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Showing papers by "Herbert Edelsbrunner published in 2011"


Book
15 Sep 2011
TL;DR: This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry and finds problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points.
Abstract: This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.

132 citations


Book
07 Aug 2011
TL;DR: A generalization of the zone theorem of [EOS], [CGL] to arrangements of curves, and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.
Abstract: Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.

123 citations


Book
17 Aug 2011
TL;DR: The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles and extend the analysis to three dimensions, where they concentrate on the case of spheres, which is relevant for the three-dimensional unit-distance problem.
Abstract: The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles. They also extend the analysis to three dimensions, where they concentrate on the case of spheres, which is relevant for the three-dimensional unit-distance problem. They obtain upper bounds for certain quantities. The authors believe that the techniques they use are of independent interest. >

89 citations


Book ChapterDOI
01 Jan 2011
TL;DR: It is demonstrated that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities.
Abstract: We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, α. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of α, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the Cosmic Web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field. Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the Cosmic Web.

86 citations


Book
26 Aug 2011
TL;DR: This paper studies the problem of calculating and storing arrangements using subquadratic space and preprocessing, so that, given any query point, the face containing p can be calculated efficiently, and reports faces in an arrangement of line segments in time.
Abstract: An arrangement of n lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists of O(n2) regions, called faces. In this paper we study the problem of calculating and storing arrangements implicitly, using subquadratic space and preprocessing, so that, given any query point p, we can calculate efficiently the face containing p. First, we consider the case of lines and show that with L(n) space1 and L(n3/2) preprocessing time, we can answer face queries in L(√n) + O(K) time, where K is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: 1) given a set of n points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, 2) given a simple polygonal path G, form a data structure from which we can find the convex hull of any subpath of G quickly, and 3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a trade-off between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in L(n1/3) time, given L(n4/3) space and L(n5/3) preprocessing time.Lastly, we note that our techniques allow us to compute m faces in an arrangement of n lines in time L(m2/3n2/3 + n), which is nearly optimal.

68 citations


Book
18 Aug 2011
TL;DR: The maximum number of faces boundingm distinct cells in an arrangement ofn planes is O(m2/3n logn +n2); the authors can calculatem such cells specified by a point in each, in worst-case timeO(m3/5−δn4/5+2δ+m+n logm), for any collection of points no three of which are collinear.
Abstract: We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m2/3n logn +n2); we can calculatem such cells specified by a point in each, in worst-case timeO(m2/3n log3n+n2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m2/3n logn+n2), but this number is onlyO(m3/5??n4/5+2?+m+n logm), for any?>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m3/4??n3/4+3?+m) log2n+n logn logm) for any?>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m3/4??n3/4+3?+m] log2n+n logn logm) for any?>0. (v) The maximum number of facets (i.e., (d?1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m2/3nd/3 logn+nd?1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.

59 citations


Proceedings ArticleDOI
06 Nov 2011
TL;DR: The concept of the regularized visual hull which reduces the effect of jittering and refraction by ensuring consistency with one 2D image is developed which guarantees connectedness through adjustments to the 3D reconstruction that minimize global error.
Abstract: We study the 3D reconstruction of plant roots from multiple 2D images. To meet the challenge caused by the delicate nature of thin branches, we make three innovations to cope with the sensitivity to image quality and calibration. First, we model the background as a harmonic function to improve the segmentation of the root in each 2D image. Second, we develop the concept of the regularized visual hull which reduces the effect of jittering and refraction by ensuring consistency with one 2D image. Third, we guarantee connectedness through adjustments to the 3D reconstruction that minimize global error. Our software is part of a biological phenotype/genotype study of agricultural root systems. It has been tested on more than 40 plant roots and results are promising in terms of reconstruction quality and efficiency.

51 citations


Book ChapterDOI
01 Jan 2011
TL;DR: The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :𝕄→ℝ2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent.
Abstract: The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :𝕄→ℝ2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming 𝕄 is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.

34 citations


Journal ArticleDOI
TL;DR: Persistent homology assigns to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then it is proved that the robustness is stable.
Abstract: By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

33 citations


Proceedings ArticleDOI
06 Nov 2011
TL;DR: It is proved that the corresponding 1-parameter family of persistence diagrams have norms that go rapidly to zero as time goes to infinity, which rationalizes experimental observations about scale-space.
Abstract: Interpreting an image as a function on a compact subset of the Euclidean plane, we get its scale-space by diffusion, spreading the image over the entire plane. This generates a 1-parameter family of functions alternatively defined as convolutions with a progressively wider Gaussian kernel. We prove that the corresponding 1-parameter family of persistence diagrams have norms that go rapidly to zero as time goes to infinity. This result rationalizes experimental observations about scale-space. We hope this will lead to targeted improvements of related computer vision methods.

32 citations


Posted Content
TL;DR: The scale dependent Betti number as mentioned in this paper is a new descriptor of the weblike pattern in the distribution of galaxies and matter, which formalizes the topological information content of the cosmic mass distribution.
Abstract: We introduce a new descriptor of the weblike pattern in the distribution of galaxies and matter: the scale dependent Betti numbers which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic used in earlier analyses of cosmological models. The richer information content of Betti numbers goes along with the availability of fast algorithms to compute them. When measured as a function of scale they provide a "Betti signature" for a point distribution that is a sensitive yet robust discriminator of structure. The signature is highly effective in revealing differences in structure arising in different cosmological models, and is exploited towards distinguishing between different dark energy models and may likewise be used to trace primordial non-Gaussianities. In this study we demonstrate the potential of Betti numbers by studying their behaviour in simulations of cosmologies differing in the nature of their dark energy.

Book ChapterDOI
01 Jan 2011
TL;DR: This work addresses the problem of covering Rn with congruent balls, while minimizing the number of balls that contain an average point, and gives a closed formula for the covering density that depends on the distortion parameter.
Abstract: We address the problem of covering Rn with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in Rn, for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3.

Posted Content
TL;DR: The robustness of the homology classes under perturbations of f is quantified using well groups, and it is shown how to read the ranks of these groups from the same extended persistence diagram.
Abstract: Given a function $f: \Xspace \to \Rspace$ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$ In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram The special case $\Xspace = \Rspace^3$ has ramifications in the fields of medical imaging and scientific visualization

Journal ArticleDOI
TL;DR: An algorithm for finding all local maxima on a smoothly embedded 2-manifold by transporting the concept from the smooth to the piecewise linear category and showing that its performance in practice is orders of magnitudes superior.
Abstract: The elevation function on a smoothly embedded 2-manifold in ℝ3 reflects the multiscale topography of cavities and protrusions as local maxima The function has been useful in identifying coarse docking configurations for protein pairs Transporting the concept from the smooth to the piecewise linear category, this article describes an algorithm for finding all local maxima While its worst-case running time is the same as of the algorithm used in prior work, its performance in practice is orders of magnitudes superior We cast light on this improvement by relating the running time to the total absolute Gaussian curvature of the 2-manifold

Posted Content
TL;DR: This note proves elementary relationships between the persistence diagrams of f restricted to Uspace, to Vspace, and to Mspace.
Abstract: This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function $f: S^{n+1} \to [0,1]$ and a decomposition $S^{n+1} = U \cup V$ such that $M = \U \cap V$ is an $n$-manifold, we prove elementary relationships between the persistence diagrams of $f$ restricted to $U$, to $V$, and to $M$.