@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{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}, } @misc{9737, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, publisher = {Public Library of Science}, title = {{Root traits computed by DynamicRoots for the maize root shown in fig 2}}, doi = {10.1371/journal.pone.0127657.s001}, year = {2015}, } @article{1792, abstract = {Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.}, author = {Pausinger, Florian and Svane, Anne}, journal = {Journal of Complexity}, number = {6}, pages = {773 -- 797}, publisher = {Academic Press}, title = {{A Koksma-Hlawka inequality for general discrepancy systems}}, doi = {10.1016/j.jco.2015.06.002}, volume = {31}, year = {2015}, } @phdthesis{1399, abstract = {This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold.}, author = {Pausinger, Florian}, issn = {2663-337X}, pages = {144}, publisher = {Institute of Science and Technology Austria}, title = {{On the approximation of intrinsic volumes}}, year = {2015}, } @inbook{10893, abstract = {Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data.}, author = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik}, booktitle = {Topological Methods in Data Analysis and Visualization III }, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {55--69}, publisher = {Springer}, title = {{Toward the extraction of saddle periodic orbits}}, doi = {10.1007/978-3-319-04099-8_4}, volume = {1}, year = {2014}, } @article{1816, abstract = {Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.}, author = {Huber, Stefan and Held, Martin and Meerwald, Peter and Kwitt, Roland}, journal = {International Journal of Computational Geometry and Applications}, number = {1}, pages = {61 -- 86}, publisher = {World Scientific Publishing}, title = {{Topology-preserving watermarking of vector graphics}}, doi = {10.1142/S0218195914500034}, volume = {24}, year = {2014}, } @article{1842, abstract = {We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices.}, author = {Cibulka, Josef and Gao, Pu and Krcál, Marek and Valla, Tomáš and Valtr, Pavel}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {64 -- 79}, publisher = {Springer}, title = {{On the geometric ramsey number of outerplanar graphs}}, doi = {10.1007/s00454-014-9646-x}, volume = {53}, year = {2014}, } @article{1876, abstract = {We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.}, author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg}, issn = {16093321}, journal = {Moscow Mathematical Journal}, number = {3}, pages = {491 -- 504}, publisher = {Independent University of Moscow}, title = {{Functionals on triangulations of delaunay sets}}, doi = {10.17323/1609-4514-2014-14-3-491-504}, volume = {14}, year = {2014}, } @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}, issn = {1573-8795}, journal = {Journal of Mathematical Sciences}, 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{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{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{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{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}, } @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}, } @inproceedings{10892, abstract = {In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.}, author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter}, booktitle = {25th International Symposium, ISAAC 2014}, isbn = {9783319130743}, issn = {1611-3349}, location = {Jeonju, Korea}, pages = {117--127}, publisher = {Springer Nature}, title = {{Planar matchings for weighted straight skeletons}}, doi = {10.1007/978-3-319-13075-0_10}, volume = {8889}, year = {2014}, }