@article{1563, abstract = {For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at {\tt http://www.pawelpilarczyk.com/combtop/}.}, author = {Graff, Grzegorz and Pilarczyk, Pawel}, journal = {Topological Methods in Nonlinear Analysis}, number = {1}, pages = {273 -- 286}, publisher = {Juliusz Schauder Center for Nonlinear Studies}, title = {{An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds}}, doi = {10.12775/TMNA.2015.014}, volume = {45}, year = {2015}, } @inproceedings{1567, abstract = {My personal journey to the fascinating world of geometric forms started more than 30 years ago with the invention of alpha shapes in the plane. It took about 10 years before we generalized the concept to higher dimensions, we produced working software with a graphics interface for the three-dimensional case. At the same time, we added homology to the computations. Needless to say that this foreshadowed the inception of persistent homology, because it suggested the study of filtrations to capture the scale of a shape or data set. Importantly, this method has fast algorithms. The arguably most useful result on persistent homology is the stability of its diagrams under perturbations.}, author = {Edelsbrunner, Herbert}, booktitle = {23rd International Symposium}, location = {Los Angeles, CA, United States}, publisher = {Springer Nature}, title = {{Shape, homology, persistence, and stability}}, volume = {9411}, year = {2015}, } @inproceedings{1568, abstract = {Aiming at the automatic diagnosis of tumors from narrow band imaging (NBI) magnifying endoscopy (ME) images of the stomach, we combine methods from image processing, computational topology, and machine learning to classify patterns into normal, tubular, vessel. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.}, author = {Dunaeva, Olga and Edelsbrunner, Herbert and Lukyanov, Anton and Machin, Michael and Malkova, Daria}, booktitle = {Proceedings - 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing}, location = {Timisoara, Romania}, pages = {7034731}, publisher = {IEEE}, title = {{The classification of endoscopy images with persistent homology}}, doi = {10.1109/SYNASC.2014.81}, year = {2015}, } @article{1578, abstract = {We prove that the dual of the digital Voronoi diagram constructed by flooding the plane from the data points gives a geometrically and topologically correct dual triangulation. This provides the proof of correctness for recently developed GPU algorithms that outperform traditional CPU algorithms for constructing two-dimensional Delaunay triangulations.}, author = {Cao, Thanhtung and Edelsbrunner, Herbert and Tan, Tiowseng}, journal = {Computational Geometry}, number = {7}, pages = {507 -- 519}, publisher = {Elsevier}, title = {{Triangulations from topologically correct digital Voronoi diagrams}}, doi = {10.1016/j.comgeo.2015.04.001}, volume = {48}, year = {2015}, } @article{1582, abstract = {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.}, author = {Biedl, Therese and Held, Martin and Huber, Stefan and Kaaser, Dominik and Palfrader, Peter}, journal = {Computational Geometry: Theory and Applications}, number = {2}, pages = {120 -- 133}, publisher = {Elsevier}, title = {{Weighted straight skeletons in the plane}}, doi = {10.1016/j.comgeo.2014.08.006}, volume = {48}, year = {2015}, } @article{1583, abstract = {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.}, author = {Biedl, Therese and Held, Martin and Huber, Stefan and Kaaser, Dominik and Palfrader, Peter}, journal = {Information Processing Letters}, number = {2}, pages = {243 -- 247}, publisher = {Elsevier}, title = {{A simple algorithm for computing positively weighted straight skeletons of monotone polygons}}, doi = {10.1016/j.ipl.2014.09.021}, volume = {115}, year = {2015}, } @article{1584, abstract = {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.}, author = {Biedl, Therese and Held, Martin and Huber, Stefan and Kaaser, Dominik and Palfrader, Peter}, journal = {Computational Geometry: Theory and Applications}, number = {5}, pages = {429 -- 442}, publisher = {Elsevier}, title = {{Reprint of: Weighted straight skeletons in the plane}}, doi = {10.1016/j.comgeo.2015.01.004}, volume = {48}, year = {2015}, } @inbook{1590, abstract = {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.}, author = {Aichholzer, Oswin and Biedl, Therese and Hackl, Thomas and Held, Martin and Huber, Stefan and Palfrader, Peter and Vogtenhuber, Birgit}, booktitle = {Graph Drawing and Network Visualization}, isbn = {978-3-319-27260-3}, location = {Los Angeles, CA, United States}, pages = {335 -- 347}, publisher = {Springer Nature}, title = {{Representing directed trees as straight skeletons}}, doi = {10.1007/978-3-319-27261-0_28}, volume = {9411}, year = {2015}, } @phdthesis{1399, abstract = {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.}, author = {Pausinger, Florian}, issn = {2663-337X}, pages = {144}, publisher = {Institute of Science and Technology Austria}, title = {{On the approximation of intrinsic volumes}}, year = {2015}, } @inproceedings{1424, abstract = {We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data.}, author = {Kwitt, Roland and Huber, Stefan and Niethammer, Marc and Lin, Weili and Bauer, Ulrich}, location = {Montreal, Canada}, pages = {3070 -- 3078}, publisher = {Neural Information Processing Systems}, title = {{Statistical topological data analysis-A kernel perspective}}, volume = {28}, year = {2015}, } @misc{9737, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, publisher = {Public Library of Science}, title = {{Root traits computed by DynamicRoots for the maize root shown in fig 2}}, doi = {10.1371/journal.pone.0127657.s001}, year = {2015}, } @inproceedings{2905, abstract = {Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions.}, author = {Edelsbrunner, Herbert and Morozovy, Dmitriy}, location = {Kraków, Poland}, pages = {31 -- 50}, publisher = {European Mathematical Society Publishing House}, title = {{Persistent homology: Theory and practice}}, doi = {10.4171/120-1/3}, year = {2014}, } @inbook{10817, abstract = {The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption.}, author = {Günther, David and Reininghaus, Jan and Seidel, Hans-Peter and Weinkauf, Tino}, booktitle = {Topological Methods in Data Analysis and Visualization III.}, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {135--150}, publisher = {Springer Nature}, title = {{Notes on the simplification of the Morse-Smale complex}}, doi = {10.1007/978-3-319-04099-8_9}, year = {2014}, } @inproceedings{10886, abstract = {We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Topological Methods in Data Analysis and Visualization III }, isbn = {9783319040981}, issn = {2197-666X}, pages = {249--262}, publisher = {Springer}, title = {{Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature}}, doi = {10.1007/978-3-319-04099-8_16}, year = {2014}, } @inproceedings{10892, abstract = {In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.}, author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter}, booktitle = {25th International Symposium, ISAAC 2014}, isbn = {9783319130743}, issn = {1611-3349}, location = {Jeonju, Korea}, pages = {117--127}, publisher = {Springer Nature}, title = {{Planar matchings for weighted straight skeletons}}, doi = {10.1007/978-3-319-13075-0_10}, volume = {8889}, year = {2014}, } @inbook{10893, abstract = {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.}, author = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik}, booktitle = {Topological Methods in Data Analysis and Visualization III }, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {55--69}, publisher = {Springer}, title = {{Toward the extraction of saddle periodic orbits}}, doi = {10.1007/978-3-319-04099-8_4}, volume = {1}, year = {2014}, } @inproceedings{10894, abstract = {PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert}, booktitle = {ICMS 2014: International Congress on Mathematical Software}, isbn = {9783662441985}, issn = {1611-3349}, location = {Seoul, South Korea}, pages = {137--143}, publisher = {Springer Berlin Heidelberg}, title = {{PHAT – Persistent Homology Algorithms Toolbox}}, doi = {10.1007/978-3-662-44199-2_24}, volume = {8592}, year = {2014}, } @article{1816, abstract = {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.}, author = {Huber, Stefan and Held, Martin and Meerwald, Peter and Kwitt, Roland}, journal = {International Journal of Computational Geometry and Applications}, number = {1}, pages = {61 -- 86}, publisher = {World Scientific Publishing}, title = {{Topology-preserving watermarking of vector graphics}}, doi = {10.1142/S0218195914500034}, volume = {24}, year = {2014}, } @article{1842, abstract = {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.}, author = {Cibulka, Josef and Gao, Pu and Krcál, Marek and Valla, Tomáš and Valtr, Pavel}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {64 -- 79}, publisher = {Springer}, title = {{On the geometric ramsey number of outerplanar graphs}}, doi = {10.1007/s00454-014-9646-x}, volume = {53}, year = {2014}, } @article{1876, 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.}, author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg}, issn = {16093321}, journal = {Moscow Mathematical Journal}, number = {3}, pages = {491 -- 504}, publisher = {Independent University of Moscow}, title = {{Functionals on triangulations of delaunay sets}}, doi = {10.17323/1609-4514-2014-14-3-491-504}, volume = {14}, year = {2014}, }