@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}, } @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}, } @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}, } @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{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}, } @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}, } @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}, } @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}, } @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}, } @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{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}, } @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}, } @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 = {1611-3349}, 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{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{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{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}, } @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}, } @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{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}, }