Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm

Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures

Efficient Distance Transformation for Path-based Metrics

In many applications, separable algorithms have demonstrated their efficiency to perform high performance volumetric processing of shape, such as distance transformation or medial axis extraction. In the literature, several authors have discussed about conditions on the metric to be considered in a separable approach. In this article, we present generic separable algorithms to efficiently compute Voronoi maps and distance transformations for a large class of metrics. Focusing on path-based norms (chamfer masks, neighborhood sequences...), we propose efficient algorithms to compute such volumetric transformation in dimension $n$. We describe a new $O(n\cdot N^n\cdot\log{N}\cdot(n+\log f))$ algorithm for shapes in a $N^n$ domain for chamfer norms with a rational ball of $f$ facets (compared to $O(f^{\lfloor\frac{n}{2}\rfloor}\cdot N^n)$ with previous approaches). Last we further investigate an even more elaborate algorithm with the same worst-case complexity, but reaching a complexity of $O(n\cdot N^n\cdot\log{f}\cdot(n+\log f))$ experimentally, under assumption of regularity distribution of the mask vectors

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let T={triangle_1,...,triangle_n} be a set of of n pairwise-disjoint triangles in R^3, and let B be a convex polytope in R^3 with a constant number of faces. For each i, let C_i = triangle_i oplus r_i B denote the Minkowski sum of triangle_i with a copy of B scaled by r_i>0. We show that if the scaling factors r_1, ..., r_n are chosen randomly then the expected complexity of the union of C_1, ..., C_n is O(n^{2+epsilon), for any epsilon > 0; the constant of proportionality depends on epsilon and the complexity of B. The worst-case bound can be Theta(n^3). We also consider a special case of this problem in which T is a set of points in R^3 and B is a unit cube in R^3, i.e., each C_i is a cube of side-length 2r_i. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log^2 n), and it improves to O(n log n) if the scaling factors are chosen randomly from a "well-behaved" probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of d, we show that the expected complexity of the union of the hypercubes is O(n^floor[d/2] log n) and the bound improves to O(n^floor[d/2]) if the scaling factors are chosen from a "well-behaved" pdf. The worst-case bounds are Theta(n^2) in R^3, and Theta(n^{ceil[d/2]}) in higher dimensions

A Systematic Review of Algorithms with Linear-time Behaviour to Generate Delaunay and Voronoi Tessellations

Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented

Constructing L∞ Voronoi diagrams in 2D and 3D

Voronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p → ∞ lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm

On Volumetric Shape Reconstruction from Implicit Forms

International audienceIn this paper we report on the evaluation of volumetric shape reconstruction methods that consider as input implicit forms in 3D. Many visual applications build implicit representations of shapes that are converted into explicit shape representations using geometric tools such as the Marching Cubes algorithm. This is the case with image based reconstructions that produce point clouds from which implicit functions are computed, with for instance a Poisson reconstruction approach. While the Marching Cubes method is a versatile solution with proven efficiency, alternative solutions exist with different and complementary properties that are of interest for shape modeling. In this paper, we propose a novel strategy that builds on Centroidal Voronoi Tessellations (CVTs). These tessellations provide volumetric and surface representations with strong regularities in addition to provably more accurate approximations of the implicit forms considered. In order to compare the existing strategies, we present an extensive evaluation that analyzes various properties of the main strategies for implicit to explicit volumetric conversions: Marching cubes, Delaunay refinement and CVTs, including accuracy and shape quality of the resulting shape mesh