@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{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{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},
}
@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 = {0302-9743},
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},
}
@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 = {Technische Universität Darmstadt},
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 = {Universite 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 = { Uspekhi Mat. Nauk},
number = {6},
pages = {203 -- 204},
publisher = {Moscow Mathematical Society },
title = {{Configuration space for 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 = {Omnipress},
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},
}
@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},
}
@article{3256,
abstract = {We use a distortion to define the dual complex of a cubical subdivision of ℝ n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝ n.},
author = {Edelsbrunner, Herbert and Kerber, Michael},
journal = {Discrete & Computational Geometry},
number = {2},
pages = {393 -- 414},
publisher = {Springer},
title = {{Dual complexes of cubical subdivisions of ℝn}},
doi = {10.1007/s00454-011-9382-4},
volume = {47},
year = {2012},
}
@inproceedings{3265,
abstract = {We propose a mid-level statistical model for image segmentation that composes multiple figure-ground hypotheses (FG) obtained by applying constraints at different locations and scales, into larger interpretations (tilings) of the entire image. Inference is cast as optimization over sets of maximal cliques sampled from a graph connecting all non-overlapping figure-ground segment hypotheses. Potential functions over cliques combine unary, Gestalt-based figure qualities, and pairwise compatibilities among spatially neighboring segments, constrained by T-junctions and the boundary interface statistics of real scenes. Learning the model parameters is based on maximum likelihood, alternating between sampling image tilings and optimizing their potential function parameters. State of the art results are reported on the Berkeley and Stanford segmentation datasets, as well as VOC2009, where a 28% improvement was achieved.},
author = {Ion, Adrian and Carreira, Joao and Sminchisescu, Cristian},
location = {Barcelona, Spain},
publisher = {IEEE},
title = {{Image segmentation by figure-ground composition into maximal cliques}},
doi = {10.1109/ICCV.2011.6126486},
year = {2012},
}
@article{3310,
abstract = {The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a real-valued function. Our account of the subject is expository, devoid of proofs, and written for non-experts in algebraic topology.},
author = {Bendich, Paul and Cabello, Sergio and Edelsbrunner, Herbert},
journal = {Pattern Recognition Letters},
number = {11},
pages = {1436 -- 1444},
publisher = {Elsevier},
title = {{A point calculus for interlevel set homology}},
doi = {10.1016/j.patrec.2011.10.007},
volume = {33},
year = {2012},
}
@article{3331,
abstract = {Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2ρ, the topology of the induced curve can be computed with bit operations ( indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n2. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve.},
author = {Kerber, Michael and Sagraloff, Michael},
journal = { Journal of Symbolic Computation},
number = {3},
pages = {239 -- 258},
publisher = {Elsevier},
title = {{A worst case bound for topology computation of algebraic curves}},
doi = {10.1016/j.jsc.2011.11.001},
volume = {47},
year = {2012},
}
@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},
}
@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},
}
@inproceedings{3266,
abstract = {We present a joint image segmentation and labeling model (JSL) which, given a bag of figure-ground segment hypotheses extracted at multiple image locations and scales, constructs a joint probability distribution over both the compatible image interpretations (tilings or image segmentations) composed from those segments, and over their labeling into categories. The process of drawing samples from the joint distribution can be interpreted as first sampling tilings, modeled as maximal cliques, from a graph connecting spatially non-overlapping segments in the bag [1], followed by sampling labels for those segments, conditioned on the choice of a particular tiling. We learn the segmentation and labeling parameters jointly, based on Maximum Likelihood with a novel Incremental Saddle Point estimation procedure. The partition function over tilings and labelings is increasingly more accurately approximated by including incorrect configurations that a not-yet-competent model rates probable during learning. We show that the proposed methodologymatches the current state of the art in the Stanford dataset [2], as well as in VOC2010, where 41.7% accuracy on the test set is achieved.},
author = {Ion, Adrian and Carreira, Joao and Sminchisescu, Cristian},
booktitle = {NIPS Proceedings},
location = {Granada, Spain},
pages = {1827 -- 1835},
publisher = {Neural Information Processing Systems Foundation},
title = {{Probabilistic joint image segmentation and labeling}},
volume = {24},
year = {2011},
}
@article{3269,
abstract = {The unintentional scattering of light between neighboring surfaces in complex projection environments increases the brightness and decreases the contrast, disrupting the appearance of the desired imagery. To achieve satisfactory projection results, the inverse problem of global illumination must be solved to cancel this secondary scattering. In this paper, we propose a global illumination cancellation method that minimizes the perceptual difference between the desired imagery and the actual total illumination in the resulting physical environment. Using Gauss-Newton and active set methods, we design a fast solver for the bound constrained nonlinear least squares problem raised by the perceptual error metrics. Our solver is further accelerated with a CUDA implementation and multi-resolution method to achieve 1–2 fps for problems with approximately 3000 variables. We demonstrate the global illumination cancellation algorithm with our multi-projector system. Results show that our method preserves the color fidelity of the desired imagery significantly better than previous methods.},
author = {Sheng, Yu and Cutler, Barbara and Chen, Chao and Nasman, Joshua},
journal = {Computer Graphics Forum},
number = {4},
pages = {1261 -- 1268},
publisher = {Wiley-Blackwell},
title = {{Perceptual global illumination cancellation in complex projection environments}},
doi = {10.1111/j.1467-8659.2011.01985.x},
volume = {30},
year = {2011},
}
@inproceedings{3270,
abstract = {The persistence diagram of a filtered simplicial com- plex is usually computed by reducing the boundary matrix of the complex. We introduce a simple op- timization technique: by processing the simplices of the complex in decreasing dimension, we can “kill” columns (i.e., set them to zero) without reducing them. This technique completely avoids reduction on roughly half of the columns. We demonstrate that this idea significantly improves the running time of the reduction algorithm in practice. We also give an output-sensitive complexity analysis for the new al- gorithm which yields to sub-cubic asymptotic bounds under certain assumptions.},
author = {Chen, Chao and Kerber, Michael},
location = {Morschach, Switzerland},
pages = {197 -- 200},
publisher = {TU Dortmund},
title = {{Persistent homology computation with a twist}},
year = {2011},
}
@inbook{3271,
abstract = {In this paper we present an efficient framework for computation of persis- tent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure de- signed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our ap- proach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.},
author = {Wagner, Hubert and Chen, Chao and Vuçini, Erald},
booktitle = {Topological Methods in Data Analysis and Visualization II},
editor = {Peikert, Ronald and Hauser, Helwig and Carr, Hamish and Fuchs, Raphael},
pages = {91 -- 106},
publisher = {Springer},
title = {{Efficient computation of persistent homology for cubical data}},
doi = {10.1007/978-3-642-23175-9_7},
year = {2011},
}
@misc{3312,
abstract = {We study the 3D reconstruction of plant roots from multiple 2D images. To meet the challenge caused by the delicate nature of thin branches, we make three innovations to cope with the sensitivity to image quality and calibration. First, we model the background as a harmonic function to improve the segmentation of the root in each 2D image. Second, we develop the concept of the regularized visual hull which reduces the effect of jittering and refraction by ensuring consistency with one 2D image. Third, we guarantee connectedness through adjustments to the 3D reconstruction that minimize global error. Our software is part of a biological phenotype/genotype study of agricultural root systems. It has been tested on more than 40 plant roots and results are promising in terms of reconstruction quality and efficiency.},
author = {Zheng, Ying and Gu, Steve and Edelsbrunner, Herbert and Tomasi, Carlo and Benfey, Philip},
booktitle = {Proceedings of the IEEE International Conference on Computer Vision},
location = {Barcelona, Spain},
publisher = {IEEE},
title = {{Detailed reconstruction of 3D plant root shape}},
doi = {10.1109/ICCV.2011.6126475},
year = {2011},
}
@inproceedings{3313,
abstract = {Interpreting an image as a function on a compact sub- set of the Euclidean plane, we get its scale-space by diffu- sion, spreading the image over the entire plane. This gener- ates a 1-parameter family of functions alternatively defined as convolutions with a progressively wider Gaussian ker- nel. We prove that the corresponding 1-parameter family of persistence diagrams have norms that go rapidly to zero as time goes to infinity. This result rationalizes experimental observations about scale-space. We hope this will lead to targeted improvements of related computer vision methods.},
author = {Chen, Chao and Edelsbrunner, Herbert},
booktitle = {Proceedings of the IEEE International Conference on Computer Vision},
location = {Barcelona, Spain},
publisher = {IEEE},
title = {{Diffusion runs low on persistence fast}},
doi = {10.1109/ICCV.2011.6126271},
year = {2011},
}
@inproceedings{3328,
abstract = {We report on a generic uni- and bivariate algebraic kernel that is publicly available with CGAL 3.7. It comprises complete, correct, though efficient state-of-the-art implementations on polynomials, roots of polynomial systems, and the support to analyze algebraic curves defined by bivariate polynomials. The kernel design is generic, that is, various number types and substeps can be exchanged. It is accompanied with a ready-to-use interface to enable arrangements induced by algebraic curves, that have already been used as basis for various geometric applications, as arrangements on Dupin cyclides or the triangulation of algebraic surfaces. We present two novel applications: arrangements of rotated algebraic curves and Boolean set operations on polygons bounded by segments of algebraic curves. We also provide experiments showing that our general implementation is competitive and even often clearly outperforms existing implementations that are explicitly tailored for specific types of non-linear curves that are available in CGAL.},
author = {Berberich, Eric and Hemmer, Michael and Kerber, Michael},
location = {Paris, France},
pages = {179 -- 186},
publisher = {ACM},
title = {{A generic algebraic kernel for non linear geometric applications}},
doi = {10.1145/1998196.1998224},
year = {2011},
}
@inproceedings{3329,
abstract = {We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance µ 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 shape P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. An alternative algorithm, based purely on rational arithmetic, answers the same deconstruction problem, up to an uncertainty parameter, and its running time depends on the parameter δ (in addition to the other input parameters: n, δ and the radius of the disk). If the input shape is found to be approximable, the rational-arithmetic algorithm also computes an approximate solution shape for the 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 shape P with at most one more vertex than a vertex-minimal one. Our study is motivated by applications from two different domains. However, since the offset operation has numerous uses, we anticipate that the reverse question that we study here will be still more broadly applicable. We present results obtained with our implementation of the rational-arithmetic algorithm.},
author = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza},
booktitle = {Proceedings of the twenty-seventh annual symposium on Computational geometry},
location = {Paris, France},
pages = {187 -- 196},
publisher = {ACM},
title = {{Deconstructing approximate offsets}},
doi = {10.1145/1998196.1998225},
year = {2011},
}
@inproceedings{3330,
abstract = {We consider the problem of approximating all real roots of a square-free polynomial f. Given isolating intervals, our algorithm refines each of them to a width at most 2-L, that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.},
author = {Kerber, Michael and Sagraloff, Michael},
location = {California, USA},
pages = {209 -- 216},
publisher = {Springer},
title = {{Root refinement for real polynomials}},
doi = {10.1145/1993886.1993920},
year = {2011},
}
@article{3332,
abstract = {Given an algebraic hypersurface O in ℝd, how many simplices are necessary for a simplicial complex isotopic to O? We address this problem and the variant where all vertices of the complex must lie on O. We give asymptotically tight worst-case bounds for algebraic plane curves. Our results gradually improve known bounds in higher dimensions; however, the question for tight bounds remains unsolved for d ≥ 3.},
author = {Kerber, Michael and Sagraloff, Michael},
journal = {Graphs and Combinatorics},
number = {3},
pages = {419 -- 430},
publisher = {Springer},
title = {{A note on the complexity of real algebraic hypersurfaces}},
doi = {10.1007/s00373-011-1020-7},
volume = {27},
year = {2011},
}
@article{3334,
author = {Edelsbrunner, Herbert and Pach, János and Ziegler, Günter},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {1 -- 2},
publisher = {Springer},
title = {{Letter from the new editors-in-chief}},
doi = {10.1007/s00454-010-9313-9},
volume = {45},
year = {2011},
}
@inbook{3335,
abstract = {We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, α. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of α, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the Cosmic Web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field. Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the Cosmic Web.},
author = {Van De Weygaert, Rien and Vegter, Gert and Edelsbrunner, Herbert and Jones, Bernard and Pranav, Pratyush and Park, Changbom and Hellwing, Wojciech and Eldering, Bob and Kruithof, Nico and Bos, Patrick and Hidding, Johan and Feldbrugge, Job and Ten Have, Eline and Van Engelen, Matti and Caroli, Manuel and Teillaud, Monique},
booktitle = {Transactions on Computational Science XIV},
editor = {Gavrilova, Marina and Tan, Kenneth and Mostafavi, Mir},
pages = {60 -- 101},
publisher = {Springer},
title = {{Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web}},
doi = {10.1007/978-3-642-25249-5_3},
volume = {6970},
year = {2011},
}
@inproceedings{3367,
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)ΓR(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, and R(n) is the complexity of computing the rank of an n x n matrix with O(n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O(C(1-δ)Γn2.376) algorithm, a O(C(1-δ)Γn2.28) Las-Vegas algorithm, or a O(C(1-δ)Γn2+ε) Monte-Carlo algorithm for an arbitrary ε>0.},
author = {Chen, Chao and Kerber, Michael},
location = {Paris, France},
pages = {207 -- 216},
publisher = {ACM},
title = {{An output sensitive algorithm for persistent homology}},
doi = {10.1145/1998196.1998228},
year = {2011},
}
@article{3377,
abstract = {By definition, transverse intersections are stable under in- finitesimal perturbations. Using persistent homology, we ex- tend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robust- ness, the magnitude of a perturbation necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for con- tours of smooth mappings.},
author = {Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
journal = {Foundations of Computational Mathematics},
number = {3},
pages = {345 -- 361},
publisher = {Springer},
title = {{Quantifying transversality by measuring the robustness of intersections}},
doi = {10.1007/s10208-011-9090-8},
volume = {11},
year = {2011},
}
@article{3378,
abstract = {The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper in- corporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersec- tion homology gives useful information about the relationship between an embedded stratified space and its singularities. We give, and prove the correctness of, an algorithm for the computa- tion of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcom- plexes. We also derive, from Poincare ́ Duality, some structural results about persistent intersection homology.},
author = {Bendich, Paul and Harer, John},
journal = {Foundations of Computational Mathematics},
number = {3},
pages = {305 -- 336},
publisher = {Springer},
title = {{Persistent intersection homology}},
doi = {10.1007/s10208-010-9081-1},
volume = {11},
year = {2011},
}
@inproceedings{9648,
abstract = {In this paper, we establish a correspondence between the incremental algorithm for computing AT-models [8,9] and the one for computing persistent homology [6,14,15]. We also present a decremental algorithm for computing AT-models that allows to extend the persistence computation to a wider setting. Finally, we show how to combine incremental and decremental techniques for persistent homology computation.},
author = {Gonzalez-Diaz, Rocio and Ion, Adrian and Jimenez, Maria Jose and Poyatos, Regina},
booktitle = {Computer Analysis of Images and Patterns},
isbn = {9783642236716},
issn = {16113349},
location = {Seville, Spain},
pages = {286--293},
publisher = {Springer Nature},
title = {{Incremental-decremental algorithm for computing AT-models and persistent homology}},
doi = {10.1007/978-3-642-23672-3_35},
volume = {6854},
year = {2011},
}
@article{3781,
abstract = {We bound the difference in length of two curves in terms of their total curvatures and the Fréchet distance. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner, and it generalizes a result by Fáry and Chakerian.},
author = {Fasy, Brittany Terese},
journal = {Acta Sci. Math. (Szeged)},
number = {1-2},
pages = {359 -- 367},
publisher = {Szegedi Tudományegyetem},
title = {{The difference in length of curves in R^n}},
volume = {77},
year = {2011},
}
@inbook{3796,
abstract = {We address the problem of covering ℝ n with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in ℝ n , for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3.},
author = {Edelsbrunner, Herbert and Kerber, Michael},
booktitle = {Rainbow of Computer Science},
editor = {Calude, Cristian and Rozenberg, Grzegorz and Salomaa, Arto},
pages = {20 -- 35},
publisher = {Springer},
title = {{Covering and packing with spheres by diagonal distortion in R^n}},
doi = {10.1007/978-3-642-19391-0_2},
volume = {6570},
year = {2011},
}
@article{3965,
abstract = {The elevation function on a smoothly embedded 2-manifold in R-3 reflects the multiscale topography of cavities and protrusions as local maxima. The function has been useful in identifying coarse docking configurations for protein pairs. Transporting the concept from the smooth to the piecewise linear category, this paper describes an algorithm for finding all local maxima. While its worst-case running time is the same as of the algorithm used in prior work, its performance in practice is orders of magnitudes superior. We cast light on this improvement by relating the running time to the total absolute Gaussian curvature of the 2-manifold.},
author = {Wang, Bei and Edelsbrunner, Herbert and Morozov, Dmitriy},
journal = {Journal of Experimental Algorithmics},
number = {2.2},
pages = {1 -- 13},
publisher = {ACM},
title = {{Computing elevation maxima by searching the Gauss sphere}},
doi = {10.1145/1963190.1970375},
volume = {16},
year = {2011},
}
@article{3267,
abstract = {We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We study the problem with different measures: volume, diameter and radius. For volume, that is, the 1-norm of a cycle, two main results are presented. First, we prove that the problem is NP-hard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NP-hard to approximate even when the Betti number is O(1). The latter result leads to the inapproximability of the problem of computing the nonbounding cycle with the smallest volume and computing cycles representing a homology basis with the minimal total volume. As for the other two measures defined by pairwise geodesic distance, diameter and radius, we show that the localization problem is NP-hard for diameter but is polynomial for radius. Our work is restricted to homology over the ℤ2 field.},
author = {Chen, Chao and Freedman, Daniel},
journal = {Discrete & Computational Geometry},
number = {3},
pages = {425 -- 448},
publisher = {Springer},
title = {{Hardness results for homology localization}},
doi = {10.1007/s00454-010-9322-8},
volume = {45},
year = {2011},
}
@inproceedings{3336,
abstract = {We introduce TopoCut: a new way to integrate knowledge about topological properties (TPs) into random field image segmentation model. Instead of including TPs as additional constraints during minimization of the energy function, we devise an efficient algorithm for modifying the unary potentials such that the resulting segmentation is guaranteed with the desired properties. Our method is more flexible in the sense that it handles more topology constraints than previous methods, which were only able to enforce pairwise or global connectivity. In particular, our method is very fast, making it for the first time possible to enforce global topological properties in practical image segmentation tasks.},
author = {Chen, Chao and Freedman, Daniel and Lampert, Christoph},
booktitle = {CVPR: Computer Vision and Pattern Recognition},
isbn = {978-1-4577-0394-2},
location = {Colorado Springs, CO, United States},
pages = {2089 -- 2096},
publisher = {IEEE},
title = {{Enforcing topological constraints in random field image segmentation}},
doi = {10.1109/CVPR.2011.5995503},
year = {2011},
}
@inbook{3311,
abstract = {Alpha shapes have been conceived in 1981 as an attempt to define the shape of a finite set of point in the plane. Since then, connections to diverse areas in the sciences and engineering have developed, including to pattern recognition, digital shape sampling and processing, and structural molecular biology. This survey begins with a historical account and discusses geometric, algorithmic, topological, and combinatorial aspects of alpha shapes in this sequence.},
author = {Edelsbrunner, Herbert},
booktitle = {Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings},
editor = {van de Weygaert, R and Vegter, G and Ritzerveld, J and Icke, V},
publisher = {Springer},
title = {{Alpha shapes - a survey}},
year = {2011},
}
@inproceedings{10907,
abstract = {This paper presents a method to create a model of an articulated object using the planar motion in an initialization video. The model consists of rigid parts connected by points of articulation. The rigid parts are described by the positions of salient feature-points tracked throughout the video. Following a filtering step that identifies points that belong to different objects, rigid parts are found by a grouping process in a graph pyramid. Valid articulation points are selected by verifying multiple hypotheses for each pair of parts.},
author = {Artner, Nicole M. and Ion, Adrian and Kropatsch, Walter G.},
booktitle = {Graph-Based Representations in Pattern Recognition},
editor = {Jiang, Xiaoyi and Ferrer, Miquel and Torsello, Andrea},
isbn = {9783642208430},
issn = {0302-9743},
location = {Münster, Germany},
pages = {215--224},
publisher = {Springer},
title = {{Spatio-temporal extraction of articulated models in a graph pyramid}},
doi = {10.1007/978-3-642-20844-7_22},
volume = {6658},
year = {2011},
}
@inproceedings{3782,
abstract = {In cortex surface segmentation, the extracted surface is required to have a particular topology, namely, a two-sphere. We present a new method for removing topology noise of a curve or surface within the level set framework, and thus produce a cortical surface with correct topology. We define a new energy term which quantifies topology noise. We then show how to minimize this term by computing its functional derivative with respect to the level set function. This method differs from existing methods in that it is inherently continuous and not digital; and in the way that our energy directly relates to the topology of the underlying curve or surface, versus existing knot-based measures which are related in a more indirect fashion. The proposed flow is validated empirically.},
author = {Chen, Chao and Freedman, Daniel},
booktitle = { Conference proceedings MCV 2010},
location = {Beijing, China},
pages = {31 -- 42},
publisher = {Springer},
title = {{Topology noise removal for curve and surface evolution}},
doi = {10.1007/978-3-642-18421-5_4},
volume = {6533},
year = {2010},
}
@inbook{3795,
abstract = {The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.},
author = {Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
booktitle = {Topological Data Analysis and Visualization: Theory, Algorithms and Applications},
pages = {27 -- 42},
publisher = {Springer},
title = {{The stability of the apparent contour of an orientable 2-manifold}},
doi = {10.1007/978-3-642-15014-2_3},
year = {2010},
}
@inproceedings{3848,
abstract = {We define the robustness of a level set homology class of a function f:XR as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X=R3 has ramifications in medical imaging and scientific visualization.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
location = {Liverpool, UK},
pages = {1 -- 10},
publisher = {Springer},
title = {{The robustness of level sets}},
doi = {10.1007/978-3-642-15775-2_1},
volume = {6346},
year = {2010},
}
@inproceedings{3849,
abstract = {Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Kerber, Michael and Patel, Amit},
location = {Brno, Czech Republic},
pages = {12 -- 23},
publisher = {Springer},
title = {{Persistent homology under non-uniform error}},
doi = {10.1007/978-3-642-15155-2_2},
volume = {6281},
year = {2010},
}
@inproceedings{3850,
abstract = {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 with a disk of fixed radius? If it does, we also seek a preferably simple solution shape P;P’s offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give a decision algorithm for fixed radius in O(nlogn) time that handles any polygonal shape. For convex shapes, the complexity drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one.},
author = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza},
location = {Dortmund, Germany},
pages = {12 -- 23},
publisher = {TU Dortmund},
title = {{Polygonal reconstruction from approximate offsets}},
year = {2010},
}
@inproceedings{3853,
abstract = {Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic meanpayoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.},
author = {Chatterjee, Krishnendu and Doyen, Laurent and Edelsbrunner, Herbert and Henzinger, Thomas A and Rannou, Philippe},
location = {Paris, France},
pages = {269 -- 283},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Mean-payoff automaton expressions}},
doi = {10.1007/978-3-642-15375-4_19},
volume = {6269},
year = {2010},
}
@article{3901,
abstract = {We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Kerber, Michael},
journal = {IEEE Transactions of Visualization and Computer Graphics},
number = {6},
pages = {1251 -- 1260},
publisher = {IEEE},
title = {{Computing robustness and persistence for images}},
doi = {10.1109/TVCG.2010.139},
volume = {16},
year = {2010},
}
@inproceedings{10909,
abstract = {We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1-norm of a cycle. Two main results are presented. First, we prove the problem is NP-hard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NP-hard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the ℤ2 field.},
author = {Chen, Chao and Freedman, Daniel},
booktitle = {Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms},
location = {Austin, TX, United States},
pages = {1594--1604},
publisher = {Society for Industrial and Applied Mathematics},
title = {{Hardness results for homology localization}},
doi = {10.1137/1.9781611973075.129},
year = {2010},
}
@inproceedings{3968,
abstract = {We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.},
author = {Edelsbrunner, Herbert and Harer, John},
location = {Zermatt, Switzerland},
pages = {36 -- 50},
publisher = {Springer},
title = {{The persistent Morse complex segmentation of a 3-manifold}},
doi = {10.1007/978-3-642-10470-1_4},
volume = {5903},
year = {2009},
}