@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},
}

@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},
}

@article{1682,
  abstract     = {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.},
  author       = {Franek, Peter and Krcál, Marek},
  journal      = {Journal of the ACM},
  number       = {4},
  publisher    = {ACM},
  title        = {{Robust satisfiability of systems of equations}},
  doi          = {10.1145/2751524},
  volume       = {62},
  year         = {2015},
}

@article{1710,
  abstract     = {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 → ∞.},
  author       = {Akopyan, Arseniy and Plakhov, Alexander},
  journal      = {Society for Industrial and Applied Mathematics},
  number       = {4},
  pages        = {2754 -- 2769},
  publisher    = {SIAM},
  title        = {{Minimal resistance of curves under the single impact assumption}},
  doi          = {10.1137/140993843},
  volume       = {47},
  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},
}

@article{1792,
  abstract     = {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.},
  author       = {Pausinger, Florian and Svane, Anne},
  journal      = {Journal of Complexity},
  number       = {6},
  pages        = {773 -- 797},
  publisher    = {Academic Press},
  title        = {{A Koksma-Hlawka inequality for general discrepancy systems}},
  doi          = {10.1016/j.jco.2015.06.002},
  volume       = {31},
  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{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},
  title        = {{Persistent homology: Theory and practice}},
  doi          = {10.4171/120-1/3},
  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},
}