@article{1180,
abstract = {In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.},
author = {Akopyan, Arseniy and Bárány, Imre and Robins, Sinai},
issn = {00018708},
journal = {Advances in Mathematics},
pages = {627 -- 644},
publisher = {Academic Press},
title = {{Algebraic vertices of non-convex polyhedra}},
doi = {10.1016/j.aim.2016.12.026},
volume = {308},
year = {2017},
}
@article{1433,
abstract = {Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology.},
author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert},
issn = { 07477171},
journal = {Journal of Symbolic Computation},
pages = {76 -- 90},
publisher = {Academic Press},
title = {{Phat - Persistent homology algorithms toolbox}},
doi = {10.1016/j.jsc.2016.03.008},
volume = {78},
year = {2017},
}
@article{1022,
abstract = {We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.},
author = {Pranav, Pratyush and Edelsbrunner, Herbert and Van De Weygaert, Rien and Vegter, Gert and Kerber, Michael and Jones, Bernard and Wintraecken, Mathijs},
issn = {00358711},
journal = {Monthly Notices of the Royal Astronomical Society},
number = {4},
pages = {4281 -- 4310},
publisher = {Oxford University Press},
title = {{The topology of the cosmic web in terms of persistent Betti numbers}},
doi = {10.1093/mnras/stw2862},
volume = {465},
year = {2017},
}
@article{1065,
abstract = {We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.},
author = {Chatterjee, Krishnendu and Osang, Georg F},
issn = {00200190},
journal = {Information Processing Letters},
pages = {25 -- 29},
publisher = {Elsevier},
title = {{Pushdown reachability with constant treewidth}},
doi = {10.1016/j.ipl.2017.02.003},
volume = {122},
year = {2017},
}
@article{1072,
abstract = {Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.},
author = {Bauer, Ulrich and Edelsbrunner, Herbert},
journal = {Transactions of the American Mathematical Society},
number = {5},
pages = {3741 -- 3762},
publisher = {American Mathematical Society},
title = {{The Morse theory of Čech and delaunay complexes}},
volume = {369},
year = {2017},
}
@article{1173,
abstract = {We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.},
author = {Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg and Nikitenko, Anton},
issn = {02099683},
journal = {Combinatorica},
number = {5},
pages = {887 -- 910},
publisher = {Springer},
title = {{The Voronoi functional is maximized by the Delaunay triangulation in the plane}},
doi = {10.1007/s00493-016-3308-y},
volume = {37},
year = {2017},
}
@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},
}
@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},
}
@article{1216,
abstract = {A framework fo r extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ2.},
author = {Kasten, Jens and Reininghaus, Jan and Hotz, Ingrid and Hege, Hans and Noack, Bernd and Daviller, Guillaume and Morzyński, Marek},
journal = {Archives of Mechanics},
number = {1},
pages = {55 -- 80},
publisher = {Polish Academy of Sciences Publishing House},
title = {{Acceleration feature points of unsteady shear flows}},
volume = {68},
year = {2016},
}
@article{1222,
abstract = {We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.},
author = {Musin, Oleg and Nikitenko, Anton},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {1 -- 20},
publisher = {Springer},
title = {{Optimal packings of congruent circles on a square flat torus}},
doi = {10.1007/s00454-015-9742-6},
volume = {55},
year = {2016},
}
@inproceedings{1237,
abstract = {Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.},
author = {Krcál, Marek and Pilarczyk, Pawel},
location = {Marseille, France},
pages = {140 -- 151},
publisher = {Springer},
title = {{Computation of cubical Steenrod squares}},
doi = {10.1007/978-3-319-39441-1_13},
volume = {9667},
year = {2016},
}
@article{1252,
abstract = {We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.},
author = {Harker, Shaun and Kokubu, Hiroshi and Mischaikow, Konstantin and Pilarczyk, Pawel},
journal = {Proceedings of the American Mathematical Society},
number = {4},
pages = {1787 -- 1801},
publisher = {American Mathematical Society},
title = {{Inducing a map on homology from a correspondence}},
doi = {10.1090/proc/12812},
volume = {144},
year = {2016},
}
@article{1254,
abstract = {We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.},
author = {Golmakani, Ali and Luzzatto, Stefano and Pilarczyk, Pawel},
journal = {Experimental Mathematics},
number = {2},
pages = {116 -- 124},
publisher = {Taylor and Francis},
title = {{Uniform expansivity outside a critical neighborhood in the quadratic family}},
doi = {10.1080/10586458.2015.1048011},
volume = {25},
year = {2016},
}
@article{1272,
abstract = {We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.},
author = {Held, Martin and Huber, Stefan and Palfrader, Peter},
journal = {Computer-Aided Design and Applications},
number = {5},
pages = {712 -- 721},
publisher = {Taylor and Francis},
title = {{Generalized offsetting of planar structures using skeletons}},
doi = {10.1080/16864360.2016.1150718},
volume = {13},
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{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{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{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{1149,
abstract = {We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.},
author = {Miyaji, Tomoyuki and Pilarczyk, Pawel and Gameiro, Marcio and Kokubu, Hiroshi and Mischaikow, Konstantin},
journal = {Applied Numerical Mathematics},
pages = {34 -- 47},
publisher = {Elsevier},
title = {{A study of rigorous ODE integrators for multi scale set oriented computations}},
doi = {10.1016/j.apnum.2016.04.005},
volume = {107},
year = {2016},
}
@article{2035,
abstract = {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.
},
author = {Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, Marian},
journal = {Foundations of Computational Mathematics},
number = {5},
pages = {1213 -- 1244},
publisher = {Springer},
title = {{The persistent homology of a self-map}},
doi = {10.1007/s10208-014-9223-y},
volume = {15},
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},
location = {Los Angeles, CA, United States},
publisher = {Springer},
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},
location = {Los Angeles, CA, United States},
pages = {335 -- 347},
publisher = {Springer},
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},
}
@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},
}
@article{1793,
abstract = {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.},
author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert},
journal = {PLoS One},
number = {6},
publisher = {Public Library of Science},
title = {{DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots}},
doi = {10.1371/journal.pone.0127657},
volume = {10},
year = {2015},
}
@article{1805,
abstract = {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.},
author = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André},
journal = {Computational Geometry: Theory and Applications},
number = {8},
pages = {606 -- 621},
publisher = {Elsevier},
title = {{Homological reconstruction and simplification in R3}},
doi = {10.1016/j.comgeo.2014.08.010},
volume = {48},
year = {2015},
}
@article{1828,
abstract = {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.},
author = {Akopyan, Arseniy and Pirogov, Sergey and Rybko, Aleksandr},
journal = {Journal of Statistical Physics},
number = {1},
pages = {163 -- 167},
publisher = {Springer},
title = {{Invariant measures of genetic recombination process}},
doi = {10.1007/s10955-015-1238-5},
volume = {160},
year = {2015},
}
@article{1938,
abstract = {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.},
author = {Pausinger, Florian and Steinerberger, Stefan},
journal = {Physics Letters, Section A},
number = {6},
pages = {535 -- 541},
publisher = {Elsevier},
title = {{On the distribution of local extrema in quantum chaos}},
doi = {10.1016/j.physleta.2014.12.010},
volume = {379},
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},
pages = {144},
publisher = {IST 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},
}
@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},
}
@article{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 Jan Reininghaus and Hotz, Ingrid},
journal = {Mathematics and Visualization},
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},
}
@inproceedings{2012,
abstract = {The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice.},
author = {Iglesias Ham, Mabel and Kerber, Michael and Uhler, Caroline},
location = {Halifax, Canada},
pages = {155 -- 161},
publisher = {Unknown},
title = {{Sphere packing with limited overlap}},
year = {2014},
}
@inproceedings{2043,
abstract = {Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node.},
author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan},
booktitle = {Proceedings of the Workshop on Algorithm Engineering and Experiments},
editor = { McGeoch, Catherine and Meyer, Ulrich},
location = {Portland, USA},
pages = {31 -- 38},
publisher = {Society of Industrial and Applied Mathematics},
title = {{Distributed computation of persistent homology}},
doi = {10.1137/1.9781611973198.4},
year = {2014},
}
@inbook{2044,
abstract = {We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation.},
author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan},
booktitle = {Topological Methods in Data Analysis and Visualization III},
editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald},
pages = {103 -- 117},
publisher = {Springer},
title = {{Clear and Compress: Computing Persistent Homology in Chunks}},
doi = {10.1007/978-3-319-04099-8_7},
year = {2014},
}
@inproceedings{2153,
abstract = {We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).},
author = {Bauer, Ulrich and Lesnick, Michael},
booktitle = {Proceedings of the Annual Symposium on Computational Geometry},
location = {Kyoto, Japan},
pages = {355 -- 364},
publisher = {ACM},
title = {{Induced matchings of barcodes and the algebraic stability of persistence}},
doi = {10.1145/2582112.2582168},
year = {2014},
}
@inproceedings{2155,
abstract = {Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).},
author = {Bauer, Ulrich and Edelsbrunner, Herbert},
booktitle = {Proceedings of the Annual Symposium on Computational Geometry},
location = {Kyoto, Japan},
pages = {484 -- 490},
publisher = {ACM},
title = {{The morse theory of Čech and Delaunay filtrations}},
doi = {10.1145/2582112.2582167},
year = {2014},
}
@inproceedings{2156,
abstract = {We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).},
author = {Bauer, Ulrich and Ge, Xiaoyin and Wang, Yusu},
booktitle = {Proceedings of the Annual Symposium on Computational Geometry},
location = {Kyoto, Japan},
pages = {464 -- 473},
publisher = {ACM},
title = {{Measuring distance between Reeb graphs}},
doi = {10.1145/2582112.2582169},
year = {2014},
}