil image. (C) 1999 Elsevier Science Ltd. All rights reserved.

Voronoi-Delaunay analysis of voids in systems of nonspherical particles.

Abstract: The Voronoi network is known to be a useful tool for the structural description of voids in the packings of spheres produced by computer simulations. In this article we extend the Voronoi-Delaunay analysis to packings of nonspherical convex objects. Main properties of the Voronoi network, which are known for systems of spheres, are valid for systems of any convex objects. A general numerical algorithm for calculation of the Voronoi network in three dimensions is proposed. It is based on the calculation of the trajectory of the imaginary empty sphere of variable size, moving inside a system (the Delaunay empty sphere method). Analysis of voids is presented for an ensemble of random straight lines and for a molecular dynamics model of liquid crystal. The spatial distribution of voids and a simple percolation analysis are obtained. The distributions of the bottleneck radii and the radii of spheres inscribed in the voids are calculated.

Accurate computation of the medial axis of a polyhedron

Abstract: We present an accurate and efficient algorithm to compute the internal Voronoi region and medial axis of a 3-D polyhedron. It uses exact arithmetic and representations for accurate computation of the medial axis. The sheets, seams, and junctions of the medial axis are represented as trimmed quadric surfaces, algebraic space curves, and points with algebraic coordinates, respectively. The algorithm works by recursively finding neighboring junctions along the seam curves. It uses spatial decomposition and linear programming to speed up the search step. We also present a new algorithm for analysis of the topology of an algebraic plane curve, which is the core of our medial axis algorithm. To speed up the computation, we have designed specialized algorithms for fast computation on implicit geometric structures. These include lazy evaluation based on multivariate Stiirm sequences, fast resultant computation, curve topology analysis, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

Hexahedral Mesh Generation using the Embedded Voronoi Graph

Abstract: This work presents a new approach for automatic hexahedral meshing, based on the embedded Voronoi graph. The embedded Voronoi graph contains the full symbolic information of the Voronoi diagram and the medial axis of the object, and a geometric approximation to the real geometry. The embedded Voronoi graph is used for decomposing the object, with the guiding principle that resulting sub-volumes are sweepable. Sub-volumes are meshed independently, and the resulting meshes are easily combined and smoothed to yield the final mesh. The approach presented here is general and automatic. It handles any volume, even if its medial axis is degenerate. The embedded Voronoi graph provides complete information regarding proximity and adjacency relationships between the entities of the volume. Hence, decomposition faces are determined unambiguously, without any further geometric computations. The sub-volumes computed by the algorithm are guaranteed to be well-defined and disjoint. The size of the decomposition is relatively small, since every sub-volume contains a different Voronoi face. Mesh quality seems high since the decomposition avoids generation of sharp angles, and sweep and other basic methods are used to mesh the sub-volumes.

Crust and anti-crust: a one-step boundary and skeleton extraction algorithm

Abstract: We wish to extract the topology from scanned maps. In previous work [15] this was done by extracting a skeleton from the Voronoi diagram, but this required vertex labelling and was only useable for polygon maps. We wished to take the crust algorithm of Amenta, Bern and Eppstein [3] and modify it to extract the skeleton from unlabelled vertices. We find that by reducing the algorithm to a local test on the simple Voronoi diagram of point sites we may extract both the crust and the skeleton simultaneously. We show that this produces the same results as the original algorithm, and illustrate its utility with various cartographic applications.

Poisson-Voronoi Spanning Trees with Applications to the Optimization of Communication Networks

Abstract: We define a family of random trees in the plane. Their nodes of level k, k = 0,...,m are the points of a homogeneous Poisson point process Πk, whereas their arcs connect nodes of level k and k + 1, according to the least distance principle: If V denotes the Voronoi cell w.r.t. Πk+1 with nucleus x, where x is a point of Πk+1, then there is an arc connecting x to all the points of Πk that belong to V. This creates a family of stationary random trees rooted in the points of Πm. These random trees are useful to model the spatial organization of several types of hierarchical communication networks. In relation to these communication networks, it is natural to associate various cost functions with such random trees. Using point process techniques, like the exchange formula between two Palm measures, and integral geometry techniques, we show how to compute these average costs as functions of the intensity parameters of the Poisson processes. The formulas derived for the average value of these cost functions can then be exploited for parametric optimization purposes. Several applications to classical and mobile cellular communication networks are presented.

Randomized Path Planning for a Rigid Body Based on Hardware Accelerated Voronoi Sampling

Abstract: Probabilistic roadmap methods have recently received considerable attention as a practical approach for motion planning in complex environments. These algorithms sample a number of con gurations in the free space and build a roadmap. Their performance varies as a function of the sampling strategies and relative con gurations of the obstacles. To improve the performance when the path of a robot has to pass through narrow passages, some researchers have proposed algorithms for sampling along or near the medial axis of the free space. However, their usage has been limited because of the practical complexity of computing the medial axis or the cost of computing such samples. In this paper, we present e cient algorithms for sampling near the medial axis and building roadmap graphs for a freeying rigid body. We use a recent algorithm for fast computation of discrete generalized Voronoi diagrams using graphics hardware [HCK99a]. We initially compute a bounded error discretized Voronoi diagram of the obstacles in the workspace and use it to generate samples in the free space. We use multi-level connection strategies and local planning algorithms to generate roadmap graphs. We also utilize the distance information provided by our Voronoi algorithm for fast proximity queries and sampling the con gurations. The resulting planner has been applied to a number of free ying rigid bodies in 2D (with 3-dof) and 3D (with 6-dof) and compared with the performance of earlier planners using a uniform sampling of the con guration space. Its performance varies with di erent environments and we obtain 25% to over 1000% speed-up.

Two generalizations of an interpolant based on voronoi diagrams

Abstract: Recently, the authors found an interpolant using Voronoi diagrams that differs from Sibson's interpolant. This paper generalizes our interpolant in two directions: one is to general-dimensional data, and the other is to data distributed continuously on curves. The Minkowski's theorem is used as the basic principle in generalization.

Critical area computation via Voronoi diagrams

Abstract: In this paper, we present a new approach for computing the critical area for shorts in a circuit layout. The critical area calculation is the main computational problem in very large scale integration yield prediction. The method is based on the concept of Voronoi diagrams and computes the critical area for shorts (for all possible defect radii, assuming square defects) accurately in O(n log n) time, where n is the size of the input. The method is presented for rectilinear layouts and layouts containing edges of slope /spl plusmn/1. As a byproduct, we briefly sketch how to speed up the grid method of Wagner and Koren [1995].

On the Existence of Similar Sublattices

Abstract: Partial answers are given to two questions When does a lattice Λ contain a sublattice Λ′ of index that is geometrically similar to Λ? When is the sublattice "clean", in the sense that the boundaries of the Voronoi cells for Λ' do not intersect Λ?

Method and system for determining critical area for circuit layouts

Abstract: A method for computing critical area for opens of a layout, which may be implemented by program storage device readable by machine, tangibly embodying a program of instructions executable by the machine, to perform the method steps includes computing Voronoi diagrams of shapes of the layout, determining core elements and weights for the core elements of the shapes, computing a weighted Voronoi diagram for the layout to arrive at a partitioning of the layout into regions, computing critical area within each region and summing the critical areas to arrive at a total critical area for opens in the layout.

Quadrilateral Meshing by Circle Packing

Abstract: We use circle-packing methods to generate quadrilateral meshes for polygonal domains, with guaranteed bounds both on the quality and the number of elements. We show that these methods can generate meshes of several types: (1) the elements form the cells of a Voronoi diagram, (2) all elements have two opposite right angles, (3) all elements are kites, or (4) all angles are at most 120 degrees. In each case the total number of elements is O(n), where n is the number of input vertices.

Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design

Abstract: In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact-computation paradigm and formalize the notion of degree of a geometric algorithm as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach where algorithm design is driven also by the degree, we consider the important classical problem of proximity queries in two and three dimensions and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique offers both low degree and fast query time and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations, and arithmetic degree.

Zonotopes, Dicings, and Voronoi's Conjecture on Parallelohedra

Abstract: In 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi?s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3).

Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a Simple Polygon in Linear Time

Abstract: In this paper, we present an $\Theta (n)$ time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.

A raster-based method for computing Voronoi diagrams of spatial objects using dynamic distance transformation

Abstract: The Voronoi diagram has been suggested as an appropriate model for the description of relations between spatial objects (Gold 1992) and it is the only possible solution (which is currently available) to dynamic (measurement-based) GIS (Wright and Goodchild 1997). A Voronoi diagram can be computed either in vector mode or in raster mode. Most existing methods are vector-based. However, vector-based methods are complex for line sets and area sets. To overcome this serious deficiency, attempts have also been made to use rasterbased methods. This paper describes a raster-based method for computing Voronoi diagrams of spatial objects (including points, line and areas) using dynamic distance transformation achieved by the dilation operator in mathematical morphology. Furthermore, an extension is presented to accommodate complex spatial objects.

Morphological iterative closest point algorithm

Abstract: This work presents a method for the registration of three-dimensional (3-D) shapes. The method is based on the iterative closest point (ICP) algorithm and improves it through the use of a 3-D volume containing the shapes to be registered. The Voronoi diagram of the "model" shape points is first constructed in the volume. Then this is used for the calculation of the closest point operator. This way a dramatic decrease of the computational cost is achieved.

A Unified Account of the Effects of Caricaturing Faces

Abstract: Explanations of the caricature advantage (caricatured faces are recognized faster than veridicals) usually involve an encoded prototypical norm face (Rhodes, Brennan, & Carey, 1987). As an alternative to these, the present study describes a mathematical framework based on Valentine's (1991) exemplar-based model, which accounts for the caricature advantage without reference to an explicitly encoded norm face. This framework encodes faces in a multidimensional Voronoi diagram based on normally distributed face-space representations. The properties of this framework are investigated geometrically and computationally. It is demonstrated how a parallel processing system can extract a Voronoi diagram from multidimensional representations. The framework was able to account for many of the empirical findings on face recognition and caricaturization without extracting a norm face.

An exact solution for the roundness evaluation problems

Abstract: Two theorems based on the minimax criterion are developed to solve the roundness evaluation problems. The Voronoi diagram is the basic component for this research. Using the Voronoi diagrams, the whole feasible plane is divided into a finite number of regions defined as the max regions. The first theorem shows that the exact minimum roundness can be obtained only at an X-type vertex on the max region. Theorem two indicates that the minimum roundness is determined by four critical measured points that lead to an efficient way to solve the minimax solution.