@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}, } @inproceedings{10897, abstract = {Taking images is an efficient way to collect data about the physical world. It can be done fast and in exquisite detail. By definition, image processing is the field that concerns itself with the computation aimed at harnessing the information contained in images [10]. This talk is concerned with topological information. Our main thesis is that persistent homology [5] is a useful method to quantify and summarize topological information, building a bridge that connects algebraic topology with applications. We provide supporting evidence for this thesis by touching upon four technical developments in the overlap between persistent homology and image processing.}, author = {Edelsbrunner, Herbert}, booktitle = {Graph-Based Representations in Pattern Recognition}, isbn = {9783642382208}, issn = {1611-3349}, location = {Vienna, Austria}, pages = {182--183}, publisher = {Springer Nature}, title = {{Persistent homology in image processing}}, doi = {10.1007/978-3-642-38221-5_19}, volume = {7877}, year = {2013}, } @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}, } @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}, } @article{3159, abstract = {The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.}, author = {Mileyko, Yuriy and Edelsbrunner, Herbert and Price, Charles and Weitz, Joshua}, journal = {PLoS One}, number = {6}, publisher = {Public Library of Science}, title = {{Hierarchical ordering of reticular networks}}, doi = {10.1371/journal.pone.0036715}, volume = {7}, year = {2012}, }