@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},
}
@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},
}
@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{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{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{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{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{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{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{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{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},
}
@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},
}
@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},
}
@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{1816,
abstract = {Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.},
author = {Huber, Stefan and Held, Martin and Meerwald, Peter and Kwitt, Roland},
journal = {International Journal of Computational Geometry and Applications},
number = {1},
pages = {61 -- 86},
publisher = {World Scientific Publishing},
title = {{Topology-preserving watermarking of vector graphics}},
doi = {10.1142/S0218195914500034},
volume = {24},
year = {2014},
}
@article{1842,
abstract = {We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices.},
author = {Cibulka, Josef and Gao, Pu and Krcál, Marek and Valla, Tomáš and Valtr, Pavel},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {64 -- 79},
publisher = {Springer},
title = {{On the geometric ramsey number of outerplanar graphs}},
doi = {10.1007/s00454-014-9646-x},
volume = {53},
year = {2014},
}
@article{1876,
abstract = {We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.},
author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg},
journal = {Moscow Mathematical Journal},
number = {3},
pages = {491 -- 504},
publisher = {Independent University of Moscow},
title = {{Functionals on triangulations of delaunay sets}},
volume = {14},
year = {2014},
}
@article{1929,
abstract = {We propose an algorithm for the generalization of cartographic objects that can be used to represent maps on different scales.},
author = {Alexeev, V V and Bogaevskaya, V G and Preobrazhenskaya, M M and Ukhalov, A Y and Edelsbrunner, Herbert and Yakimova, Olga},
journal = {Journal of Mathematical Sciences (United States)},
number = {6},
pages = {754 -- 760},
publisher = {Springer},
title = {{An algorithm for cartographic generalization that preserves global topology}},
doi = {10.1007/s10958-014-2165-8},
volume = {203},
year = {2014},
}
@article{1930,
abstract = {(Figure Presented) Data acquisition, numerical inaccuracies, and sampling often introduce noise in measurements and simulations. Removing this noise is often necessary for efficient analysis and visualization of this data, yet many denoising techniques change the minima and maxima of a scalar field. For example, the extrema can appear or disappear, spatially move, and change their value. This can lead to wrong interpretations of the data, e.g., when the maximum temperature over an area is falsely reported being a few degrees cooler because the denoising method is unaware of these features. Recently, a topological denoising technique based on a global energy optimization was proposed, which allows the topology-controlled denoising of 2D scalar fields. While this method preserves the minima and maxima, it is constrained by the size of the data. We extend this work to large 2D data and medium-sized 3D data by introducing a novel domain decomposition approach. It allows processing small patches of the domain independently while still avoiding the introduction of new critical points. Furthermore, we propose an iterative refinement of the solution, which decreases the optimization energy compared to the previous approach and therefore gives smoother results that are closer to the input. We illustrate our technique on synthetic and real-world 2D and 3D data sets that highlight potential applications.},
author = {Günther, David and Jacobson, Alec and Reininghaus, Jan and Seidel, Hans and Sorkine Hornung, Olga and Weinkauf, Tino},
journal = {IEEE Transactions on Visualization and Computer Graphics},
number = {12},
pages = {2585 -- 2594},
publisher = {IEEE},
title = {{Fast and memory-efficient topological denoising of 2D and 3D scalar fields}},
doi = {10.1109/TVCG.2014.2346432},
volume = {20},
year = {2014},
}
@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},
}
@inproceedings{2177,
abstract = {We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.},
author = {Edelsbrunner, Herbert and Parsa, Salman},
booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms},
location = {Portland, USA},
pages = {152 -- 160},
publisher = {SIAM},
title = {{On the computational complexity of betti numbers reductions from matrix rank}},
doi = {10.1137/1.9781611973402.11},
year = {2014},
}
@article{2184,
abstract = {Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.},
author = {Čadek, Martin and Krcál, Marek and Matoušek, Jiří and Sergeraert, Francis and Vokřínek, Lukáš and Wagner, Uli},
journal = {Journal of the ACM},
number = {3},
publisher = {ACM},
title = {{Computing all maps into a sphere}},
doi = {10.1145/2597629},
volume = {61},
year = {2014},
}
@article{2255,
abstract = {Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.},
author = {Edelsbrunner, Herbert and Pausinger, Florian},
issn = {09249907},
journal = {Journal of Mathematical Imaging and Vision},
number = {1},
pages = {164 -- 177},
publisher = {Springer},
title = {{Stable length estimates of tube-like shapes}},
doi = {10.1007/s10851-013-0468-x},
volume = {50},
year = {2014},
}
@inproceedings{2905,
abstract = {Persistent homology is a recent grandchild of homology that has found use in
science and engineering as well as in mathematics. This paper surveys the method as well
as the applications, neglecting completeness in favor of highlighting ideas and directions.},
author = {Edelsbrunner, Herbert and Morozovy, Dmitriy},
location = {Kraków, Poland},
pages = {31 -- 50},
publisher = {European Mathematical Society Publishing House},
title = {{Persistent homology: Theory and practice}},
doi = {10.4171/120-1/3},
year = {2014},
}
@book{6853,
abstract = {This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields.},
author = {Edelsbrunner, Herbert},
isbn = {9783319059563},
issn = {2191-530X},
pages = {IX, 110},
publisher = {Springer International Publishing},
title = {{A Short Course in Computational Geometry and Topology}},
doi = {10.1007/978-3-319-05957-0},
year = {2014},
}
@inproceedings{2209,
abstract = {A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985.
},
author = {Biedl, Therese and Held, Martin and Huber, Stefan},
location = {St. Petersburg, Russia},
pages = {37 -- 46},
publisher = {IEEE},
title = {{Recognizing straight skeletons and Voronoi diagrams and reconstructing their input}},
doi = {10.1109/ISVD.2013.11},
year = {2013},
}
@inproceedings{2210,
abstract = {A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon. In this paper, we ask the reverse question: Given the straight skeleton (in form of a tree with a drawing in the plane, but with the exact position of the leaves unspecified), can we reconstruct the polygon? We show that in most cases there exists at most one polygon; in the remaining case there is an infinite number of polygons determined by one angle that can range in an interval. We can find this (set of) polygon(s) in linear time in the Real RAM computer model.},
author = {Biedl, Therese and Held, Martin and Huber, Stefan},
booktitle = {29th European Workshop on Computational Geometry},
location = {Braunschweig, Germany},
pages = {95 -- 98},
publisher = {TU Braunschweig},
title = {{Reconstructing polygons from embedded straight skeletons}},
year = {2013},
}
@article{2304,
abstract = {This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter.},
author = {Pausinger, Florian},
journal = {Electronic Notes in Discrete Mathematics},
pages = {43 -- 50},
publisher = {Elsevier},
title = {{Van der Corput sequences and linear permutations}},
doi = {10.1016/j.endm.2013.07.008},
volume = {43},
year = {2013},
}
@inproceedings{2807,
abstract = {We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex. },
author = {Čadek, Martin and Krcál, Marek and Matoušek, Jiří and Vokřínek, Lukáš and Wagner, Uli},
booktitle = {45th Annual ACM Symposium on theory of computing},
location = {Palo Alto, CA, United States},
pages = {595 -- 604},
publisher = {ACM},
title = {{Extending continuous maps: Polynomiality and undecidability}},
doi = {10.1145/2488608.2488683},
year = {2013},
}
@inproceedings{2812,
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 ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on 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},
booktitle = {Proceedings of the 29th annual symposium on Computational Geometry},
location = {Rio de Janeiro, Brazil},
pages = {117 -- 125},
publisher = {ACM},
title = {{Homological reconstruction and simplification in R3}},
doi = {10.1145/2462356.2462373},
year = {2013},
}
@article{2822,
abstract = {Identification of genes that control root system architecture in crop plants requires innovations that enable high-throughput and accurate measurements of root system architecture through time. We demonstrate the ability of a semiautomated 3D in vivo imaging and digital phenotyping pipeline to interrogate the quantitative genetic basis of root system growth in a rice biparental mapping population, Bala x Azucena. We phenotyped >1,400 3D root models and >57,000 2D images for a suite of 25 traits that quantified the distribution, shape, extent of exploration, and the intrinsic size of root networks at days 12, 14, and 16 of growth in a gellan gum medium. From these data we identified 89 quantitative trait loci, some of which correspond to those found previously in soil-grown plants, and provide evidence for genetic tradeoffs in root growth allocations, such as between the extent and thoroughness of exploration. We also developed a multivariate method for generating and mapping central root architecture phenotypes and used it to identify five major quantitative trait loci (r2 = 24-37%), two of which were not identified by our univariate analysis. Our imaging and analytical platform provides a means to identify genes with high potential for improving root traits and agronomic qualities of crops.},
author = {Topp, Christopher and Iyer Pascuzzi, Anjali and Anderson, Jill and Lee, Cheng and Zurek, Paul and Symonova, Olga and Zheng, Ying and Bucksch, Alexander and Mileyko, Yuriy and Galkovskyi, Taras and Moore, Brad and Harer, John and Edelsbrunner, Herbert and Mitchell Olds, Thomas and Weitz, Joshua and Benfey, Philip},
journal = {PNAS},
number = {18},
pages = {E1695 -- E1704},
publisher = {National Academy of Sciences},
title = {{3D phenotyping and quantitative trait locus mapping identify core regions of the rice genome controlling root architecture}},
doi = {10.1073/pnas.1304354110},
volume = {110},
year = {2013},
}
@inproceedings{2843,
abstract = {Mathematical objects can be measured unambiguously, but not so objects from our physical world. Even the total length of tubelike shapes has its difficulties. We introduce a combination of geometric, probabilistic, and topological methods to design a stable length estimate for tube-like shapes; that is: one that is insensitive to small shape changes.},
author = {Edelsbrunner, Herbert and Pausinger, Florian},
booktitle = {17th IAPR International Conference on Discrete Geometry for Computer Imagery},
location = {Seville, Spain},
pages = {XV -- XIX},
publisher = {Springer},
title = {{Stable length estimates of tube-like shapes}},
doi = {10.1007/978-3-642-37067-0},
volume = {7749},
year = {2013},
}