@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}, } @inproceedings{2906, abstract = {Motivated by an application in cell biology, we describe an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. Our algorithm is implemented using CGAL, following the exact geometric computation paradigm. We report on several techniques to accelerate the computation that turn our implementation applicable to the underlying biological problem.}, author = {Kerber, Michael and Edelsbrunner, Herbert}, booktitle = {2013 Proceedings of the 15th Workshop on Algorithm Engineering and Experiments}, location = {New Orleans, LA, United States}, pages = {70 -- 77}, publisher = {Society of Industrial and Applied Mathematics}, title = {{3D kinetic alpha complexes and their implementation}}, doi = {10.1137/1.9781611972931.6}, year = {2013}, } @article{2939, abstract = {In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ > 0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0, 1), the running time is O (C (1 - δ) Γ R d (n) log n), where C (1 - δ) Γ is the number of homology classes with persistence at least (1 - δ) Γ, n is the total number of simplices in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} > 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n).}, author = {Chen, Chao and Kerber, Michael}, journal = {Computational Geometry: Theory and Applications}, number = {4}, pages = {435 -- 447}, publisher = {Elsevier}, title = {{An output sensitive algorithm for persistent homology}}, doi = {10.1016/j.comgeo.2012.02.010}, volume = {46}, year = {2013}, } @article{6588, abstract = {First we note that the best polynomial approximation to vertical bar x vertical bar on the set, which consists of an interval on the positive half-axis and a point on the negative half-axis, can be given by means of the classical Chebyshev polynomials. Then we explore the cases when a solution of the related problem on two intervals can be given in elementary functions.}, author = {Pausinger, Florian}, issn = {1812-9471}, journal = {Journal of Mathematical Physics, Analysis, Geometry}, number = {1}, pages = {63--78}, publisher = {B. Verkin Institute for Low Temperature Physics and Engineering}, title = {{Elementary solutions of the Bernstein problem on two intervals}}, volume = {8}, year = {2012}, } @article{2849, author = {Edelsbrunner, Herbert and Strelkova, Nataliya}, journal = {Russian Mathematical Surveys}, number = {6}, pages = {1167 -- 1168}, publisher = {IOP Publishing Ltd.}, title = {{On the configuration space of Steiner minimal trees}}, doi = {10.1070/RM2012v067n06ABEH004820}, volume = {67}, year = {2012}, } @article{2902, abstract = {We present an algorithm for simplifying linear cartographic objects and results obtained with a computer program implementing this algorithm. }, author = {Edelsbrunner, Herbert and Musin, Oleg and Ukhalov, Alexey and Yakimova, Olga and Alexeev, Vladislav and Bogaevskaya, Victoriya and Gorohov, Andrey and Preobrazhenskaya, Margarita}, journal = {Modeling and Analysis of Information Systems}, number = {6}, pages = {152 -- 160}, publisher = {Russian Academy of Sciences}, title = {{Fractal and computational geometry for generalizing cartographic objects}}, volume = {19}, year = {2012}, } @inproceedings{2903, abstract = {In order to enjoy a digital version of the Jordan Curve Theorem, it is common to use the closed topology for the foreground and the open topology for the background of a 2-dimensional binary image. In this paper, we introduce a single topology that enjoys this theorem for all thresholds decomposing a real-valued image into foreground and background. This topology is easy to construct and it generalizes to n-dimensional images.}, author = {Edelsbrunner, Herbert and Symonova, Olga}, location = {New Brunswick, NJ, USA }, pages = {41 -- 48}, publisher = {IEEE}, title = {{The adaptive topology of a digital image}}, doi = {10.1109/ISVD.2012.11}, year = {2012}, } @article{2904, abstract = {Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P(i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b - 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.}, author = {Pausinger, Florian}, issn = {2118-8572}, journal = {Journal de Theorie des Nombres des Bordeaux}, number = {3}, pages = {729 -- 749}, publisher = {Université de Bordeaux}, title = {{Weak multipliers for generalized van der Corput sequences}}, doi = {10.5802/jtnb.819}, volume = {24}, year = {2012}, } @article{2912, author = {Edelsbrunner, Herbert and Strelkova, Nataliya}, journal = {Russian Mathematical Surveys}, number = {6}, pages = {1167–1168}, publisher = {Russian Academy of Sciences}, title = {{On the configuration space for the shortest networks}}, doi = {10.4213/rm9503}, volume = {67}, year = {2012}, } @article{2941, author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Musin, Oleg}, journal = {Russian Mathematical Surveys}, number = {4}, pages = {781 -- 783}, publisher = {IOP Publishing}, title = {{On the optimality of functionals over triangulations of Delaunay sets}}, doi = {10.1070/RM2012v067n04ABEH004807}, volume = {67}, year = {2012}, } @inproceedings{2971, abstract = {We study the task of interactive semantic labeling of a segmentation hierarchy. To this end we propose a framework interleaving two components: an automatic labeling step, based on a Conditional Random Field whose dependencies are defined by the inclusion tree of the segmentation hierarchy, and an interaction step that integrates incremental input from a human user. Evaluated on two distinct datasets, the proposed interactive approach efficiently integrates human interventions and illustrates the advantages of structured prediction in an interactive framework. }, author = {Zankl, Georg and Haxhimusa, Yll and Ion, Adrian}, location = {Graz, Austria}, pages = {11 -- 20}, publisher = {Springer}, title = {{Interactive labeling of image segmentation hierarchies}}, doi = {10.1007/978-3-642-32717-9_2}, volume = {7476}, year = {2012}, } @article{3115, abstract = {We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(nlogn)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.}, author = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza}, journal = {Discrete & Computational Geometry}, number = {4}, pages = {964 -- 989}, publisher = {Springer}, title = {{Deconstructing approximate offsets}}, doi = {10.1007/s00454-012-9441-5}, volume = {48}, year = {2012}, } @article{3120, abstract = {We introduce a strategy based on Kustin-Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9 × 16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology. © 2012 Copyright Foundation Compositio Mathematica.}, author = {Brown, Gavin and Kerber, Michael and Reid, Miles}, journal = {Compositio Mathematica}, number = {4}, pages = {1171 -- 1194}, publisher = {Cambridge University Press}, title = {{Fano 3 folds in codimension 4 Tom and Jerry Part I}}, doi = {10.1112/S0010437X11007226}, volume = {148}, year = {2012}, } @inproceedings{3127, abstract = {When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when different graph instances have different edge sets pose a serious challenge. In this work we address this challenge for the problem of finding maximum weighted cliques. We introduce the concept of most persistent soft-clique. This is subset of vertices, that 1) is almost fully or at least densely connected, 2) occurs in all or almost all graph instances, and 3) has the maximum weight. We present a measure of clique-ness, that essentially counts the number of edge missing to make a subset of vertices into a clique. With this measure, we show that the problem of finding the most persistent soft-clique problem can be cast either as: a) a max-min two person game optimization problem, or b) a min-min soft margin optimization problem. Both formulations lead to the same solution when using a partial Lagrangian method to solve the optimization problems. By experiments on synthetic data and on real social network data, we show that the proposed method is able to reliably find soft cliques in graph data, even if that is distorted by random noise or unreliable observations.}, author = {Quadrianto, Novi and Lampert, Christoph and Chen, Chao}, booktitle = {Proceedings of the 29th International Conference on Machine Learning}, location = {Edinburgh, United Kingdom}, pages = {211--218}, publisher = {ML Research Press}, title = {{The most persistent soft-clique in a set of sampled graphs}}, year = {2012}, } @inproceedings{3129, abstract = {Let K be a simplicial complex and g the rank of its p-th homology group Hp(K) defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n×n matrices can be multiplied in O(n ω ) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently. Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of H1(K) , we improve the previously known time complexity from O(n 4) to O(n ω  + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω ) + 2 O(g) n 2logn time using annotations. }, author = {Busaryev, Oleksiy and Cabello, Sergio and Chen, Chao and Dey, Tamal and Wang, Yusu}, location = {Helsinki, Finland}, pages = {189 -- 200}, publisher = {Springer}, title = {{Annotating simplices with a homology basis and its applications}}, doi = {10.1007/978-3-642-31155-0_17}, volume = {7357}, year = {2012}, } @inproceedings{3133, abstract = {This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: S n+1 →[0, 1 and a decomposition S n+1 = U ∪ V into two (n + 1)-manifolds with common boundary M, we prove elementary relationships between the persistence diagrams of f restricted to U, to V, and to M. }, author = {Edelsbrunner, Herbert and Kerber, Michael}, booktitle = {Proceedings of the twenty-eighth annual symposium on Computational geometry }, location = {Chapel Hill, NC, USA}, pages = {249 -- 258}, publisher = {ACM}, title = {{Alexander duality for functions: The persistent behavior of land and water and shore}}, doi = {10.1145/2261250.2261287}, year = {2012}, } @inproceedings{3134, abstract = {It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n ≥ 2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpiñán and Williams exhibited n +1 isotropic Gaussian kernels in ℝ n with n + 2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points and that the resilience of the extra mode grows like √n. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes. }, author = {Edelsbrunner, Herbert and Fasy, Brittany and Rote, Günter}, booktitle = {Proceedings of the twenty-eighth annual symposium on Computational geometry }, location = {Chapel Hill, NC, USA}, pages = {91 -- 100}, publisher = {ACM}, title = {{Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions}}, doi = {10.1145/2261250.2261265}, year = {2012}, }