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