Showing papers by "Herbert Edelsbrunner published in 2012"

Hierarchical ordering of reticular networks.

Abstract: The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

Dual Complexes of Cubical Subdivisions of ℝ n

Abstract: We use a distortion to define the dual complex of a cubical subdivision of ℝn as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝn .

Add isotropic Gaussian kernels at own risk: more and more resilient modes in higher dimensions

Abstract: It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n ≥ 2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpinan and Williams exhibited n+1 isotropic Gaussian kernels in Rn with n+2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points and that the resilience of the extra mode grows like √n. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.

Alexander duality for functions: the persistent behavior of land and water and shore

Abstract: This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: Sspacen+1 -> [0,1] and a decomposition Sspacen+1 = Uspace ∪ Vspace into two (n+1)-manifolds with common boundary Mspace, we prove elementary relationships between the persistence diagrams of f restricted to Uspace, to Vspace, and to Mspace.

The Adaptive Topology of a Digital Image

Abstract: In order to enjoy a digital version of the Jordan Curve Theorem, it is common to use the closed topology for the foreground and the open topology for the background of a 2-dimensional binary image. In this paper, we introduce a single topology that enjoys this theorem for all thresholds decomposing a real-valued image into foreground and background. This topology is easy to construct and it generalizes to n-dimensional images.

The Medusa of Spatial Sorting: Topological Construction

Abstract: We consider the simultaneous movement of finitely many colored points in space, calling it a spatial sorting process. The name suggests a purpose that drives the collection to a configuration of increased or decreased order. Mapping such a process to a subset of space-time, we use persistent homology measurements of the time function to characterize the process topologically.

The Medusa of Spatial Sorting: 3D Kinetic Alpha Complexes and Implementation

Abstract: Motivated by an application in cell biology, we consider spatial sorting processes defined by particles moving from an initial to a final configuration. We describe an algorithm for constructing a cell complex in space-time, called the medusa, that measures topological properties of the sorting process. The algorithm requires an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. We report on several techniques to accelerate the computation.

On the optimality of functionals over triangulations of Delaunay sets

Abstract: In this short paper we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations. A Delaunay set in E is a set of points X for which there are positive numbers r and R such that every open d-ball of radius r contains at most one point and every closed d-ball of radius R contains at least one point of X. In this paper we consider Delaunay sets in general position, that is, no d + 2 points in X lie on a common (d− 1)-sphere. By a triangulation of X we mean a simplicial complex whose vertex set is X. For finite sets the simplices decompose the convex hull of the set, while for Delaunay sets X the simplices decompose E. We say that a triangulation T is uniformly bounded if there exists a positive number q = q(T ) that is greater than or equal to the circumradii of all d-simplices in the triangulation: R(S) 6 q for all d-simplices S of T . We denote the family of all uniformly bounded triangulations of X by Θ(X). Delaunay sets were introduced byBorisDelaunay (1924), who called them (r, R)-systems. He proved that for any Delaunay set X there exists a unique Delaunay tesselation DT (X) (see, for instance, [1]). If X is in general position, then DT (X) is a triangulation of X in the sense defined above. Since the circumradius of any simplex is at most R, the Delaunay triangulation is uniformly bounded with q = R, that is, DT (X) ∈ Θ(X). We note that every Delaunay set also has triangulations that are not uniformly bounded, and it is not difficult to construct them. We want to remind the reader of a related open problem about Delaunay sets: is it true that for every planar Delaunay set X and every positive number C there exists a triangle ∆ that contains none of the points in X and has area greater than C? While we heard of this question from Michael Boshernitzan, it is sometimes referred to as Danzer’s problem. Let F be a functional defined on d-simplices S. (For instance, F (S) may be the sum of squares of edge lengths multiplied by the volume of S.) We only consider functionals that are continuous with respect to the parameters describing the simplices, for example, the lengths of their edges. Let X be a finite set in E and T any triangulation of X. Then F can be defined on T as F (T ) = ∑ S∈T F (S). It is clear that this definition cannot be used for infinite sets. We therefore define the (lower) density of F for a uniformly bounded triangulation T of a Delaunay set X as

Modes of gaussian mixtures and an inequality for the distance between curves in space

Abstract: This dissertation studies high dimensional problems from a low dimensional perspective. First, we explore rectifiable curves in high-dimensional space by using the Frechet distance between and total curvatures of the two curves to bound the difference of their lengths. We create this bound by mapping the curves into R3 while preserving the length between the curves and increasing neither the total curvature of the curves nor the Frechet distance between them. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner for dimensions greater than three and it generalizes a result by Fary and Chakerian. In the second half of the dissertation, we analyze Gaussian mixtures. In particular, we consider the sum of n + 1 identical isotropic Gaussians, where each Gaussian is centered at the vertex of a regular n-simplex. We prove that all critical points are located on a set of one-dimensional lines (axes) connecting barycenters of complementary faces of the simplex. Fixing the width of the Gaussians and varying the diameter of the simplex from zero to infinity by increasing a parameter that we call the scale factor, we find the window of scale factors for which the Gaussian mixture has more modes, or local maxima, than components. We analyze these modes using the one-dimensional axes that contain the critical points. We see that the extra mode created is subtle, but becomes more pronounced as the dimension increases.

On the Optimality of Functionals over Triangulations of Delaunay Sets

Abstract: In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations.

On the Configuration Space of Steiner Minimal Trees

Abstract: Among other results, we prove the following theorem about Steiner minimal trees in ddimensional Euclidean space: if two finite sets in R have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.

Functionals on Triangulations of Delaunay Sets

Abstract: We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.