TY - JOUR AB - We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights. AU - Biedl, Therese AU - Held, Martin AU - Huber, Stefan AU - Kaaser, Dominik AU - Palfrader, Peter ID - 1584 IS - 5 JF - Computational Geometry: Theory and Applications TI - Reprint of: Weighted straight skeletons in the plane VL - 48 ER - TY - JOUR AB - We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights. AU - Biedl, Therese AU - Held, Martin AU - Huber, Stefan AU - Kaaser, Dominik AU - Palfrader, Peter ID - 1582 IS - 2 JF - Computational Geometry: Theory and Applications TI - Weighted straight skeletons in the plane VL - 48 ER - TY - JOUR AB - We study the characteristics of straight skeletons of monotone polygonal chains and use them to devise an algorithm for computing positively weighted straight skeletons of monotone polygons. Our algorithm runs in O(nlogn) time and O(n) space, where n denotes the number of vertices of the polygon. AU - Biedl, Therese AU - Held, Martin AU - Huber, Stefan AU - Kaaser, Dominik AU - Palfrader, Peter ID - 1583 IS - 2 JF - Information Processing Letters TI - A simple algorithm for computing positively weighted straight skeletons of monotone polygons VL - 115 ER - TY - CHAP AB - The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed. AU - Aichholzer, Oswin AU - Biedl, Therese AU - Hackl, Thomas AU - Held, Martin AU - Huber, Stefan AU - Palfrader, Peter AU - Vogtenhuber, Birgit ID - 1590 SN - 978-3-319-27260-3 T2 - Graph Drawing and Network Visualization TI - Representing directed trees as straight skeletons VL - 9411 ER - TY - JOUR AB - We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K→ ℝn on a finite simplicial complex K and α > 0, it holds that each function g: K → ℝn such that ||g - f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K > 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings. AU - Franek, Peter AU - Krcál, Marek ID - 1682 IS - 4 JF - Journal of the ACM TI - Robust satisfiability of systems of equations VL - 62 ER - TY - JOUR AB - We consider the hollow on the half-plane {(x, y) : y ≤ 0} ⊂ ℝ2 defined by a function u : (-1, 1) → ℝ, u(x) < 0, and a vertical flow of point particles incident on the hollow. It is assumed that u satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph(u) and goes away without hitting the graph of u anymore. We solve the problem: find the function u minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the y-axis. Assuming that the resistance of u ≡ 0 equals 1, we show that the minimal resistance equals π/2 - 2arctan(1/2) ≈ 0.6435. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730-2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space {(x1,...,xd,y) : y ≤ 0} ⊂ ℝd+1 is defined by a radial function U satisfying the SIC, U(x) = u(|x|), with x = (x1,...,xd), u(ξ) < 0 for 0 ≤ ξ < 1, and u(ξ) = 0 for ξ ≥ 1, and the flow is parallel to the y-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as d → ∞. AU - Akopyan, Arseniy AU - Plakhov, Alexander ID - 1710 IS - 4 JF - Society for Industrial and Applied Mathematics TI - Minimal resistance of curves under the single impact assumption VL - 47 ER - TY - JOUR AB - We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory. AU - Akopyan, Arseniy AU - Pirogov, Sergey AU - Rybko, Aleksandr ID - 1828 IS - 1 JF - Journal of Statistical Physics TI - Invariant measures of genetic recombination process VL - 160 ER - TY - JOUR AB - We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid Z2. AU - Pausinger, Florian AU - Steinerberger, Stefan ID - 1938 IS - 6 JF - Physics Letters, Section A TI - On the distribution of local extrema in quantum chaos VL - 379 ER - TY - JOUR AB - Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility. AU - Edelsbrunner, Herbert AU - Jablonski, Grzegorz AU - Mrozek, Marian ID - 2035 IS - 5 JF - Foundations of Computational Mathematics TI - The persistent homology of a self-map VL - 15 ER - TY - JOUR AB - We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard. AU - Attali, Dominique AU - Bauer, Ulrich AU - Devillers, Olivier AU - Glisse, Marc AU - Lieutier, André ID - 1805 IS - 8 JF - Computational Geometry: Theory and Applications TI - Homological reconstruction and simplification in R3 VL - 48 ER - TY - JOUR AB - We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth. AU - Symonova, Olga AU - Topp, Christopher AU - Edelsbrunner, Herbert ID - 1793 IS - 6 JF - PLoS One TI - DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots VL - 10 ER - TY - GEN AU - Symonova, Olga AU - Topp, Christopher AU - Edelsbrunner, Herbert ID - 9737 TI - Root traits computed by DynamicRoots for the maize root shown in fig 2 ER - TY - JOUR AB - Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology. AU - Pausinger, Florian AU - Svane, Anne ID - 1792 IS - 6 JF - Journal of Complexity TI - A Koksma-Hlawka inequality for general discrepancy systems VL - 31 ER - TY - THES AB - This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold. AU - Pausinger, Florian ID - 1399 SN - 2663-337X TI - On the approximation of intrinsic volumes ER - TY - CHAP AB - Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data. AU - Kasten, Jens AU - Reininghaus, Jan AU - Reich, Wieland AU - Scheuermann, Gerik ED - Bremer, Peer-Timo ED - Hotz, Ingrid ED - Pascucci, Valerio ED - Peikert, Ronald ID - 10893 SN - 1612-3786 T2 - Topological Methods in Data Analysis and Visualization III TI - Toward the extraction of saddle periodic orbits VL - 1 ER - TY - JOUR AB - Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques. AU - Huber, Stefan AU - Held, Martin AU - Meerwald, Peter AU - Kwitt, Roland ID - 1816 IS - 1 JF - International Journal of Computational Geometry and Applications TI - Topology-preserving watermarking of vector graphics VL - 24 ER - TY - JOUR AB - We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices. AU - Cibulka, Josef AU - Gao, Pu AU - Krcál, Marek AU - Valla, Tomáš AU - Valtr, Pavel ID - 1842 IS - 1 JF - Discrete & Computational Geometry TI - On the geometric ramsey number of outerplanar graphs VL - 53 ER - TY - JOUR AB - 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. AU - Dolbilin, Nikolai AU - Edelsbrunner, Herbert AU - Glazyrin, Alexey AU - Musin, Oleg ID - 1876 IS - 3 JF - Moscow Mathematical Journal SN - 16093321 TI - Functionals on triangulations of delaunay sets VL - 14 ER - TY - JOUR AB - We propose an algorithm for the generalization of cartographic objects that can be used to represent maps on different scales. AU - Alexeev, V V AU - Bogaevskaya, V G AU - Preobrazhenskaya, M M AU - Ukhalov, A Y AU - Edelsbrunner, Herbert AU - Yakimova, Olga ID - 1929 IS - 6 JF - Journal of Mathematical Sciences SN - 1072-3374 TI - An algorithm for cartographic generalization that preserves global topology VL - 203 ER - TY - JOUR AB - (Figure Presented) Data acquisition, numerical inaccuracies, and sampling often introduce noise in measurements and simulations. Removing this noise is often necessary for efficient analysis and visualization of this data, yet many denoising techniques change the minima and maxima of a scalar field. For example, the extrema can appear or disappear, spatially move, and change their value. This can lead to wrong interpretations of the data, e.g., when the maximum temperature over an area is falsely reported being a few degrees cooler because the denoising method is unaware of these features. Recently, a topological denoising technique based on a global energy optimization was proposed, which allows the topology-controlled denoising of 2D scalar fields. While this method preserves the minima and maxima, it is constrained by the size of the data. We extend this work to large 2D data and medium-sized 3D data by introducing a novel domain decomposition approach. It allows processing small patches of the domain independently while still avoiding the introduction of new critical points. Furthermore, we propose an iterative refinement of the solution, which decreases the optimization energy compared to the previous approach and therefore gives smoother results that are closer to the input. We illustrate our technique on synthetic and real-world 2D and 3D data sets that highlight potential applications. AU - Günther, David AU - Jacobson, Alec AU - Reininghaus, Jan AU - Seidel, Hans AU - Sorkine Hornung, Olga AU - Weinkauf, Tino ID - 1930 IS - 12 JF - IEEE Transactions on Visualization and Computer Graphics TI - Fast and memory-efficient topological denoising of 2D and 3D scalar fields VL - 20 ER -