---
_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.
arxiv: 1
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. <i>Homology, Homotopy and Applications</i>. 2013;15(1):51-72.
    doi:<a href="https://doi.org/10.4310/HHA.2013.v15.n1.a3">10.4310/HHA.2013.v15.n1.a3</a>
  apa: Bendich, P., Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2013). Homology
    and robustness of level and interlevel sets. <i>Homology, Homotopy and Applications</i>.
    International Press. <a href="https://doi.org/10.4310/HHA.2013.v15.n1.a3">https://doi.org/10.4310/HHA.2013.v15.n1.a3</a>
  chicago: Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “Homology
    and Robustness of Level and Interlevel Sets.” <i>Homology, Homotopy and Applications</i>.
    International Press, 2013. <a href="https://doi.org/10.4310/HHA.2013.v15.n1.a3">https://doi.org/10.4310/HHA.2013.v15.n1.a3</a>.
  ieee: P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, “Homology and robustness
    of level and interlevel sets,” <i>Homology, Homotopy and Applications</i>, 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.”
    <i>Homology, Homotopy and Applications</i>, vol. 15, no. 1, International Press,
    2013, pp. 51–72, doi:<a href="https://doi.org/10.4310/HHA.2013.v15.n1.a3">10.4310/HHA.2013.v15.n1.a3</a>.
  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: '3377'
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.
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.
author:
- 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: 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>
  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>
  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>.
  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>.
  short: H. Edelsbrunner, D. Morozov, A. Patel, Foundations of Computational Mathematics
    11 (2011) 345–361.
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/s10208-011-9090-8
intvolume: '        11'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://arxiv.org/abs/0911.2142
month: '06'
oa: 1
oa_version: Submitted Version
page: 345 - 361
publication: Foundations of Computational Mathematics
publication_status: published
publisher: Springer
publist_id: '3230'
quality_controlled: '1'
scopus_import: 1
status: public
title: Quantifying transversality by measuring the robustness of intersections
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2011'
...
---
_id: '3795'
abstract:
- lang: eng
  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.'
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.
alternative_title:
- Mathematics and Visualization
author:
- 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: '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>'
  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>'
  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>.'
  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.'
  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.'
  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>.'
  short: 'H. Edelsbrunner, D. Morozov, A. Patel, in:, Topological Data Analysis and
    Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42.'
date_created: 2018-12-11T12:05:13Z
date_published: 2010-12-22T00:00:00Z
date_updated: 2021-01-12T07:52:15Z
day: '22'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/978-3-642-15014-2_3
file:
- access_level: open_access
  checksum: f03a44c3d1c3e2d4fedb3b94404f3fd5
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:11:40Z
  date_updated: 2020-07-14T12:46:16Z
  file_id: '4896'
  file_name: IST-2016-538-v1+1_2011-B-02-ApparentContour.pdf
  file_size: 210710
  relation: main_file
file_date_updated: 2020-07-14T12:46:16Z
has_accepted_license: '1'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Submitted Version
page: 27 - 42
publication: 'Topological Data Analysis and Visualization: Theory, Algorithms and
  Applications'
publication_status: published
publisher: Springer
publist_id: '2428'
pubrep_id: '538'
quality_controlled: '1'
scopus_import: 1
status: public
title: The stability of the apparent contour of an orientable 2-manifold
type: book_chapter
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2010'
...
---
_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:<a href="https://doi.org/10.1007/978-3-642-15775-2_1">10.1007/978-3-642-15775-2_1</a>'
  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>'
  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>.
  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. <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>.
  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:<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>'
  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>.
  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. <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>.
  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: '3974'
abstract:
- lang: eng
  text: Generalizing the concept of a Reeb graph, the Reeb space of a multivariate
    continuous mapping identifies points of the domain that belong to a common component
    of the preimage of a point in the range. We study the local and global structure
    of this space for generic, piecewise linear mappings on a combinatorial manifold.
author:
- 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: Amit
  full_name: Amit Patel
  id: 34A254A0-F248-11E8-B48F-1D18A9856A87
  last_name: Patel
citation:
  ama: 'Edelsbrunner H, Harer J, Patel A. Reeb spaces of piecewise linear mappings.
    In: ACM; 2008:242-250. doi:<a href="https://doi.org/10.1145/1377676.1377720">10.1145/1377676.1377720</a>'
  apa: 'Edelsbrunner, H., Harer, J., &#38; Patel, A. (2008). Reeb spaces of piecewise
    linear mappings (pp. 242–250). Presented at the SCG: Symposium on Computational
    Geometry, ACM. <a href="https://doi.org/10.1145/1377676.1377720">https://doi.org/10.1145/1377676.1377720</a>'
  chicago: Edelsbrunner, Herbert, John Harer, and Amit Patel. “Reeb Spaces of Piecewise
    Linear Mappings,” 242–50. ACM, 2008. <a href="https://doi.org/10.1145/1377676.1377720">https://doi.org/10.1145/1377676.1377720</a>.
  ieee: 'H. Edelsbrunner, J. Harer, and A. Patel, “Reeb spaces of piecewise linear
    mappings,” presented at the SCG: Symposium on Computational Geometry, 2008, pp.
    242–250.'
  ista: 'Edelsbrunner H, Harer J, Patel A. 2008. Reeb spaces of piecewise linear mappings.
    SCG: Symposium on Computational Geometry, 242–250.'
  mla: Edelsbrunner, Herbert, et al. <i>Reeb Spaces of Piecewise Linear Mappings</i>.
    ACM, 2008, pp. 242–50, doi:<a href="https://doi.org/10.1145/1377676.1377720">10.1145/1377676.1377720</a>.
  short: H. Edelsbrunner, J. Harer, A. Patel, in:, ACM, 2008, pp. 242–250.
conference:
  name: 'SCG: Symposium on Computational Geometry'
date_created: 2018-12-11T12:06:13Z
date_published: 2008-01-01T00:00:00Z
date_updated: 2021-01-12T07:53:35Z
day: '01'
doi: 10.1145/1377676.1377720
extern: 1
month: '01'
page: 242 - 250
publication_status: published
publisher: ACM
publist_id: '2155'
quality_controlled: 0
status: public
title: Reeb spaces of piecewise linear mappings
type: conference
year: '2008'
...