[{"publication_status":"published","series_title":"Special Issue on Voronoi Diagrams and Delaunay Triangulation","external_id":{"arxiv":["1306.3640"]},"abstract":[{"lang":"eng","text":"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."}],"publisher":"Springer","date_published":"2011-11-09T00:00:00Z","page":"60 - 101","department":[{"_id":"HeEd"}],"volume":6970,"quality_controlled":"1","year":"2011","date_created":"2018-12-11T12:02:44Z","citation":{"apa":"Van De Weygaert, R., Vegter, G., Edelsbrunner, H., Jones, B., Pranav, P., Park, C., … Teillaud, M. (2011). Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web. In M. Gavrilova, K. Tan, &#38; M. Mostafavi (Eds.), <i>Transactions on Computational Science XIV</i> (Vol. 6970, pp. 60–101). Springer. <a href=\"https://doi.org/10.1007/978-3-642-25249-5_3\">https://doi.org/10.1007/978-3-642-25249-5_3</a>","short":"R. Van De Weygaert, G. Vegter, H. Edelsbrunner, B. Jones, P. Pranav, C. Park, W. Hellwing, B. Eldering, N. Kruithof, P. Bos, J. Hidding, J. Feldbrugge, E. Ten Have, M. Van Engelen, M. Caroli, M. Teillaud, in:, M. Gavrilova, K. Tan, M. Mostafavi (Eds.), Transactions on Computational Science XIV, Springer, 2011, pp. 60–101.","ieee":"R. Van De Weygaert <i>et al.</i>, “Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web,” in <i>Transactions on Computational Science XIV</i>, vol. 6970, M. Gavrilova, K. Tan, and M. Mostafavi, Eds. Springer, 2011, pp. 60–101.","ama":"Van De Weygaert R, Vegter G, Edelsbrunner H, et al. Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web. In: Gavrilova M, Tan K, Mostafavi M, eds. <i>Transactions on Computational Science XIV</i>. Vol 6970. Special Issue on Voronoi Diagrams and Delaunay Triangulation. Springer; 2011:60-101. doi:<a href=\"https://doi.org/10.1007/978-3-642-25249-5_3\">10.1007/978-3-642-25249-5_3</a>","ista":"Van De Weygaert R, Vegter G, Edelsbrunner H, Jones B, Pranav P, Park C, Hellwing W, Eldering B, Kruithof N, Bos P, Hidding J, Feldbrugge J, Ten Have E, Van Engelen M, Caroli M, Teillaud M. 2011.Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web. In: Transactions on Computational Science XIV. LNCS, vol. 6970, 60–101.","chicago":"Van De Weygaert, Rien, Gert Vegter, Herbert Edelsbrunner, Bernard Jones, Pratyush Pranav, Changbom Park, Wojciech Hellwing, et al. “Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web.” In <i>Transactions on Computational Science XIV</i>, edited by Marina Gavrilova, Kenneth Tan, and Mir Mostafavi, 6970:60–101. Special Issue on Voronoi Diagrams and Delaunay Triangulation. Springer, 2011. <a href=\"https://doi.org/10.1007/978-3-642-25249-5_3\">https://doi.org/10.1007/978-3-642-25249-5_3</a>.","mla":"Van De Weygaert, Rien, et al. “Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web.” <i>Transactions on Computational Science XIV</i>, edited by Marina Gavrilova et al., vol. 6970, Springer, 2011, pp. 60–101, doi:<a href=\"https://doi.org/10.1007/978-3-642-25249-5_3\">10.1007/978-3-642-25249-5_3</a>."},"publist_id":"3295","_id":"3335","doi":"10.1007/978-3-642-25249-5_3","author":[{"last_name":"Van De Weygaert","full_name":"Van De Weygaert, Rien","first_name":"Rien"},{"full_name":"Vegter, Gert","first_name":"Gert","last_name":"Vegter"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"first_name":"Bernard","full_name":"Jones, Bernard","last_name":"Jones"},{"last_name":"Pranav","full_name":"Pranav, Pratyush","first_name":"Pratyush"},{"last_name":"Park","full_name":"Park, Changbom","first_name":"Changbom"},{"last_name":"Hellwing","full_name":"Hellwing, Wojciech","first_name":"Wojciech"},{"last_name":"Eldering","first_name":"Bob","full_name":"Eldering, Bob"},{"full_name":"Kruithof, Nico","first_name":"Nico","last_name":"Kruithof"},{"last_name":"Bos","full_name":"Bos, Patrick","first_name":"Patrick"},{"full_name":"Hidding, Johan","first_name":"Johan","last_name":"Hidding"},{"first_name":"Job","full_name":"Feldbrugge, Job","last_name":"Feldbrugge"},{"first_name":"Eline","full_name":"Ten Have, Eline","last_name":"Ten Have"},{"last_name":"Van Engelen","full_name":"Van Engelen, Matti","first_name":"Matti"},{"last_name":"Caroli","first_name":"Manuel","full_name":"Caroli, Manuel"},{"last_name":"Teillaud","first_name":"Monique","full_name":"Teillaud, Monique"}],"arxiv":1,"month":"11","oa_version":"Preprint","main_file_link":[{"url":"http://arxiv.org/abs/1306.3640","open_access":"1"}],"day":"09","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","alternative_title":["LNCS"],"status":"public","type":"book_chapter","date_updated":"2021-01-12T07:42:44Z","oa":1,"title":"Alpha, Betti and the Megaparsec Universe: On the topology of the Cosmic Web","editor":[{"first_name":"Marina","full_name":"Gavrilova, Marina","last_name":"Gavrilova"},{"last_name":"Tan","first_name":"Kenneth","full_name":"Tan, Kenneth"},{"full_name":"Mostafavi, Mir","first_name":"Mir","last_name":"Mostafavi"}],"intvolume":"      6970","scopus_import":1,"language":[{"iso":"eng"}],"publication":"Transactions on Computational Science XIV"},{"quality_controlled":"1","page":"2089 - 2096","department":[{"_id":"HeEd"},{"_id":"ChLa"}],"date_published":"2011-07-22T00:00:00Z","type":"conference","status":"public","publisher":"IEEE","publication_status":"published","author":[{"full_name":"Chen, Chao","first_name":"Chao","last_name":"Chen","id":"3E92416E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Daniel","full_name":"Freedman, Daniel","last_name":"Freedman"},{"first_name":"Christoph","full_name":"Lampert, Christoph","last_name":"Lampert","id":"40C20FD2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-8622-7887"}],"abstract":[{"text":"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.","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"22","oa_version":"None","month":"07","publication_identifier":{"isbn":["978-1-4577-0394-2"],"eisbn":["978-1-4577-0395-9"]},"conference":{"start_date":"2011-06-20","name":"CVPR: Conference on Computer Vision and Pattern Recognition","end_date":"2011-06-25","location":"Colorado Springs, CO, United States"},"_id":"3336","language":[{"iso":"eng"}],"doi":"10.1109/CVPR.2011.5995503","publication":"CVPR: Computer Vision and Pattern Recognition","date_created":"2018-12-11T12:02:45Z","publist_id":"3294","title":"Enforcing topological constraints in random field image segmentation","citation":{"apa":"Chen, C., Freedman, D., &#38; Lampert, C. (2011). Enforcing topological constraints in random field image segmentation. In <i>CVPR: Computer Vision and Pattern Recognition</i> (pp. 2089–2096). Colorado Springs, CO, United States: IEEE. <a href=\"https://doi.org/10.1109/CVPR.2011.5995503\">https://doi.org/10.1109/CVPR.2011.5995503</a>","short":"C. Chen, D. Freedman, C. Lampert, in:, CVPR: Computer Vision and Pattern Recognition, IEEE, 2011, pp. 2089–2096.","ama":"Chen C, Freedman D, Lampert C. Enforcing topological constraints in random field image segmentation. In: <i>CVPR: Computer Vision and Pattern Recognition</i>. IEEE; 2011:2089-2096. doi:<a href=\"https://doi.org/10.1109/CVPR.2011.5995503\">10.1109/CVPR.2011.5995503</a>","ieee":"C. Chen, D. Freedman, and C. Lampert, “Enforcing topological constraints in random field image segmentation,” in <i>CVPR: Computer Vision and Pattern Recognition</i>, Colorado Springs, CO, United States, 2011, pp. 2089–2096.","ista":"Chen C, Freedman D, Lampert C. 2011. Enforcing topological constraints in random field image segmentation. CVPR: Computer Vision and Pattern Recognition. CVPR: Conference on Computer Vision and Pattern Recognition, 2089–2096.","mla":"Chen, Chao, et al. “Enforcing Topological Constraints in Random Field Image Segmentation.” <i>CVPR: Computer Vision and Pattern Recognition</i>, IEEE, 2011, pp. 2089–96, doi:<a href=\"https://doi.org/10.1109/CVPR.2011.5995503\">10.1109/CVPR.2011.5995503</a>.","chicago":"Chen, Chao, Daniel Freedman, and Christoph Lampert. “Enforcing Topological Constraints in Random Field Image Segmentation.” In <i>CVPR: Computer Vision and Pattern Recognition</i>, 2089–96. IEEE, 2011. <a href=\"https://doi.org/10.1109/CVPR.2011.5995503\">https://doi.org/10.1109/CVPR.2011.5995503</a>."},"year":"2011","article_processing_charge":"No","date_updated":"2023-02-23T12:23:56Z","acknowledgement":"The first author is supported by the Austrian Science Fund (FWF) grant No. P20134-N13. The authors would like to thank Sebastian Nowozin for helpful discussions.","related_material":{"record":[{"status":"public","id":"5386","relation":"earlier_version"}]},"scopus_import":"1"},{"date_updated":"2023-02-23T11:05:04Z","year":"2011","article_processing_charge":"No","publist_id":"3245","title":"An output sensitive algorithm for persistent homology","citation":{"apa":"Chen, C., &#38; Kerber, M. (2011). An output sensitive algorithm for persistent homology (pp. 207–216). Presented at the SoCG: Symposium on Computational Geometry, Paris, France: ACM. <a href=\"https://doi.org/10.1145/1998196.1998228\">https://doi.org/10.1145/1998196.1998228</a>","short":"C. Chen, M. Kerber, in:, ACM, 2011, pp. 207–216.","ieee":"C. Chen and M. Kerber, “An output sensitive algorithm for persistent homology,” presented at the SoCG: Symposium on Computational Geometry, Paris, France, 2011, pp. 207–216.","ama":"Chen C, Kerber M. An output sensitive algorithm for persistent homology. In: ACM; 2011:207-216. doi:<a href=\"https://doi.org/10.1145/1998196.1998228\">10.1145/1998196.1998228</a>","mla":"Chen, Chao, and Michael Kerber. <i>An Output Sensitive Algorithm for Persistent Homology</i>. ACM, 2011, pp. 207–16, doi:<a href=\"https://doi.org/10.1145/1998196.1998228\">10.1145/1998196.1998228</a>.","chicago":"Chen, Chao, and Michael Kerber. “An Output Sensitive Algorithm for Persistent Homology,” 207–16. ACM, 2011. <a href=\"https://doi.org/10.1145/1998196.1998228\">https://doi.org/10.1145/1998196.1998228</a>.","ista":"Chen C, Kerber M. 2011. An output sensitive algorithm for persistent homology. SoCG: Symposium on Computational Geometry, 207–216."},"date_created":"2018-12-11T12:02:56Z","related_material":{"record":[{"relation":"later_version","id":"2939","status":"public"}]},"scopus_import":1,"_id":"3367","language":[{"iso":"eng"}],"doi":"10.1145/1998196.1998228","publication_status":"published","author":[{"id":"3E92416E-F248-11E8-B48F-1D18A9856A87","last_name":"Chen","first_name":"Chao","full_name":"Chen, Chao"},{"orcid":"0000-0002-8030-9299","last_name":"Kerber","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","first_name":"Michael","full_name":"Kerber, Michael"}],"conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2011-06-13","location":"Paris, France","end_date":"2011-06-15"},"month":"06","oa_version":"None","day":"13","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ&gt;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&gt;0.","lang":"eng"}],"publisher":"ACM","date_published":"2011-06-13T00:00:00Z","status":"public","type":"conference","department":[{"_id":"HeEd"}],"page":"207 - 216","quality_controlled":"1"},{"quality_controlled":"1","volume":11,"page":"345 - 361","department":[{"_id":"HeEd"}],"date_published":"2011-06-01T00:00:00Z","publisher":"Springer","publication_status":"published","abstract":[{"lang":"eng","text":"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."}],"_id":"3377","doi":"10.1007/s10208-011-9090-8","citation":{"short":"H. Edelsbrunner, D. Morozov, A. Patel, Foundations of Computational Mathematics 11 (2011) 345–361.","ama":"Edelsbrunner H, Morozov D, Patel A. Quantifying transversality by measuring the robustness of intersections. <i>Foundations of Computational Mathematics</i>. 2011;11(3):345-361. doi:<a href=\"https://doi.org/10.1007/s10208-011-9090-8\">10.1007/s10208-011-9090-8</a>","ieee":"H. Edelsbrunner, D. Morozov, and A. Patel, “Quantifying transversality by measuring the robustness of intersections,” <i>Foundations of Computational Mathematics</i>, vol. 11, no. 3. Springer, pp. 345–361, 2011.","ista":"Edelsbrunner H, Morozov D, Patel A. 2011. Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics. 11(3), 345–361.","mla":"Edelsbrunner, Herbert, et al. “Quantifying Transversality by Measuring the Robustness of Intersections.” <i>Foundations of Computational Mathematics</i>, vol. 11, no. 3, Springer, 2011, pp. 345–61, doi:<a href=\"https://doi.org/10.1007/s10208-011-9090-8\">10.1007/s10208-011-9090-8</a>.","chicago":"Edelsbrunner, Herbert, Dmitriy Morozov, and Amit Patel. “Quantifying Transversality by Measuring the Robustness of Intersections.” <i>Foundations of Computational Mathematics</i>. Springer, 2011. <a href=\"https://doi.org/10.1007/s10208-011-9090-8\">https://doi.org/10.1007/s10208-011-9090-8</a>.","apa":"Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2011). Quantifying transversality by measuring the robustness of intersections. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-011-9090-8\">https://doi.org/10.1007/s10208-011-9090-8</a>"},"publist_id":"3230","date_created":"2018-12-11T12:02:59Z","year":"2011","acknowledgement":"This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","type":"journal_article","status":"public","author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"first_name":"Amit","full_name":"Patel, Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","last_name":"Patel"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"01","oa_version":"Submitted Version","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/0911.2142"}],"month":"06","language":[{"iso":"eng"}],"publication":"Foundations of Computational Mathematics","title":"Quantifying transversality by measuring the robustness of intersections","issue":"3","date_updated":"2021-01-12T07:43:04Z","oa":1,"scopus_import":1,"intvolume":"        11"},{"volume":11,"quality_controlled":"1","department":[{"_id":"HeEd"}],"page":"305 - 336","type":"journal_article","status":"public","date_published":"2011-06-01T00:00:00Z","publisher":"Springer","abstract":[{"text":"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.","lang":"eng"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"01","oa_version":"None","month":"06","publication_status":"published","author":[{"id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","last_name":"Bendich","first_name":"Paul","full_name":"Bendich, Paul"},{"first_name":"John","full_name":"Harer, John","last_name":"Harer"}],"doi":"10.1007/s10208-010-9081-1","publication":"Foundations of Computational Mathematics","_id":"3378","language":[{"iso":"eng"}],"scopus_import":1,"intvolume":"        11","citation":{"ista":"Bendich P, Harer J. 2011. Persistent intersection homology. Foundations of Computational Mathematics. 11(3), 305–336.","mla":"Bendich, Paul, and John Harer. “Persistent Intersection Homology.” <i>Foundations of Computational Mathematics</i>, vol. 11, no. 3, Springer, 2011, pp. 305–36, doi:<a href=\"https://doi.org/10.1007/s10208-010-9081-1\">10.1007/s10208-010-9081-1</a>.","chicago":"Bendich, Paul, and John Harer. “Persistent Intersection Homology.” <i>Foundations of Computational Mathematics</i>. Springer, 2011. <a href=\"https://doi.org/10.1007/s10208-010-9081-1\">https://doi.org/10.1007/s10208-010-9081-1</a>.","short":"P. Bendich, J. Harer, Foundations of Computational Mathematics 11 (2011) 305–336.","ama":"Bendich P, Harer J. Persistent intersection homology. <i>Foundations of Computational Mathematics</i>. 2011;11(3):305-336. doi:<a href=\"https://doi.org/10.1007/s10208-010-9081-1\">10.1007/s10208-010-9081-1</a>","ieee":"P. Bendich and J. Harer, “Persistent intersection homology,” <i>Foundations of Computational Mathematics</i>, vol. 11, no. 3. Springer, pp. 305–336, 2011.","apa":"Bendich, P., &#38; Harer, J. (2011). Persistent intersection homology. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-010-9081-1\">https://doi.org/10.1007/s10208-010-9081-1</a>"},"date_created":"2018-12-11T12:02:59Z","title":"Persistent intersection homology","publist_id":"3229","year":"2011","issue":"3","acknowledgement":"This research was partially supported by the Defense Advanced Research Projects Agency (DARPA) under grant HR0011-05-1-0007.","date_updated":"2021-01-12T07:43:04Z"},{"department":[{"_id":"HeEd"}],"page":"359 - 367","quality_controlled":"1","volume":77,"publisher":"Szegedi Tudományegyetem","type":"journal_article","status":"public","date_published":"2011-01-01T00:00:00Z","day":"01","oa_version":"None","abstract":[{"lang":"eng","text":"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."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"01","author":[{"last_name":"Fasy","id":"F65D502E-E68D-11E9-9252-C644099818F6","full_name":"Fasy, Brittany Terese","first_name":"Brittany Terese"}],"publication_status":"published","publication":"Acta Sci. Math. (Szeged)","language":[{"iso":"eng"}],"_id":"3781","intvolume":"        77","date_created":"2018-12-11T12:05:08Z","publist_id":"2446","title":"The difference in length of curves in R^n","citation":{"apa":"Fasy, B. T. (2011). The difference in length of curves in R^n. <i>Acta Sci. Math. (Szeged)</i>. Szegedi Tudományegyetem.","ista":"Fasy BT. 2011. The difference in length of curves in R^n. Acta Sci. Math. (Szeged). 77(1–2), 359–367.","mla":"Fasy, Brittany Terese. “The Difference in Length of Curves in R^n.” <i>Acta Sci. Math. (Szeged)</i>, vol. 77, no. 1–2, Szegedi Tudományegyetem, 2011, pp. 359–67.","chicago":"Fasy, Brittany Terese. “The Difference in Length of Curves in R^n.” <i>Acta Sci. Math. (Szeged)</i>. Szegedi Tudományegyetem, 2011.","short":"B.T. Fasy, Acta Sci. Math. (Szeged) 77 (2011) 359–367.","ieee":"B. T. Fasy, “The difference in length of curves in R^n,” <i>Acta Sci. Math. (Szeged)</i>, vol. 77, no. 1–2. Szegedi Tudományegyetem, pp. 359–367, 2011.","ama":"Fasy BT. The difference in length of curves in R^n. <i>Acta Sci Math (Szeged)</i>. 2011;77(1-2):359-367."},"date_updated":"2021-01-12T07:52:09Z","acknowledgement":"Funded by Graduate Aid in Areas of National Need (GAANN) Fellowship.","issue":"1-2","year":"2011"},{"doi":"10.1007/978-3-642-19391-0_2","_id":"3796","file_date_updated":"2020-07-14T12:46:16Z","citation":{"apa":"Edelsbrunner, H., &#38; Kerber, M. (2011). Covering and packing with spheres by diagonal distortion in R^n. In C. Calude, G. Rozenberg, &#38; A. Salomaa (Eds.), <i>Rainbow of Computer Science</i> (Vol. 6570, pp. 20–35). Springer. <a href=\"https://doi.org/10.1007/978-3-642-19391-0_2\">https://doi.org/10.1007/978-3-642-19391-0_2</a>","short":"H. Edelsbrunner, M. Kerber, in:, C. Calude, G. Rozenberg, A. Salomaa (Eds.), Rainbow of Computer Science, Springer, 2011, pp. 20–35.","ieee":"H. Edelsbrunner and M. Kerber, “Covering and packing with spheres by diagonal distortion in R^n,” in <i>Rainbow of Computer Science</i>, vol. 6570, C. Calude, G. Rozenberg, and A. Salomaa, Eds. Springer, 2011, pp. 20–35.","ama":"Edelsbrunner H, Kerber M. Covering and packing with spheres by diagonal distortion in R^n. In: Calude C, Rozenberg G, Salomaa A, eds. <i>Rainbow of Computer Science</i>. Vol 6570. Dedicated to Hermann Maurer on the Occasion of His 70th Birthday. Springer; 2011:20-35. doi:<a href=\"https://doi.org/10.1007/978-3-642-19391-0_2\">10.1007/978-3-642-19391-0_2</a>","mla":"Edelsbrunner, Herbert, and Michael Kerber. “Covering and Packing with Spheres by Diagonal Distortion in R^n.” <i>Rainbow of Computer Science</i>, edited by Cristian Calude et al., vol. 6570, Springer, 2011, pp. 20–35, doi:<a href=\"https://doi.org/10.1007/978-3-642-19391-0_2\">10.1007/978-3-642-19391-0_2</a>.","ista":"Edelsbrunner H, Kerber M. 2011.Covering and packing with spheres by diagonal distortion in R^n. In: Rainbow of Computer Science. LNCS, vol. 6570, 20–35.","chicago":"Edelsbrunner, Herbert, and Michael Kerber. “Covering and Packing with Spheres by Diagonal Distortion in R^n.” In <i>Rainbow of Computer Science</i>, edited by Cristian Calude, Grzegorz Rozenberg, and Arto Salomaa, 6570:20–35. Dedicated to Hermann Maurer on the Occasion of His 70th Birthday. Springer, 2011. <a href=\"https://doi.org/10.1007/978-3-642-19391-0_2\">https://doi.org/10.1007/978-3-642-19391-0_2</a>."},"date_created":"2018-12-11T12:05:13Z","publist_id":"2427","year":"2011","volume":6570,"quality_controlled":"1","page":"20 - 35","department":[{"_id":"HeEd"}],"date_published":"2011-05-03T00:00:00Z","publisher":"Springer","abstract":[{"text":"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.","lang":"eng"}],"series_title":"Dedicated to Hermann Maurer on the Occasion of His 70th Birthday","ddc":["000"],"publication_status":"published","publication":"Rainbow of Computer Science","language":[{"iso":"eng"}],"has_accepted_license":"1","intvolume":"      6570","editor":[{"last_name":"Calude","first_name":"Cristian","full_name":"Calude, Cristian"},{"last_name":"Rozenberg","full_name":"Rozenberg, Grzegorz","first_name":"Grzegorz"},{"full_name":"Salomaa, Arto","first_name":"Arto","last_name":"Salomaa"}],"title":"Covering and packing with spheres by diagonal distortion in R^n","pubrep_id":"539","oa":1,"date_updated":"2021-01-12T07:52:15Z","status":"public","type":"book_chapter","alternative_title":["LNCS"],"file":[{"creator":"system","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_size":436875,"file_name":"IST-2016-539-v1+1_2011-B-01-CoveringPacking.pdf","file_id":"4640","checksum":"aaf22b4d7bd4277ffe8db532119cf474","date_created":"2018-12-12T10:07:42Z","date_updated":"2020-07-14T12:46:16Z"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"03","oa_version":"Submitted Version","month":"05","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Kerber","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8030-9299","first_name":"Michael","full_name":"Kerber, Michael"}]},{"volume":16,"quality_controlled":"1","page":"1 - 13","department":[{"_id":"HeEd"}],"date_published":"2011-05-01T00:00:00Z","type":"journal_article","status":"public","publisher":"ACM","author":[{"full_name":"Wang, Bei","first_name":"Bei","last_name":"Wang"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"full_name":"Morozov, Dmitriy","first_name":"Dmitriy","last_name":"Morozov"}],"publication_status":"published","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"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.","lang":"eng"}],"day":"01","oa_version":"None","month":"05","language":[{"iso":"eng"}],"_id":"3965","doi":"10.1145/1963190.1970375","publication":"Journal of Experimental Algorithmics","publist_id":"2161","date_created":"2018-12-11T12:06:09Z","title":"Computing elevation maxima by searching the Gauss sphere","citation":{"apa":"Wang, B., Edelsbrunner, H., &#38; Morozov, D. (2011). Computing elevation maxima by searching the Gauss sphere. <i>Journal of Experimental Algorithmics</i>. ACM. <a href=\"https://doi.org/10.1145/1963190.1970375\">https://doi.org/10.1145/1963190.1970375</a>","ieee":"B. Wang, H. Edelsbrunner, and D. Morozov, “Computing elevation maxima by searching the Gauss sphere,” <i>Journal of Experimental Algorithmics</i>, vol. 16, no. 2.2. ACM, pp. 1–13, 2011.","ama":"Wang B, Edelsbrunner H, Morozov D. Computing elevation maxima by searching the Gauss sphere. <i>Journal of Experimental Algorithmics</i>. 2011;16(2.2):1-13. doi:<a href=\"https://doi.org/10.1145/1963190.1970375\">10.1145/1963190.1970375</a>","short":"B. Wang, H. Edelsbrunner, D. Morozov, Journal of Experimental Algorithmics 16 (2011) 1–13.","mla":"Wang, Bei, et al. “Computing Elevation Maxima by Searching the Gauss Sphere.” <i>Journal of Experimental Algorithmics</i>, vol. 16, no. 2.2, ACM, 2011, pp. 1–13, doi:<a href=\"https://doi.org/10.1145/1963190.1970375\">10.1145/1963190.1970375</a>.","ista":"Wang B, Edelsbrunner H, Morozov D. 2011. Computing elevation maxima by searching the Gauss sphere. Journal of Experimental Algorithmics. 16(2.2), 1–13.","chicago":"Wang, Bei, Herbert Edelsbrunner, and Dmitriy Morozov. “Computing Elevation Maxima by Searching the Gauss Sphere.” <i>Journal of Experimental Algorithmics</i>. ACM, 2011. <a href=\"https://doi.org/10.1145/1963190.1970375\">https://doi.org/10.1145/1963190.1970375</a>."},"year":"2011","issue":"2.2","date_updated":"2021-01-12T07:53:31Z","scopus_import":1,"intvolume":"        16"},{"quality_controlled":"1","page":"1594-1604","department":[{"_id":"HeEd"}],"status":"public","date_published":"2010-02-01T00:00:00Z","type":"conference","publisher":"Society for Industrial and Applied Mathematics","author":[{"full_name":"Chen, Chao","first_name":"Chao","id":"3E92416E-F248-11E8-B48F-1D18A9856A87","last_name":"Chen"},{"last_name":"Freedman","first_name":"Daniel","full_name":"Freedman, Daniel"}],"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","text":"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."}],"oa_version":"None","day":"01","publication_identifier":{"eisbn":["9781611973075"]},"month":"02","conference":{"name":"SODA: Symposium on Discrete Algorithms","start_date":"2010-01-17","location":"Austin, TX, United States","end_date":"2010-01-19"},"language":[{"iso":"eng"}],"_id":"10909","doi":"10.1137/1.9781611973075.129","publication":"Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms","citation":{"chicago":"Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” In <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 1594–1604. Society for Industrial and Applied Mathematics, 2010. <a href=\"https://doi.org/10.1137/1.9781611973075.129\">https://doi.org/10.1137/1.9781611973075.129</a>.","ista":"Chen C, Freedman D. 2010. Hardness results for homology localization. Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 1594–1604.","mla":"Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Society for Industrial and Applied Mathematics, 2010, pp. 1594–604, doi:<a href=\"https://doi.org/10.1137/1.9781611973075.129\">10.1137/1.9781611973075.129</a>.","short":"C. Chen, D. Freedman, in:, Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2010, pp. 1594–1604.","ama":"Chen C, Freedman D. Hardness results for homology localization. In: <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Society for Industrial and Applied Mathematics; 2010:1594-1604. doi:<a href=\"https://doi.org/10.1137/1.9781611973075.129\">10.1137/1.9781611973075.129</a>","ieee":"C. Chen and D. Freedman, “Hardness results for homology localization,” in <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Austin, TX, United States, 2010, pp. 1594–1604.","apa":"Chen, C., &#38; Freedman, D. (2010). Hardness results for homology localization. In <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (pp. 1594–1604). Austin, TX, United States: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611973075.129\">https://doi.org/10.1137/1.9781611973075.129</a>"},"title":"Hardness results for homology localization","date_created":"2022-03-21T08:24:07Z","year":"2010","article_processing_charge":"No","acknowledgement":"Partially supported by the Austrian Science Fund under grantFSP-S9103-N04 and P20134-N13.","date_updated":"2023-02-23T11:19:46Z","related_material":{"record":[{"id":"3267","relation":"later_version","status":"public"}]},"scopus_import":"1"},{"language":[{"iso":"eng"}],"_id":"3782","publication":" Conference proceedings MCV 2010","doi":"10.1007/978-3-642-18421-5_4","acknowledgement":"Partially supported by the Austri an Science Fund unde r grant P20134-N13.\r\nWe thank Helena Molina-Abril for very helpful discussion. We thank anonymous reviewers for helpful comments.","date_updated":"2021-01-12T07:52:10Z","year":"2010","publist_id":"2445","date_created":"2018-12-11T12:05:08Z","title":"Topology noise removal for curve  and surface evolution","citation":{"apa":"Chen, C., &#38; Freedman, D. (2010). Topology noise removal for curve  and surface evolution. In <i> Conference proceedings MCV 2010</i> (Vol. 6533, pp. 31–42). Beijing, China: Springer. <a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">https://doi.org/10.1007/978-3-642-18421-5_4</a>","ieee":"C. Chen and D. Freedman, “Topology noise removal for curve  and surface evolution,” in <i> Conference proceedings MCV 2010</i>, Beijing, China, 2010, vol. 6533, pp. 31–42.","ama":"Chen C, Freedman D. Topology noise removal for curve  and surface evolution. In: <i> Conference Proceedings MCV 2010</i>. Vol 6533. Springer; 2010:31-42. doi:<a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">10.1007/978-3-642-18421-5_4</a>","short":"C. Chen, D. Freedman, in:,  Conference Proceedings MCV 2010, Springer, 2010, pp. 31–42.","ista":"Chen C, Freedman D. 2010. Topology noise removal for curve  and surface evolution.  Conference proceedings MCV 2010. MCV: Medical Computer Vision, LNCS, vol. 6533, 31–42.","mla":"Chen, Chao, and Daniel Freedman. “Topology Noise Removal for Curve  and Surface Evolution.” <i> Conference Proceedings MCV 2010</i>, vol. 6533, Springer, 2010, pp. 31–42, doi:<a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">10.1007/978-3-642-18421-5_4</a>.","chicago":"Chen, Chao, and Daniel Freedman. “Topology Noise Removal for Curve  and Surface Evolution.” In <i> Conference Proceedings MCV 2010</i>, 6533:31–42. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">https://doi.org/10.1007/978-3-642-18421-5_4</a>."},"scopus_import":1,"intvolume":"      6533","publisher":"Springer","alternative_title":["LNCS"],"type":"conference","date_published":"2010-12-31T00:00:00Z","status":"public","department":[{"_id":"HeEd"}],"page":"31 - 42","volume":6533,"quality_controlled":"1","author":[{"id":"3E92416E-F248-11E8-B48F-1D18A9856A87","last_name":"Chen","first_name":"Chao","full_name":"Chen, Chao"},{"last_name":"Freedman","full_name":"Freedman, Daniel","first_name":"Daniel"}],"publication_status":"published","conference":{"start_date":"2010-09-20","name":"MCV: Medical Computer Vision","end_date":"2010-09-20","location":"Beijing, China"},"month":"12","day":"31","oa_version":"None","abstract":[{"lang":"eng","text":"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."}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87"},{"page":"27 - 42","department":[{"_id":"HeEd"}],"quality_controlled":"1","publisher":"Springer","date_published":"2010-12-22T00:00:00Z","abstract":[{"text":"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.","lang":"eng"}],"ddc":["000"],"publication_status":"published","doi":"10.1007/978-3-642-15014-2_3","file_date_updated":"2020-07-14T12:46:16Z","_id":"3795","date_created":"2018-12-11T12:05:13Z","publist_id":"2428","citation":{"apa":"Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2010). The stability of the apparent contour of an orientable 2-manifold. In <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i> (pp. 27–42). Springer. <a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">https://doi.org/10.1007/978-3-642-15014-2_3</a>","ama":"Edelsbrunner H, Morozov D, Patel A. The stability of the apparent contour of an orientable 2-manifold. In: <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>. Springer; 2010:27-42. doi:<a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">10.1007/978-3-642-15014-2_3</a>","ieee":"H. Edelsbrunner, D. Morozov, and A. Patel, “The stability of the apparent contour of an orientable 2-manifold,” in <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, Springer, 2010, pp. 27–42.","short":"H. Edelsbrunner, D. Morozov, A. Patel, in:, Topological Data Analysis and Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42.","chicago":"Edelsbrunner, Herbert, Dmitriy Morozov, and Amit Patel. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” In <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, 27–42. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">https://doi.org/10.1007/978-3-642-15014-2_3</a>.","mla":"Edelsbrunner, Herbert, et al. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, Springer, 2010, pp. 27–42, doi:<a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">10.1007/978-3-642-15014-2_3</a>.","ista":"Edelsbrunner H, Morozov D, Patel A. 2010.The stability of the apparent contour of an orientable 2-manifold. In: Topological Data Analysis and Visualization: Theory, Algorithms and Applications. Mathematics and Visualization, , 27–42."},"acknowledgement":"This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","year":"2010","alternative_title":["Mathematics and Visualization"],"status":"public","type":"book_chapter","day":"22","oa_version":"Submitted Version","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","file":[{"creator":"system","access_level":"open_access","file_size":210710,"relation":"main_file","content_type":"application/pdf","checksum":"f03a44c3d1c3e2d4fedb3b94404f3fd5","file_name":"IST-2016-538-v1+1_2011-B-02-ApparentContour.pdf","file_id":"4896","date_created":"2018-12-12T10:11:40Z","date_updated":"2020-07-14T12:46:16Z"}],"month":"12","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"last_name":"Patel","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","full_name":"Patel, Amit","first_name":"Amit"}],"publication":"Topological Data Analysis and Visualization: Theory, Algorithms and Applications","language":[{"iso":"eng"}],"has_accepted_license":"1","scopus_import":1,"title":"The stability of the apparent contour of an orientable 2-manifold","oa":1,"pubrep_id":"538","date_updated":"2021-01-12T07:52:15Z"},{"language":[{"iso":"eng"}],"_id":"3848","doi":"10.1007/978-3-642-15775-2_1","citation":{"apa":"Bendich, P., Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2010). The robustness of level sets (Vol. 6346, pp. 1–10). Presented at the ESA: European Symposium on Algorithms, Liverpool, UK: Springer. <a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">https://doi.org/10.1007/978-3-642-15775-2_1</a>","mla":"Bendich, Paul, et al. <i>The Robustness of Level Sets</i>. Vol. 6346, Springer, 2010, pp. 1–10, doi:<a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">10.1007/978-3-642-15775-2_1</a>.","chicago":"Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “The Robustness of Level Sets,” 6346:1–10. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">https://doi.org/10.1007/978-3-642-15775-2_1</a>.","ista":"Bendich P, Edelsbrunner H, Morozov D, Patel A. 2010. The robustness of level sets. ESA: European Symposium on Algorithms, LNCS, vol. 6346, 1–10.","short":"P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, in:, Springer, 2010, pp. 1–10.","ieee":"P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, “The robustness of level sets,” presented at the ESA: European Symposium on Algorithms, Liverpool, UK, 2010, vol. 6346, pp. 1–10.","ama":"Bendich P, Edelsbrunner H, Morozov D, Patel A. The robustness of level sets. In: Vol 6346. Springer; 2010:1-10. doi:<a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">10.1007/978-3-642-15775-2_1</a>"},"title":"The robustness of level sets","publist_id":"2336","date_created":"2018-12-11T12:05:30Z","date_updated":"2021-01-12T07:52:38Z","year":"2010","intvolume":"      6346","scopus_import":1,"department":[{"_id":"HeEd"}],"page":"1 - 10","quality_controlled":"1","volume":6346,"publisher":"Springer","alternative_title":["LNCS"],"status":"public","date_published":"2010-09-01T00:00:00Z","type":"conference","author":[{"first_name":"Paul","full_name":"Bendich, Paul","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","last_name":"Bendich"},{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Morozov, Dmitriy","first_name":"Dmitriy","last_name":"Morozov"},{"first_name":"Amit","full_name":"Patel, Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","last_name":"Patel"}],"publication_status":"published","day":"01","oa_version":"None","abstract":[{"text":"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.","lang":"eng"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","conference":{"end_date":"2010-09-08","location":"Liverpool, UK","start_date":"2010-09-06","name":"ESA: European Symposium on Algorithms"},"month":"09"},{"year":"2010","publist_id":"2333","citation":{"mla":"Bendich, Paul, et al. <i>Persistent Homology under Non-Uniform Error</i>. Vol. 6281, Springer, 2010, pp. 12–23, doi:<a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">10.1007/978-3-642-15155-2_2</a>.","chicago":"Bendich, Paul, Herbert Edelsbrunner, Michael Kerber, and Amit Patel. “Persistent Homology under Non-Uniform Error,” 6281:12–23. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">https://doi.org/10.1007/978-3-642-15155-2_2</a>.","ista":"Bendich P, Edelsbrunner H, Kerber M, Patel A. 2010. Persistent homology under non-uniform error. MFCS: Mathematical Foundations of Computer Science, LNCS, vol. 6281, 12–23.","short":"P. Bendich, H. Edelsbrunner, M. Kerber, A. Patel, in:, Springer, 2010, pp. 12–23.","ieee":"P. Bendich, H. Edelsbrunner, M. Kerber, and A. Patel, “Persistent homology under non-uniform error,” presented at the MFCS: Mathematical Foundations of Computer Science, Brno, Czech Republic, 2010, vol. 6281, pp. 12–23.","ama":"Bendich P, Edelsbrunner H, Kerber M, Patel A. Persistent homology under non-uniform error. In: Vol 6281. Springer; 2010:12-23. doi:<a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">10.1007/978-3-642-15155-2_2</a>","apa":"Bendich, P., Edelsbrunner, H., Kerber, M., &#38; Patel, A. (2010). Persistent homology under non-uniform error (Vol. 6281, pp. 12–23). Presented at the MFCS: Mathematical Foundations of Computer Science, Brno, Czech Republic: Springer. <a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">https://doi.org/10.1007/978-3-642-15155-2_2</a>"},"date_created":"2018-12-11T12:05:30Z","doi":"10.1007/978-3-642-15155-2_2","file_date_updated":"2020-07-14T12:46:17Z","_id":"3849","ddc":["000"],"abstract":[{"lang":"eng","text":"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."}],"publication_status":"published","publisher":"Springer","date_published":"2010-08-10T00:00:00Z","page":"12 - 23","department":[{"_id":"HeEd"}],"volume":6281,"quality_controlled":"1","has_accepted_license":"1","scopus_import":1,"intvolume":"      6281","oa":1,"pubrep_id":"537","date_updated":"2021-01-12T07:52:38Z","title":"Persistent homology under non-uniform error","language":[{"iso":"eng"}],"conference":{"end_date":"2010-08-27","location":"Brno, Czech Republic","start_date":"2010-08-23","name":"MFCS: Mathematical Foundations of Computer Science"},"month":"08","oa_version":"Submitted Version","day":"10","file":[{"file_name":"IST-2016-537-v1+1_2010-P-05-NonuniformError.pdf","checksum":"af61e1c2bb42f3d556179d4692caeb1b","file_id":"4994","date_created":"2018-12-12T10:13:13Z","date_updated":"2020-07-14T12:46:17Z","creator":"system","access_level":"open_access","relation":"main_file","file_size":142357,"content_type":"application/pdf"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Paul","full_name":"Bendich, Paul","last_name":"Bendich","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"orcid":"0000-0002-8030-9299","last_name":"Kerber","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","first_name":"Michael","full_name":"Kerber, Michael"},{"first_name":"Amit","full_name":"Patel, Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","last_name":"Patel"}],"alternative_title":["LNCS"],"status":"public","type":"conference"},{"_id":"3850","language":[{"iso":"eng"}],"citation":{"apa":"Berberich, E., Halperin, D., Kerber, M., &#38; Pogalnikova, R. (2010). Polygonal reconstruction from approximate offsets (pp. 12–23). Presented at the EuroCG: European Workshop on Computational Geometry, Dortmund, Germany: TU Dortmund.","ama":"Berberich E, Halperin D, Kerber M, Pogalnikova R. Polygonal reconstruction from approximate offsets. In: TU Dortmund; 2010:12-23.","ieee":"E. Berberich, D. Halperin, M. Kerber, and R. Pogalnikova, “Polygonal reconstruction from approximate offsets,” presented at the EuroCG: European Workshop on Computational Geometry, Dortmund, Germany, 2010, pp. 12–23.","short":"E. Berberich, D. Halperin, M. Kerber, R. Pogalnikova, in:, TU Dortmund, 2010, pp. 12–23.","ista":"Berberich E, Halperin D, Kerber M, Pogalnikova R. 2010. Polygonal reconstruction from approximate offsets. EuroCG: European Workshop on Computational Geometry, 12–23.","mla":"Berberich, Eric, et al. <i>Polygonal Reconstruction from Approximate Offsets</i>. TU Dortmund, 2010, pp. 12–23.","chicago":"Berberich, Eric, Dan Halperin, Michael Kerber, and Roza Pogalnikova. “Polygonal Reconstruction from Approximate Offsets,” 12–23. TU Dortmund, 2010."},"publist_id":"2334","title":"Polygonal reconstruction from approximate offsets","date_created":"2018-12-11T12:05:30Z","year":"2010","date_updated":"2021-01-12T07:52:39Z","quality_controlled":"1","department":[{"_id":"HeEd"}],"page":"12 - 23","type":"conference","status":"public","date_published":"2010-01-01T00:00:00Z","publisher":"TU Dortmund","author":[{"last_name":"Berberich","full_name":"Berberich, Eric","first_name":"Eric"},{"last_name":"Halperin","full_name":"Halperin, Dan","first_name":"Dan"},{"first_name":"Michael","full_name":"Kerber, Michael","orcid":"0000-0002-8030-9299","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","last_name":"Kerber"},{"last_name":"Pogalnikova","full_name":"Pogalnikova, Roza","first_name":"Roza"}],"publication_status":"published","abstract":[{"lang":"eng","text":"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."}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"01","oa_version":"None","month":"01","conference":{"location":"Dortmund, Germany","end_date":"2010-03-24","name":"EuroCG: European Workshop on Computational Geometry","start_date":"2010-03-22"}},{"day":"18","oa_version":"Submitted Version","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","file":[{"date_created":"2018-12-12T10:15:41Z","date_updated":"2020-07-14T12:46:17Z","file_id":"5163","file_name":"IST-2012-62-v1+1_Mean-payoff_automaton_expressions.pdf","checksum":"4f753ae99d076553fb8733e2c8b390e2","file_size":233260,"relation":"main_file","access_level":"open_access","content_type":"application/pdf","creator":"system"}],"conference":{"end_date":"2010-09-03","location":"Paris, France","start_date":"2010-08-31","name":"CONCUR: Concurrency Theory"},"month":"11","author":[{"first_name":"Krishnendu","full_name":"Chatterjee, Krishnendu","last_name":"Chatterjee","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4561-241X"},{"last_name":"Doyen","full_name":"Doyen, Laurent","first_name":"Laurent"},{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"full_name":"Henzinger, Thomas A","first_name":"Thomas A","orcid":"0000−0002−2985−7724","last_name":"Henzinger","id":"40876CD8-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Philippe","full_name":"Rannou, Philippe","last_name":"Rannou"}],"alternative_title":["LNCS"],"type":"conference","status":"public","scopus_import":1,"has_accepted_license":"1","intvolume":"      6269","title":"Mean-payoff automaton expressions","oa":1,"pubrep_id":"62","date_updated":"2021-01-12T07:52:40Z","language":[{"iso":"eng"}],"abstract":[{"text":"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.","lang":"eng"}],"ddc":["000","005"],"publication_status":"published","department":[{"_id":"KrCh"},{"_id":"HeEd"},{"_id":"ToHe"}],"page":"269 - 283","quality_controlled":"1","volume":6269,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_published":"2010-11-18T00:00:00Z","project":[{"call_identifier":"FP7","name":"COMponent-Based Embedded Systems design Techniques","grant_number":"215543","_id":"25EFB36C-B435-11E9-9278-68D0E5697425"},{"_id":"25F1337C-B435-11E9-9278-68D0E5697425","grant_number":"214373","name":"Design for Embedded Systems","call_identifier":"FP7"}],"ec_funded":1,"citation":{"apa":"Chatterjee, K., Doyen, L., Edelsbrunner, H., Henzinger, T. A., &#38; Rannou, P. (2010). Mean-payoff automaton expressions (Vol. 6269, pp. 269–283). Presented at the CONCUR: Concurrency Theory, Paris, France: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">https://doi.org/10.1007/978-3-642-15375-4_19</a>","ista":"Chatterjee K, Doyen L, Edelsbrunner H, Henzinger TA, Rannou P. 2010. Mean-payoff automaton expressions. CONCUR: Concurrency Theory, LNCS, vol. 6269, 269–283.","mla":"Chatterjee, Krishnendu, et al. <i>Mean-Payoff Automaton Expressions</i>. Vol. 6269, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010, pp. 269–83, doi:<a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">10.1007/978-3-642-15375-4_19</a>.","chicago":"Chatterjee, Krishnendu, Laurent Doyen, Herbert Edelsbrunner, Thomas A Henzinger, and Philippe Rannou. “Mean-Payoff Automaton Expressions,” 6269:269–83. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">https://doi.org/10.1007/978-3-642-15375-4_19</a>.","ama":"Chatterjee K, Doyen L, Edelsbrunner H, Henzinger TA, Rannou P. Mean-payoff automaton expressions. In: Vol 6269. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2010:269-283. doi:<a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">10.1007/978-3-642-15375-4_19</a>","ieee":"K. Chatterjee, L. Doyen, H. Edelsbrunner, T. A. Henzinger, and P. Rannou, “Mean-payoff automaton expressions,” presented at the CONCUR: Concurrency Theory, Paris, France, 2010, vol. 6269, pp. 269–283.","short":"K. Chatterjee, L. Doyen, H. Edelsbrunner, T.A. Henzinger, P. Rannou, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010, pp. 269–283."},"publist_id":"2328","date_created":"2018-12-11T12:05:31Z","year":"2010","doi":"10.1007/978-3-642-15375-4_19","file_date_updated":"2020-07-14T12:46:17Z","_id":"3853"},{"language":[{"iso":"eng"}],"publication":"IEEE Transactions of Visualization and Computer Graphics","title":"Computing robustness and persistence for images","oa":1,"date_updated":"2021-01-12T07:53:04Z","pubrep_id":"536","issue":"6","scopus_import":1,"has_accepted_license":"1","intvolume":"        16","status":"public","type":"journal_article","author":[{"last_name":"Bendich","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","first_name":"Paul","full_name":"Bendich, Paul"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"full_name":"Kerber, Michael","first_name":"Michael","last_name":"Kerber","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8030-9299"}],"day":"28","oa_version":"Submitted Version","file":[{"checksum":"f6d813c04f4b46023cec6b9a17f15472","file_id":"5262","file_name":"IST-2016-536-v1+1_2010-J-02-PersistenceforImages.pdf","date_created":"2018-12-12T10:17:10Z","date_updated":"2020-07-14T12:46:21Z","creator":"system","file_size":721994,"relation":"main_file","access_level":"open_access","content_type":"application/pdf"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","month":"10","file_date_updated":"2020-07-14T12:46:21Z","_id":"3901","doi":"10.1109/TVCG.2010.139","publist_id":"2253","citation":{"apa":"Bendich, P., Edelsbrunner, H., &#38; Kerber, M. (2010). Computing robustness and persistence for images. <i>IEEE Transactions of Visualization and Computer Graphics</i>. IEEE. <a href=\"https://doi.org/10.1109/TVCG.2010.139\">https://doi.org/10.1109/TVCG.2010.139</a>","ista":"Bendich P, Edelsbrunner H, Kerber M. 2010. Computing robustness and persistence for images. IEEE Transactions of Visualization and Computer Graphics. 16(6), 1251–1260.","mla":"Bendich, Paul, et al. “Computing Robustness and Persistence for Images.” <i>IEEE Transactions of Visualization and Computer Graphics</i>, vol. 16, no. 6, IEEE, 2010, pp. 1251–60, doi:<a href=\"https://doi.org/10.1109/TVCG.2010.139\">10.1109/TVCG.2010.139</a>.","chicago":"Bendich, Paul, Herbert Edelsbrunner, and Michael Kerber. “Computing Robustness and Persistence for Images.” <i>IEEE Transactions of Visualization and Computer Graphics</i>. IEEE, 2010. <a href=\"https://doi.org/10.1109/TVCG.2010.139\">https://doi.org/10.1109/TVCG.2010.139</a>.","short":"P. Bendich, H. Edelsbrunner, M. Kerber, IEEE Transactions of Visualization and Computer Graphics 16 (2010) 1251–1260.","ama":"Bendich P, Edelsbrunner H, Kerber M. Computing robustness and persistence for images. <i>IEEE Transactions of Visualization and Computer Graphics</i>. 2010;16(6):1251-1260. doi:<a href=\"https://doi.org/10.1109/TVCG.2010.139\">10.1109/TVCG.2010.139</a>","ieee":"P. Bendich, H. Edelsbrunner, and M. Kerber, “Computing robustness and persistence for images,” <i>IEEE Transactions of Visualization and Computer Graphics</i>, vol. 16, no. 6. IEEE, pp. 1251–1260, 2010."},"date_created":"2018-12-11T12:05:47Z","year":"2010","page":"1251 - 1260","department":[{"_id":"HeEd"}],"volume":16,"quality_controlled":"1","publisher":"IEEE","date_published":"2010-10-28T00:00:00Z","publication_status":"published","abstract":[{"lang":"eng","text":"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."}],"ddc":["000"]},{"year":"2009","acknowledgement":"This research was partially supported by Geomagic, Inc., and by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","publist_id":"2160","date_created":"2018-12-11T12:06:10Z","citation":{"short":"H. Edelsbrunner, J. Harer, in:, Springer, 2009, pp. 36–50.","ama":"Edelsbrunner H, Harer J. The persistent Morse complex segmentation of a 3-manifold. In: Vol 5903. Springer; 2009:36-50. doi:<a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">10.1007/978-3-642-10470-1_4</a>","ieee":"H. Edelsbrunner and J. Harer, “The persistent Morse complex segmentation of a 3-manifold,” presented at the 3DPH: Modelling the Physiological Human, Zermatt, Switzerland, 2009, vol. 5903, pp. 36–50.","ista":"Edelsbrunner H, Harer J. 2009. The persistent Morse complex segmentation of a 3-manifold. 3DPH: Modelling the Physiological Human, LNCS, vol. 5903, 36–50.","mla":"Edelsbrunner, Herbert, and John Harer. <i>The Persistent Morse Complex Segmentation of a 3-Manifold</i>. Vol. 5903, Springer, 2009, pp. 36–50, doi:<a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">10.1007/978-3-642-10470-1_4</a>.","chicago":"Edelsbrunner, Herbert, and John Harer. “The Persistent Morse Complex Segmentation of a 3-Manifold,” 5903:36–50. Springer, 2009. <a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">https://doi.org/10.1007/978-3-642-10470-1_4</a>.","apa":"Edelsbrunner, H., &#38; Harer, J. (2009). The persistent Morse complex segmentation of a 3-manifold (Vol. 5903, pp. 36–50). Presented at the 3DPH: Modelling the Physiological Human, Zermatt, Switzerland: Springer. <a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">https://doi.org/10.1007/978-3-642-10470-1_4</a>"},"_id":"3968","file_date_updated":"2020-07-14T12:46:21Z","doi":"10.1007/978-3-642-10470-1_4","publication_status":"published","ddc":["000"],"abstract":[{"lang":"eng","text":"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."}],"date_published":"2009-11-17T00:00:00Z","publisher":"Springer","volume":5903,"quality_controlled":"1","department":[{"_id":"HeEd"}],"page":"36 - 50","oa":1,"date_updated":"2021-01-12T07:53:32Z","pubrep_id":"535","title":"The persistent Morse complex segmentation of a 3-manifold","scopus_import":1,"has_accepted_license":"1","intvolume":"      5903","language":[{"iso":"eng"}],"author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Harer, John","first_name":"John","last_name":"Harer"}],"month":"11","conference":{"start_date":"2009-11-29","name":"3DPH: Modelling the Physiological Human","end_date":"2009-12-02","location":"Zermatt, Switzerland"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"creator":"system","content_type":"application/pdf","access_level":"open_access","file_size":165090,"relation":"main_file","file_id":"4694","file_name":"IST-2016-535-v1+1_2009-P-04-3ManifoldSegmentation.pdf","checksum":"11fc85bcc19bab1f020e706a4b8a4660","date_updated":"2020-07-14T12:46:21Z","date_created":"2018-12-12T10:08:33Z"}],"oa_version":"Submitted Version","day":"17","type":"conference","status":"public","alternative_title":["LNCS"]}]