--- _id: '2859' abstract: - lang: eng text: Given a continuous function f:X-R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization. author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Dmitriy full_name: Morozov, Dmitriy last_name: Morozov - first_name: Amit full_name: Patel, Amit id: 34A254A0-F248-11E8-B48F-1D18A9856A87 last_name: Patel citation: ama: Bendich P, Edelsbrunner H, Morozov D, Patel A. Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. 2013;15(1):51-72. doi:10.4310/HHA.2013.v15.n1.a3 apa: Bendich, P., Edelsbrunner, H., Morozov, D., & Patel, A. (2013). Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. International Press. https://doi.org/10.4310/HHA.2013.v15.n1.a3 chicago: Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “Homology and Robustness of Level and Interlevel Sets.” Homology, Homotopy and Applications. International Press, 2013. https://doi.org/10.4310/HHA.2013.v15.n1.a3. ieee: P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, “Homology and robustness of level and interlevel sets,” Homology, Homotopy and Applications, vol. 15, no. 1. International Press, pp. 51–72, 2013. ista: Bendich P, Edelsbrunner H, Morozov D, Patel A. 2013. Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. 15(1), 51–72. mla: Bendich, Paul, et al. “Homology and Robustness of Level and Interlevel Sets.” Homology, Homotopy and Applications, vol. 15, no. 1, International Press, 2013, pp. 51–72, doi:10.4310/HHA.2013.v15.n1.a3. short: P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, Homology, Homotopy and Applications 15 (2013) 51–72. date_created: 2018-12-11T11:59:58Z date_published: 2013-05-01T00:00:00Z date_updated: 2021-01-12T07:00:18Z day: '01' department: - _id: HeEd doi: 10.4310/HHA.2013.v15.n1.a3 external_id: arxiv: - '1102.3389' intvolume: ' 15' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1102.3389v1 month: '05' oa: 1 oa_version: Preprint page: 51 - 72 publication: Homology, Homotopy and Applications publication_status: published publisher: International Press publist_id: '3930' quality_controlled: '1' scopus_import: 1 status: public title: Homology and robustness of level and interlevel sets type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2013' ... --- _id: '3310' abstract: - lang: eng text: The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a real-valued function. Our account of the subject is expository, devoid of proofs, and written for non-experts in algebraic topology. acknowledgement: Research by the third author is partially supported by the National Science Foundation (NSF) under grant DBI-0820624. author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: Sergio full_name: Cabello, Sergio last_name: Cabello - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 citation: ama: Bendich P, Cabello S, Edelsbrunner H. A point calculus for interlevel set homology. Pattern Recognition Letters. 2012;33(11):1436-1444. doi:10.1016/j.patrec.2011.10.007 apa: Bendich, P., Cabello, S., & Edelsbrunner, H. (2012). A point calculus for interlevel set homology. Pattern Recognition Letters. Elsevier. https://doi.org/10.1016/j.patrec.2011.10.007 chicago: Bendich, Paul, Sergio Cabello, and Herbert Edelsbrunner. “A Point Calculus for Interlevel Set Homology.” Pattern Recognition Letters. Elsevier, 2012. https://doi.org/10.1016/j.patrec.2011.10.007. ieee: P. Bendich, S. Cabello, and H. Edelsbrunner, “A point calculus for interlevel set homology,” Pattern Recognition Letters, vol. 33, no. 11. Elsevier, pp. 1436–1444, 2012. ista: Bendich P, Cabello S, Edelsbrunner H. 2012. A point calculus for interlevel set homology. Pattern Recognition Letters. 33(11), 1436–1444. mla: Bendich, Paul, et al. “A Point Calculus for Interlevel Set Homology.” Pattern Recognition Letters, vol. 33, no. 11, Elsevier, 2012, pp. 1436–44, doi:10.1016/j.patrec.2011.10.007. short: P. Bendich, S. Cabello, H. Edelsbrunner, Pattern Recognition Letters 33 (2012) 1436–1444. date_created: 2018-12-11T12:02:36Z date_published: 2012-08-01T00:00:00Z date_updated: 2021-01-12T07:42:34Z day: '01' ddc: - '000' department: - _id: HeEd doi: 10.1016/j.patrec.2011.10.007 file: - access_level: open_access checksum: d65f79775b51258a604ca5ec741297cc content_type: application/pdf creator: system date_created: 2018-12-12T10:15:00Z date_updated: 2020-07-14T12:46:06Z file_id: '5116' file_name: IST-2016-542-v1+1_2012-J-01-Poinculus.pdf file_size: 280280 relation: main_file file_date_updated: 2020-07-14T12:46:06Z has_accepted_license: '1' intvolume: ' 33' issue: '11' language: - iso: eng month: '08' oa: 1 oa_version: Submitted Version page: 1436 - 1444 publication: Pattern Recognition Letters publication_status: published publisher: Elsevier publist_id: '3330' pubrep_id: '542' quality_controlled: '1' scopus_import: 1 status: public title: A point calculus for interlevel set homology type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 33 year: '2012' ... --- _id: '3378' abstract: - lang: eng 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. acknowledgement: This research was partially supported by the Defense Advanced Research Projects Agency (DARPA) under grant HR0011-05-1-0007. author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: John full_name: Harer, John last_name: Harer citation: ama: Bendich P, Harer J. Persistent intersection homology. Foundations of Computational Mathematics. 2011;11(3):305-336. doi:10.1007/s10208-010-9081-1 apa: Bendich, P., & Harer, J. (2011). Persistent intersection homology. Foundations of Computational Mathematics. Springer. https://doi.org/10.1007/s10208-010-9081-1 chicago: Bendich, Paul, and John Harer. “Persistent Intersection Homology.” Foundations of Computational Mathematics. Springer, 2011. https://doi.org/10.1007/s10208-010-9081-1. ieee: P. Bendich and J. Harer, “Persistent intersection homology,” Foundations of Computational Mathematics, vol. 11, no. 3. Springer, pp. 305–336, 2011. 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.” Foundations of Computational Mathematics, vol. 11, no. 3, Springer, 2011, pp. 305–36, doi:10.1007/s10208-010-9081-1. short: P. Bendich, J. Harer, Foundations of Computational Mathematics 11 (2011) 305–336. date_created: 2018-12-11T12:02:59Z date_published: 2011-06-01T00:00:00Z date_updated: 2021-01-12T07:43:04Z day: '01' department: - _id: HeEd doi: 10.1007/s10208-010-9081-1 intvolume: ' 11' issue: '3' language: - iso: eng month: '06' oa_version: None page: 305 - 336 publication: Foundations of Computational Mathematics publication_status: published publisher: Springer publist_id: '3229' quality_controlled: '1' scopus_import: 1 status: public title: Persistent intersection homology type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 11 year: '2011' ... --- _id: '3848' abstract: - lang: eng 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. alternative_title: - LNCS author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Dmitriy full_name: Morozov, Dmitriy last_name: Morozov - first_name: Amit full_name: Patel, Amit id: 34A254A0-F248-11E8-B48F-1D18A9856A87 last_name: Patel citation: ama: 'Bendich P, Edelsbrunner H, Morozov D, Patel A. The robustness of level sets. In: Vol 6346. Springer; 2010:1-10. doi:10.1007/978-3-642-15775-2_1' apa: 'Bendich, P., Edelsbrunner, H., Morozov, D., & Patel, A. (2010). The robustness of level sets (Vol. 6346, pp. 1–10). Presented at the ESA: European Symposium on Algorithms, Liverpool, UK: Springer. https://doi.org/10.1007/978-3-642-15775-2_1' chicago: Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “The Robustness of Level Sets,” 6346:1–10. Springer, 2010. https://doi.org/10.1007/978-3-642-15775-2_1. 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.' 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.' mla: Bendich, Paul, et al. The Robustness of Level Sets. Vol. 6346, Springer, 2010, pp. 1–10, doi:10.1007/978-3-642-15775-2_1. short: P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, in:, Springer, 2010, pp. 1–10. conference: end_date: 2010-09-08 location: Liverpool, UK name: 'ESA: European Symposium on Algorithms' start_date: 2010-09-06 date_created: 2018-12-11T12:05:30Z date_published: 2010-09-01T00:00:00Z date_updated: 2021-01-12T07:52:38Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-642-15775-2_1 intvolume: ' 6346' language: - iso: eng month: '09' oa_version: None page: 1 - 10 publication_status: published publisher: Springer publist_id: '2336' quality_controlled: '1' scopus_import: 1 status: public title: The robustness of level sets type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 6346 year: '2010' ... --- _id: '3849' 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. alternative_title: - LNCS author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Michael full_name: Kerber, Michael id: 36E4574A-F248-11E8-B48F-1D18A9856A87 last_name: Kerber orcid: 0000-0002-8030-9299 - first_name: Amit full_name: Patel, Amit id: 34A254A0-F248-11E8-B48F-1D18A9856A87 last_name: Patel citation: ama: 'Bendich P, Edelsbrunner H, Kerber M, Patel A. Persistent homology under non-uniform error. In: Vol 6281. Springer; 2010:12-23. doi:10.1007/978-3-642-15155-2_2' apa: 'Bendich, P., Edelsbrunner, H., Kerber, M., & 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. https://doi.org/10.1007/978-3-642-15155-2_2' chicago: Bendich, Paul, Herbert Edelsbrunner, Michael Kerber, and Amit Patel. “Persistent Homology under Non-Uniform Error,” 6281:12–23. Springer, 2010. https://doi.org/10.1007/978-3-642-15155-2_2. 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.' 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.' mla: Bendich, Paul, et al. Persistent Homology under Non-Uniform Error. Vol. 6281, Springer, 2010, pp. 12–23, doi:10.1007/978-3-642-15155-2_2. short: P. Bendich, H. Edelsbrunner, M. Kerber, A. Patel, in:, Springer, 2010, pp. 12–23. conference: end_date: 2010-08-27 location: Brno, Czech Republic name: 'MFCS: Mathematical Foundations of Computer Science' start_date: 2010-08-23 date_created: 2018-12-11T12:05:30Z date_published: 2010-08-10T00:00:00Z date_updated: 2021-01-12T07:52:38Z day: '10' ddc: - '000' department: - _id: HeEd doi: 10.1007/978-3-642-15155-2_2 file: - access_level: open_access checksum: af61e1c2bb42f3d556179d4692caeb1b content_type: application/pdf creator: system date_created: 2018-12-12T10:13:13Z date_updated: 2020-07-14T12:46:17Z file_id: '4994' file_name: IST-2016-537-v1+1_2010-P-05-NonuniformError.pdf file_size: 142357 relation: main_file file_date_updated: 2020-07-14T12:46:17Z has_accepted_license: '1' intvolume: ' 6281' language: - iso: eng month: '08' oa: 1 oa_version: Submitted Version page: 12 - 23 publication_status: published publisher: Springer publist_id: '2333' pubrep_id: '537' quality_controlled: '1' scopus_import: 1 status: public title: Persistent homology under non-uniform error type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 6281 year: '2010' ... --- _id: '3901' 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. author: - first_name: Paul full_name: Bendich, Paul id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Michael full_name: Kerber, Michael id: 36E4574A-F248-11E8-B48F-1D18A9856A87 last_name: Kerber orcid: 0000-0002-8030-9299 citation: ama: Bendich P, Edelsbrunner H, Kerber M. Computing robustness and persistence for images. IEEE Transactions of Visualization and Computer Graphics. 2010;16(6):1251-1260. doi:10.1109/TVCG.2010.139 apa: Bendich, P., Edelsbrunner, H., & Kerber, M. (2010). Computing robustness and persistence for images. IEEE Transactions of Visualization and Computer Graphics. IEEE. https://doi.org/10.1109/TVCG.2010.139 chicago: Bendich, Paul, Herbert Edelsbrunner, and Michael Kerber. “Computing Robustness and Persistence for Images.” IEEE Transactions of Visualization and Computer Graphics. IEEE, 2010. https://doi.org/10.1109/TVCG.2010.139. ieee: P. Bendich, H. Edelsbrunner, and M. Kerber, “Computing robustness and persistence for images,” IEEE Transactions of Visualization and Computer Graphics, vol. 16, no. 6. IEEE, pp. 1251–1260, 2010. 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.” IEEE Transactions of Visualization and Computer Graphics, vol. 16, no. 6, IEEE, 2010, pp. 1251–60, doi:10.1109/TVCG.2010.139. short: P. Bendich, H. Edelsbrunner, M. Kerber, IEEE Transactions of Visualization and Computer Graphics 16 (2010) 1251–1260. date_created: 2018-12-11T12:05:47Z date_published: 2010-10-28T00:00:00Z date_updated: 2021-01-12T07:53:04Z day: '28' ddc: - '000' department: - _id: HeEd doi: 10.1109/TVCG.2010.139 file: - access_level: open_access checksum: f6d813c04f4b46023cec6b9a17f15472 content_type: application/pdf creator: system date_created: 2018-12-12T10:17:10Z date_updated: 2020-07-14T12:46:21Z file_id: '5262' file_name: IST-2016-536-v1+1_2010-J-02-PersistenceforImages.pdf file_size: 721994 relation: main_file file_date_updated: 2020-07-14T12:46:21Z has_accepted_license: '1' intvolume: ' 16' issue: '6' language: - iso: eng month: '10' oa: 1 oa_version: Submitted Version page: 1251 - 1260 publication: IEEE Transactions of Visualization and Computer Graphics publication_status: published publisher: IEEE publist_id: '2253' pubrep_id: '536' quality_controlled: '1' scopus_import: 1 status: public title: Computing robustness and persistence for images type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 16 year: '2010' ... --- _id: '3975' abstract: - lang: eng text: We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a I-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample. author: - first_name: Paul full_name: Paul Bendich id: 43F6EC54-F248-11E8-B48F-1D18A9856A87 last_name: Bendich - first_name: David full_name: Cohen-Steiner, David last_name: Cohen Steiner - first_name: Herbert full_name: Herbert Edelsbrunner id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: John full_name: Harer, John last_name: Harer - first_name: Dmitriy full_name: Morozov, Dmitriy last_name: Morozov citation: ama: 'Bendich P, Cohen Steiner D, Edelsbrunner H, Harer J, Morozov D. Inferring local homology from sampled stratified spaces. In: IEEE; 2007:536-546. doi:10.1109/FOCS.2007.33' apa: 'Bendich, P., Cohen Steiner, D., Edelsbrunner, H., Harer, J., & Morozov, D. (2007). Inferring local homology from sampled stratified spaces (pp. 536–546). Presented at the FOCS: Foundations of Computer Science, IEEE. https://doi.org/10.1109/FOCS.2007.33' chicago: Bendich, Paul, David Cohen Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. “Inferring Local Homology from Sampled Stratified Spaces,” 536–46. IEEE, 2007. https://doi.org/10.1109/FOCS.2007.33. ieee: 'P. Bendich, D. Cohen Steiner, H. Edelsbrunner, J. Harer, and D. Morozov, “Inferring local homology from sampled stratified spaces,” presented at the FOCS: Foundations of Computer Science, 2007, pp. 536–546.' ista: 'Bendich P, Cohen Steiner D, Edelsbrunner H, Harer J, Morozov D. 2007. Inferring local homology from sampled stratified spaces. FOCS: Foundations of Computer Science, 536–546.' mla: Bendich, Paul, et al. Inferring Local Homology from Sampled Stratified Spaces. IEEE, 2007, pp. 536–46, doi:10.1109/FOCS.2007.33. short: P. Bendich, D. Cohen Steiner, H. Edelsbrunner, J. Harer, D. Morozov, in:, IEEE, 2007, pp. 536–546. conference: name: 'FOCS: Foundations of Computer Science' date_created: 2018-12-11T12:06:13Z date_published: 2007-01-01T00:00:00Z date_updated: 2021-01-12T07:53:35Z day: '01' doi: 10.1109/FOCS.2007.33 extern: 1 month: '01' page: 536 - 546 publication_status: published publisher: IEEE publist_id: '2150' quality_controlled: 0 status: public title: Inferring local homology from sampled stratified spaces type: conference year: '2007' ...