Abstract: The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other by convex duality or embedding. Moreover, we can always compute them indirectly as power diagrams in primal or dual spaces, or directly after linearization in an extra-dimensional space as the projection of a Euclidean polytope. Finally, our paper proposes to generalize Bregman divergences to higher-order terms, called κ-jet Bregman divergences, and touch upon their Voronoi diagrams.

Abstract: It is a well-established fact that the witness complex is closelyrelated to the restricted Delaunay triangulation in lowdimensions. Specifically, it has been proved that the witness complexcoincides with the restricted Delaunay triangulation on curves, and isstill a subset of it on surfaces, under mild samplingassumptions. Unfortunately, these results do not extend tohigher-dimensional manifolds, even under stronger samplingconditions. In this paper, we show how the sets of witnesses andlandmarks can be enriched, so that the nice relations that existbetween both complexes still hold on higher-dimensional manifolds. Wealso use our structural results to devise an algorithm thatreconstructs manifolds of any arbitrary dimension or co-dimension atdifferent scales. The algorithm combines a farthest-point refinementscheme with a vertex pumping strategy. It is very simple conceptually,and it does not require the input point sample W to be sparse. Itstime complexity is bounded by c(d) |W|2, where c(d) is a constantdepending solely on the dimension d of the ambient space.

Abstract: This paper presents a new method of specific cavity analysis in protein molecules. Long-term biochemical research has the discovery that protein molecule behaviour depends on the existence of cavities (tunnels) leading from the inside of the molecule to its surface. Previous methods of tunnel computation were based on space rasterization. Our approach is based on computational geometry and uses Voronoi diagram and Delaunay triangulation. Our method computes tunnels with better quality in reasonable computational time. The proposed algorithm was implemented and tested on several real protein molecules and is expected to be used in various applications in protein modelling and analysis. This is an interesting example of applying computational geometry principles to practical problems.

Abstract: This paper presents a practical method for obtaining the global minimum to the least-squares (L2) triangulation problem. Although optimal algorithms for the triangulation problem under L8-norm have been given, finding an optimal solution to the L2 triangulation problem is difficult. This is because the cost function under L2-norm is not convex. Since there are no ideal techniques for initialization, traditional iterative methods that are sensitive to initialization may be trapped in local minima. A branch-and-bound algorithm was introduced in [1] for finding the optimal solution and it theoretically guarantees the global optimality within a chosen tolerance. However, this algorithm is complicated and too slow for large-scale use. In this paper, we propose a simpler branch-and-bound algorithm to approach the global estimate. Linear programming algorithms plus iterative techniques are all we need in implementing our method. Experiments on a large data set of 277,887 points show that it only takes on average 0.02s for each triangulation problem.

Abstract: For hydrologic applications, terrain models should have few local minima, and drainage lines should coincide with edges. We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heuristics are presented to reduce the number of local minima for higher-order Delaunay triangulations, which start out with the Delaunay triangulation. We give efficient algorithms for their implementation, and test on real-world data how well they perform. We also study another desirable drainage characteristic, few valley components, and how to obtain it for higher-order Delaunay triangulations. This gives rise to a third heuristic. Tables and visualizations show how the heuristics perform for the drainage characteristics on real-world data.

Abstract: Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.

Abstract: In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines

Abstract: We present a parallel Delaunay refinement algorithm for generating well-shaped meshes in both two and three dimensions. Like its sequential counterparts, the parallel algorithm iteratively improves the quality of a mesh by inserting new points, the Steiner points, into the input domain while maintaining the Delaunay triangulation. The Steiner points are carefully chosen from a set of candidates that includes the circumcenters of poorly-shaped triangular elements. We introduce a notion of independence among possible Steiner points that can be inserted simultaneously during Delaunay refinements and show that such a set of independent points can be constructed efficiently and that the number of parallel iterations is O(log2Δ), where Δ is the spread of the input — the ratio of the longest to the shortest pairwise distances among input features. In addition, we show that the parallel insertion of these set of points can be realized by sequential Delaunay refinement algorithms such as by Ruppert's algorithm in ...

Abstract: Adding motion to geometric data structures creates a host of new computational issues. This thesis explores the issues encountered when using two different motion models---smooth, continuous motion of objects moving along algebraic trajectories and the discrete steps characteristic of physical simulations and optimization processes. The focus in this thesis is on how to perform the computations efficiently while maintaining exactness, rather than asymptotic complexity or other theoretical concerns. The work presented in this thesis has two main threads which correspond to the two motion models. The first thread concerns techniques for implementing kinetic data structures, a class of geometric data structures constructed on top of smoothly moving primitives. We developed a framework for kinetic data structures which was the first to support the necessary algebraic operations exactly in addition to providing a host of new tools for easing implementation and debugging. This framework has been incorporated into a software module that have been released as part of CGAL, the Computational Geometry Algorithms Library. The second thread looks at the problem of updating Delaunay triangulations, an important subproblem in a variety of simulations. After each step of the simulation, we have the Delaunay triangulation from the last time step and a perturbation of the underlying points as dictated by the computed forces. We need the Delaunay triangulation of the new, perturbed, points. We first present a kinetic data structures based update scheme that updates the prior triangulation to the desired one. This scheme, which required the development of several arithmetic filters, produces the desired triangulation significantly faster than computing it from scratch on a variety of real data sets. We then relax the kinetic data structure by processing events out of order, which results in a significantly faster update algorithm at the cost of occasional backtracking.

Abstract: This paper discusses optimization of quality measures over first order Delaunay triangulations. Unlike most previous work, our measures relate to edge-adjacent or vertex-adjacent triangles instead of only to single triangles. We give efficient algorithms to optimize certain measures, whereas other measures are shown to be NP-hard. For two of the NP-hard maximization problems we provide for any constant e > 0, factor (1-e) approximation algorithms that run in 2O(1/e)ċn and 2O(1/e2)ċn time (when the Delaunay triangulation is given). For a third NP-hard problem the NP-hardness proof provides an inapproximability result. Our results are presented for the class of first-order Delaunay triangulations, but also apply to triangulations where every triangle has at most one flippable edge. One of the approximation results is also extended to k-th order Delaunay triangulations.

Abstract: We define a Delaunay mesh to be a manifold triangle mesh whose edges form an intrinsic Delaunay triangulation or iDT of its vertices, where the triangulated domain is the piecewise flat mesh surface. We show that meshes constructed from a smooth surface by taking an iDT or a restricted Delaunay triangulation, do not in general yield a Delaunay mesh.We establish a precise dual relationship between the iDT and the Voronoi tessellation of the vertices of a piecewise flat (pwf) surface and exploit this duality to demonstrate criteria which ensure the existence of a proper Delaunay triangulation.

Abstract: We present a streaming algorithm for reconstructing closed surfaces from large non-uniform point sets based on a geometric convection technique. Assuming that the sample points are organized into slices stacked along one coordinate axis, a triangle mesh can be efficiently reconstructed in a streamable layout with a controlled memory footprint. Our algorithm associates a streaming 3D Delaunay triangulation data-structure with a multilayer version of the geometric convection algorithm. Our method can process millions of sample points at the rate of 50k points per minute with 350 MB of main memory.

Abstract: We present two new Delaunay refinement algorithms, the second an extension of the first For a given input domain (a set of points in the plane or a planar straight line graph), and a threshold angle α, the Delaunay refinement algorithms compute triangulations that have all angles at least α Our algorithms have the same theoretical guarantees as the previous Delaunay refinement algorithms The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for α ≤ 207° In practice, it generally works for α ≤ 34° and fails to terminate for larger constraint angles The new Delaunay refinement algorithm generally terminates for constraint angles up to 42° Experiments also indicate that our algorithm computes significantly (almost by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms

Abstract: In this paper, we propose a robust and efficient Lagrangian approach for modeling dynamic interfaces between different materials undergoing large deformations and topology changes, in two dimensions. Our work brings an interesting alternative to popular techniques such as the level set method and the particle level set method, for two-dimensional and axisymmetric simulations. The principle of our approach is to maintain a two-dimensional triangulation which embeds the one-dimensional polygonal description of the interfaces. Topology changes can then be detected as inversions of the faces of this triangulation. Each triangular face is labeled with the type of material it contains. The connectivity of the triangulation and the labels of the faces are updated consistently during deformation, within a neat framework developed in computational geometry: kinetic data structures. Thanks to the exact computation paradigm, the reliability of our algorithm, even in difficult situations such as shocks and topology changes, can be certified. We demonstrate the applicability and the efficiency of our approach with a series of numerical experiments in two dimensions. Finally, we discuss the feasibility of an extension to three dimensions.

Abstract: A new algorithm is presented for surface reconstruction from unorganized points. Unlike many previous algorithms, this algorithm does not select a subcomplex of the Delaunay Triangulation of the points. Instead, it uses an incremental algorithm, adding one simplex of the surface at a time. As a result, the algorithm does not require the surface's embedding space to be R^3; the dimension of the embedding space may vary arbitrarily without substantially affecting the complexity of the algorithm. One result of using this incremental algorithmic technique is that very little can be proven about the reconstruction; nonetheless, it is interesting from an experimental viewpoint, as it allows for a wider variety of surfaces to be reconstructed. In particular, the class of non-orientable surfaces, such as the Klein Bottle, may be reconstructed. Results are shown for surfaces of varying genus.

Abstract: Purpose – In GIS applications for a realistic representation of a terrain a great number of triangles are needed that ultimately increases the data size. For online GIS interactive programs it has become highly essential to reduce the number of triangles in order to save more storing space. Therefore, there is need to visualize terrains at different levels of detail, for example, a region of high interest should be in higher resolution than a region of low or no interest. Wavelet technology provides an efficient approach to achieve this. Using this technology, one can decompose a terrain data into hierarchy. On the other hand, the reduction of the number of triangles in subsequent levels should not be too small; otherwise leading to poor representation of terrain.Design/methodology/approach – This paper proposes a new computational code (please see Appendix for the flow chart and pseudo code) for triangulated irregular network (TIN) using Delaunay triangulation methods. The algorithms have proved to be ef...

Abstract: This paper presents a method of skeletonization for high resolution images The proposed method is based on Delaunay triangulation to segment a shape into triangular mesh The triangles are modified and filtered Internal edges are rearranged such that the edge vertexes become smoothed local symmetry in the shape Symmetric points are extracted to be skeletal junctures They are interpolated by piecewise Bezier interpolation to form the skeletal description of a shape Experimental results have demonstrated that the algorithm can extract smooth skeleton The algorithm was also compared with other methods Result has shown that the proposed method has high computational efficiency and superior performance

Abstract: A direction map or an orientation field is widely used in automatic fingerprint verification and recognition systems. In most existing methods, the block-wise orientation field is computed from the squared gradients. Since a fingerprint image can contain areas that may be smeared or scarcely distributed, faulty orientations can be produced during the process of orientation field estimation. In this paper, we propose a constrained Delaunay triangulation (CDT) based orientation field interpolation method. In this method, the orientation coherence is used to segment an image into foreground and background, valid regions and invalid regions, and the contours of the invalid regions are used to construct the CDT that is used to interpolate the orientations of the invalid regions. Compared with the commonly used filter and diffusion based methods, the advantages of our method include that no pre-defined neighborhood is needed, that no iteration is needed, and that the orientations still conform well to the fingerprints’ ridges and valleys. Our experiment results show the efficiency of the proposed approach.

Abstract: In this paper, we propose a novel single-electron device for computation of a Voronoi diagram (VD). A cellular-automaton model of VD formation [18] was used to construct the device that consisted of three layers of a 2-D array of single-electron oscillators. Through extensive numerical simulations, we show pattern formation on the proposed device and demonstrate the VD computation.

Abstract: Digital data that consist of discrete points are frequently captured and processed by scientific and engineering applications. Due to the rapid advance of new data gathering technologies, data set sizes are increasing, and the data distributions are becoming more irregular. These trends call for new computational tools that are both efficient enough to handle large data sets and flexible enough to accommodate irregularity. A mathematical foundation that is well-suited for developing such tools is triangulation, which can be defined for discrete point sets with little assumption about their distribution. The potential benefits from using triangulation are not fully exploited. The challenges fundamentally stem from the complexity of the triangulation structure, which generally takes more space to represent than the input points. This complexity makes developing a triangulation program a delicate task, particularly when it is important that the program runs fast and robustly over large data. This thesis addresses these challenges in two parts. The first part concentrates on techniques designed for efficiently and robustly computing Delaunay triangulations of three kinds of practical data: the terrain data from LIDAR sensors commonly found in GIS, the atom coordinate data used for biological applications, and the time varying volume data generated from scientific simulations. The second part addresses the problem of defining spline spaces over triangulations in two dimensions. It does so by generalizing Delaunay configurations, defined as follows. For a given point set P in two dimensions, a Delaunay configuration is a pair of subsets (T,I ) from P, where T, called the boundary set, is a triplet and I, called the interior set, is the set of points that fall in the circumcircle through T. The size of the interior set is the degree of the configuration. As recently discovered by Neamtu (2004), for a chosen point set, the set of all degree k Delaunay configurations can be associated with a set of degree k+1 splines that form the basis of a spline space. In particular, for the trivial case of k=0, the spline space coincides with the PL interpolation functions over the Delaunay triangulation. Neamtu's definition of the spline space relies only on a few structural properties of the Delaunay configurations. This raises the question whether there exist other sets of configurations with identical structural properties. If there are, then these sets of configurations—let us call them generalized configurations from hereon—can be substituted for Delaunay configurations in Neamtu's definition of spline space thereby yielding a family of splines over the same point set.

Abstract: Roughly speaking, the rank of a Delaunay polytope is its number of degrees of freedom. In [M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Verlag, Berlin, Heidelberg, 1997], a method for computing the rank of a Delaunay polytope P, using the hypermetrics related to P, is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics, is given. This method is applied to the classical Delaunay polytopes: cross-polytopes and half-cubes. Then, we give an example of a Delaunay polytope, which does not have any affine basis.

Abstract: Given a set of points called sites, the Voronoi diagram is a partition of the plane into sets of points having the same closest site. Several generalizations of the Voronoi diagram have been studied, mainly Voronoi diagrams for different distances (other than the Euclidean one), and Voronoi diagrams for sites that are not necessarily points (line segments for example). In this paper we present a new generalization of the Voronoi diagram in the plane, in which we shift our interest from points to lines, that is, we compute the partition of the set of lines in the plane into sets of lines having the same closest site (where sites are points in the plane). We first define formally this diagram and give first properties. Then we use a duality relationship between points and lines to visualize this data structure and give more properties. We show that the size of this line space Voronoi diagram for n sites is in Theta(n2) and give an optimal algorithm for its explicit computation. We then show a remarkable relationship between this diagram and the dual arrangement of the sites and give a new result on an arrangement of lines: we show that the size of the zone of a line augmented with its incident faces is still in O(n). We finally apply this result to show that the size of the zone of a line in the line space Voronoi diagram is in O(n).

Abstract: Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several properties of point set triangulations extend to segment triangulations. Thus, the number of their faces is an invariant of S. In the same way, if S is in general position, there exists a unique segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. This triangulation, called segment Delaunay triangulation, is dual to the segment Voronoi diagram. The main result of this paper is that the local optimality which characterizes point set Delaunay triangulations [10] extends to segment Delaunay triangulations. A similar result holds for segment triangulations with same topology as the Delaunay one.

Abstract: This chapter discusses geometric models of biomolecules and geometric constructs, including the union of ball model, the weigthed Voronoi diagram, the weighted Delaunay triangulation, and the alpha shapes. These geometric constructs enable fast and analytical computaton of shapes of biomoleculres (including features such as voids and pockets) and metric properties (such as area and volume). The algorithms of Delaunay triangulation, computation of voids and pockets, as well volume/area computation are also described. In addition, applications in packing analysis of protein structures and protein function prediction are also discussed.

Abstract: We present a Delaunay based algorithm for simplifying vector field datasets. Our aim is to reduce the size of the mesh on which the vector field is defined while preserving topological features of the original vector field. We leverage a simple paradigm, vertex deletion in Delaunay triangulations, to achieve this goal. This technique is effective for two reasons. First, we guide deletions by a local error metric that bounds the change of the vectors at the affected simplices and maintains regions near critical points to prevent topological changes. Second, piecewise-linear interpolation over Delaunay triangulations is known to give good approximations of scalar fields. Since a vector field can be regarded as a collection of component scalar fields, a Delaunay triangulation can preserve each component and thus the structure of the vector field as a whole. We provide experimental evidence showing the effectiveness of our technique and its ability to preserve features of both two and three dimensional vector fields.

Abstract: When a triangulation of a set of points and edges is required, the constrained Delaunay triangulation is often the preferred choice because of its well-shaped triangles. However, in applications like terrain modeling, it is sometimes necessary to have flexibility to optimize some other aspect of the triangulation, while still having nicely-shaped triangles and including a set of constraints. Higher order Delaunay triangulations were introduced to provide a class of well-shaped triangulations, flexible enough to allow the optimization of some extra criterion. But they are not able to handle constraints: a single constraining edge may cause that all triangulations with that edge have high order, allowing ill-shaped triangles at any part of the triangulation. In this paper we generalize the concept of the constrained Delaunay triangulation to higher order constrained Delaunay triangulations. We study several possible definitions that assure that an order-k constrained Delaunay triangulation exists for any k>=0, while maintaining the character of higher order Delaunay triangulations of point sets. Several properties of these definitions are studied, and efficient algorithms to support computations with order-k constrained Delaunay triangulations are discussed. For the special case of k=1, we show that many criteria can be optimized efficiently in the presence of constraints.

Abstract: This paper describes an approach based on Zernike moments and Delaunay triangulation for localization of hand-written text in machine printed text documents. The Zernike moments of the image are first evaluated and we classify the text as hand-written using the nearest neighbor classifier. These features are independent of size, slant, orientation, translation and other variations in handwritten text. We then use Delaunay triangulation to reclassify the misclassified text regions. When imposing Delaunay triangulation on the centroid points of the connected components, we extract features based on the triangles and reclassify the text. We remove the noise components in the document as part of the preprocessing step so this method works well on noisy documents. The success rate of the method is found to be 86%. Also for specific hand-written elements such as signatures or similar text the accuracy is found to be even higher at 93%.

Abstract: This article presents a matching algorithm developed for a generic object tracking system. Matching is a critical part for the effectiveness of tracking. The proposed method is a probabilistic algorithm inspired from the emerging "discriminative random fields". Points are associated according to their visual similarity and to spatial relations in their neighborhood, based on a Delaunay triangulation. Experimental results are presented to validate this contribution.