@article{1295, abstract = {Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls.}, author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel}, journal = {Electronic Notes in Discrete Mathematics}, pages = {169 -- 174}, publisher = {Elsevier}, title = {{Multiple covers with balls II: Weighted averages}}, doi = {10.1016/j.endm.2016.09.030}, volume = {54}, year = {2016}, } @article{1292, abstract = {We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.}, author = {Durst, Sebastian and Kegel, Marc and Klukas, Mirko D}, journal = {Acta Mathematica Hungarica}, number = {2}, pages = {441 -- 455}, publisher = {Springer}, title = {{Computing the Thurston–Bennequin invariant in open books}}, doi = {10.1007/s10474-016-0648-4}, volume = {150}, year = {2016}, } @article{1330, abstract = {In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.}, author = {Akopyan, Arseniy and Balitskiy, Alexey}, journal = {Israel Journal of Mathematics}, number = {2}, pages = {833 -- 845}, publisher = {Springer}, title = {{Billiards in convex bodies with acute angles}}, doi = {10.1007/s11856-016-1429-z}, volume = {216}, year = {2016}, } @article{1360, abstract = {We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. }, author = {Akopyan, Arseniy and Balitskiy, Alexey and Karasev, Roman and Sharipova, Anastasia}, journal = {Proceedings of the American Mathematical Society}, number = {10}, pages = {4501 -- 4513}, publisher = {American Mathematical Society}, title = {{Elementary approach to closed billiard trajectories in asymmetric normed spaces}}, doi = {10.1090/proc/13062}, volume = {144}, year = {2016}, } @article{1408, abstract = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.}, author = {Franek, Peter and Krcál, Marek}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {126 -- 164}, publisher = {Springer}, title = {{On computability and triviality of well groups}}, doi = {10.1007/s00454-016-9794-2}, volume = {56}, year = {2016}, } @article{1289, abstract = {Aiming at the automatic diagnosis of tumors using narrow band imaging (NBI) magnifying endoscopic (ME) images of the stomach, we combine methods from image processing, topology, geometry, and machine learning to classify patterns into three classes: oval, tubular and irregular. 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 and Kuvaev, Roman and Kashin, Sergey}, journal = {Pattern Recognition Letters}, number = {1}, pages = {13 -- 22}, publisher = {Elsevier}, title = {{The classification of endoscopy images with persistent homology}}, doi = {10.1016/j.patrec.2015.12.012}, volume = {83}, year = {2016}, } @article{1617, abstract = {We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]d is generated for fixed m∈ℕ by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 1/10 d/N1/2+1/2d ≤EDN∗(P)≤ √d(log N) 1/2/N1/2+1/2d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points.}, author = {Pausinger, Florian and Steinerberger, Stefan}, journal = {Journal of Complexity}, pages = {199 -- 216}, publisher = {Academic Press}, title = {{On the discrepancy of jittered sampling}}, doi = {10.1016/j.jco.2015.11.003}, volume = {33}, year = {2016}, } @inproceedings{5806, abstract = {Although the concept of functional plane for naive plane is studied and reported in the literature in great detail, no similar study is yet found for naive sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is ‘mono-jumping’ but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane.}, author = {Biswas, Ranita and Bhowmick, Partha}, booktitle = {Discrete Geometry for Computer Imagery}, isbn = {978-3-319-32359-6}, issn = {0302-9743}, location = {Nantes, France}, pages = {256--267}, publisher = {Springer Nature}, title = {{On functionality of quadraginta octants of naive sphere with application to circle drawing}}, doi = {10.1007/978-3-319-32360-2_20}, volume = {9647}, year = {2016}, } @inbook{5805, abstract = {Discretization of sphere in the integer space follows a particular discretization scheme, which, in principle, conforms to some topological model. This eventually gives rise to interesting topological properties of a discrete spherical surface, which need to be investigated for its analytical characterization. This paper presents some novel results on the local topological properties of the naive model of discrete sphere. They follow from the bijection of each quadraginta octant of naive sphere with its projection map called f -map on the corresponding functional plane and from the characterization of certain jumps in the f-map. As an application, we have shown how these properties can be used in designing an efficient reconstruction algorithm for a naive spherical surface from an input voxel set when it is sparse or noisy.}, author = {Sen, Nabhasmita and Biswas, Ranita and Bhowmick, Partha}, booktitle = {Computational Topology in Image Context}, isbn = {978-3-319-39440-4}, issn = {1611-3349}, location = {Marseille, France}, pages = {253--264}, publisher = {Springer Nature}, title = {{On some local topological properties of naive discrete sphere}}, doi = {10.1007/978-3-319-39441-1_23}, volume = {9667}, year = {2016}, } @inbook{5809, abstract = {A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness.}, author = {Biswas, Ranita and Bhowmick, Partha and Brimkov, Valentin E.}, booktitle = {Combinatorial image analysis}, isbn = {978-3-319-26144-7}, issn = {1611-3349}, location = {Kolkata, India}, pages = {86--100}, publisher = {Springer Nature}, title = {{On the connectivity and smoothness of discrete spherical circles}}, doi = {10.1007/978-3-319-26145-4_7}, volume = {9448}, year = {2016}, } @article{1662, abstract = {We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.}, author = {Edelsbrunner, Herbert and Pausinger, Florian}, journal = {Advances in Mathematics}, pages = {674 -- 703}, publisher = {Academic Press}, title = {{Approximation and convergence of the intrinsic volume}}, doi = {10.1016/j.aim.2015.10.004}, volume = {287}, year = {2016}, } @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}, } @inproceedings{1483, abstract = {Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.}, author = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland}, location = {Boston, MA, USA}, pages = {4741 -- 4748}, publisher = {IEEE}, title = {{A stable multi-scale kernel for topological machine learning}}, doi = {10.1109/CVPR.2015.7299106}, year = {2015}, } @inproceedings{1495, abstract = {Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations. }, author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel and Kurlin, Vitaliy}, booktitle = {Proceedings of the 27th Canadian Conference on Computational Geometry}, location = {Ontario, Canada}, pages = {128--135}, publisher = {Queen's University}, title = {{Relaxed disk packing}}, volume = {2015-August}, year = {2015}, } @inproceedings{1510, abstract = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. }, author = {Franek, Peter and Krcál, Marek}, location = {Eindhoven, Netherlands}, pages = {842 -- 856}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{On computability and triviality of well groups}}, doi = {10.4230/LIPIcs.SOCG.2015.842}, volume = {34}, year = {2015}, } @inbook{1531, abstract = {The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Visualization and Processing of Higher Order Descriptors for Multi-Valued Data}, editor = {Hotz, Ingrid and Schultz, Thomas}, isbn = {978-3-319-15089-5}, pages = {257 -- 267}, publisher = {Springer}, title = {{Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature}}, doi = {10.1007/978-3-319-15090-1_13}, volume = {40}, year = {2015}, } @article{1555, abstract = {We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.}, author = {Knipl, Diána and Pilarczyk, Pawel and Röst, Gergely}, issn = {1536-0040}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {2}, pages = {980 -- 1017}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Rich bifurcation structure in a two patch vaccination model}}, doi = {10.1137/140993934}, volume = {14}, 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}, } @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}, } @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}, }