@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topological denoising of 2D and 3D scalar fields}}, doi = {10.1109/TVCG.2014.2346432}, volume = {20}, year = {2014}, } @unpublished{2012, abstract = {The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as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}, booktitle = {arXiv}, title = {{Sphere packing with limited overlap}}, doi = {10.48550/arXiv.1401.0468}, 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}, } @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 = {9-783-3190-5956-3}, issn = {2191-5318}, pages = {IX, 110}, publisher = {Springer Nature}, title = {{A Short Course in Computational Geometry and Topology}}, doi = {10.1007/978-3-319-05957-0}, year = {2014}, } @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{2815, abstract = {The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in ℝ1 and are generally not well studied in higher dimensions. We study a configuration of n+1 Gaussian kernels for which there are exactly n+2 modes. We show that all modes lie on a finite set of lines, which we call axes, and study the restriction of the Gaussian mixture to these axes in order to discover that there are an exponential number of critical points in this configuration. Although the existence of ghost modes remained unknown due to the difficulty of finding examples in ℝ2, we show that the resilience of ghost modes grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.}, author = {Edelsbrunner, Herbert and Fasy, Brittany Terese and Rote, Günter}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {4}, pages = {797 -- 822}, publisher = {Springer}, title = {{Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions}}, doi = {10.1007/s00454-013-9517-x}, volume = {49}, 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}, } @article{2859, abstract = {Given a continuous function f:X-R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization.}, author = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit}, journal = {Homology, Homotopy and Applications}, number = {1}, pages = {51 -- 72}, publisher = {International Press}, title = {{Homology and robustness of level and interlevel sets}}, doi = {10.4310/HHA.2013.v15.n1.a3}, volume = {15}, year = {2013}, } @article{2887, abstract = {Root system growth and development is highly plastic and is influenced by the surrounding environment. Roots frequently grow in heterogeneous environments that include interactions from neighboring plants and physical impediments in the rhizosphere. To investigate how planting density and physical objects affect root system growth, we grew rice in a transparent gel system in close proximity with another plant or a physical object. Root systems were imaged and reconstructed in three dimensions. Root-root interaction strength was calculated using quantitative metrics that characterize the extent towhich the reconstructed root systems overlap each other. Surprisingly, we found the overlap of root systems of the same genotype was significantly higher than that of root systems of different genotypes. Root systems of the same genotype tended to grow toward each other but those of different genotypes appeared to avoid each other. Shoot separation experiments excluded the possibility of aerial interactions, suggesting root communication. Staggered plantings indicated that interactions likely occur at root tips in close proximity. Recognition of obstacles also occurred through root tips, but through physical contact in a size-dependent manner. These results indicate that root systems use two different forms of communication to recognize objects and alter root architecture: root-root recognition, possibly mediated through root exudates, and root-object recognition mediated by physical contact at the root tips. This finding suggests that root tips act as local sensors that integrate rhizosphere information into global root architectural changes.}, author = {Fang, Suqin and Clark, Randy and Zheng, Ying and Iyer Pascuzzi, Anjali and Weitz, Joshua and Kochian, Leon and Edelsbrunner, Herbert and Liao, Hong and Benfey, Philip}, journal = {PNAS}, number = {7}, pages = {2670 -- 2675}, publisher = {National Academy of Sciences}, title = {{Genotypic recognition and spatial responses by rice roots}}, doi = {10.1073/pnas.1222821110}, volume = {110}, year = {2013}, } @inproceedings{2901, abstract = { We introduce the M-modes problem for graphical models: predicting the M label configurations of highest probability that are at the same time local maxima of the probability landscape. M-modes have multiple possible applications: because they are intrinsically diverse, they provide a principled alternative to non-maximum suppression techniques for structured prediction, they can act as codebook vectors for quantizing the configuration space, or they can form component centers for mixture model approximation. We present two algorithms for solving the M-modes problem. The first algorithm solves the problem in polynomial time when the underlying graphical model is a simple chain. The second algorithm solves the problem for junction chains. In synthetic and real dataset, we demonstrate how M-modes can improve the performance of prediction. We also use the generated modes as a tool to understand the topography of the probability distribution of configurations, for example with relation to the training set size and amount of noise in the data. }, author = {Chen, Chao and Kolmogorov, Vladimir and Yan, Zhu and Metaxas, Dimitris and Lampert, Christoph}, location = {Scottsdale, AZ, United States}, pages = {161 -- 169}, publisher = {JMLR}, title = {{Computing the M most probable modes of a graphical model}}, volume = {31}, year = {2013}, }