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Cited by 1,274Topological and Geometric Data AnalysisHomotopy and Cohomology in Algebraic TopologyGeometric and Algebraic Topology

Barcodes: the persistent topology of data

Open access
Ghrist, Robert·2007·Bulletin of the American Mathematical Society
DOI 10.1090/s0273-0979-07-01191-3
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Abstract

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— persistent homology —and a novel representation of this algebraic characterization— barcodes . We sketch an application of these techniques to the classification of natural images.

Integrity signals

Registry

Crossref · 10.1090/s0273-0979-07-01191-3

Cites

32 of 32 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

8 of 28 cited sources checked · 5 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    For any positive parameter, there is a chain of inclusion maps from a Rips complex to a Čech complex at scaled parameter and back to a Rips complex at scaled parameter.

    “R ↩→ C √ 2 ↩→ R √ 2 .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [10] DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358. source
  • Premisecitation

    Barcodes are stable under noise added to a Morse filtration.

    “Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [7] CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271. source
  • Premisecitation

    For a finite persistence module with field coefficients, the persistent homology decomposes as a direct sum of free parts and torsional parts.

    “H * (C; F ) ∼ = i x t i • F [x] ⊕ ⎛ ⎝ j x r j • (F [x]/(x s j • F [x])) ⎞ ⎠ .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
    • The authors state this as an assumption: Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
  • Premisecitation

    The Čech complex and topologically equivalent subcomplexes are delicate to compute because they rely on precise distances between points.

    “However, the Čech complex and various topologically equivalent subcomplexes (e.g., the alpha complex of [13]) are delicate objects to compute, relying on the precise distances between the nodes in E n .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [13] EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72. source
  • Premisecitation

    The full data set consists of roughly 8,000,000 points in nine-dimensional Euclidean space.

    “The full data set consists of roughly 8,000,000 points in E 9 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. source

      Section 2 (page unavailable) reports about 4.2 million optical patches and 0.79 million range patches—roughly 5 million total, not 8 million. Section 3 (page unavailable) represents each original 3 × 3 patch as a vector in nine-dimensional space, although the preprocessed data lie on a seven-dimensional sphere. (LLM, conf=0.90)

      Experimental — accuracy not yet measured
  • Classified as probably not a substantial claim. 10
  • May combine two claims. 16

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf.

    “A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [20] HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366. source

      The Images section describes a collection photographed with a digital camera, but gives its size as 4,212 images, not 4,167. It says the images came from woods, open landscapes and urban areas, often in consecutive series, rather than describing them as random outdoor scenes. The paper was published in 1998, but the passage does not specify when the photographs were taken. (LLM, conf=0.90)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    In the Figure 4 example, the point cloud likely represents a connected object with one or two significant holes as measured by first homology and no significant higher homology.

    “For example, in Figure 4, one sees (from a very coarse sampling) that the point cloud likely represents a connected object with one or two significant 'holes' as measured by H 1 and no significant higher homology.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    In the annulus example, an ideal choice of proximity parameter, if it exists, is rare: increasing the parameter enough to remove small holes fills the large hole.

    “From the figure, it certainly appears as though an ideal choice of , if it exists, is rare: by the time is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    The coarse reduction of natural images still yields a point cloud too high-dimensional to visualize, and its structure appears blurred and initially distributed over the entire seven-sphere.

    “So coarse a reduction of natural images (into three-by-three squares of greyscale intensities) still leads to a point cloud of too high a dimension to visualize.Worse still, what structure there is is blurred and foggy: points appear at first to be distributed over the entire S 7 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    The resulting homology retains the structure of a module over the polynomial ring but is not necessarily free.

    “The resulting homology H * (C) retains the structure of an R[x]-module, but, unlike the chain module, is not necessarily free.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The combinatorics of the 1-skeleton completely determines the Rips complex, so it can be stored as a graph and reconstituted.

    “Thus, the combinatorics of the 1-skeleton completely determines the complex, and the Rips complex can be stored as a graph and reconstituted instead of storing the entire boundary operator needed for a Čech complex.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Premisenone given

    The Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.

    “The reason is that the Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Premisenone given

    For sufficiently small proximity parameter, the complex is a discrete set; for sufficiently large parameter, it is a single high-dimensional simplex.

    “For sufficiently small, the complex is a discrete set; for sufficiently large, the complex is a single highdimensional simplex.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisereasoning

    Classifying finite-type persistence modules is equivalent to classifying finitely generated nonnegatively graded modules over the polynomial ring, and this is very difficult for integer coefficients.

    “Nor is it easily classified: the Artin-Rees theory from commutative algebra implies that the problem of classifying (finite-type) persistence modules such as H * (C) is equivalent to classifying finitely generated nonnegatively graded R[x]-modules.This is known to be very difficult in, say, Z[x].”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    The Rips complex is generally neither a subcomplex of Euclidean space nor necessarily behaves like an n-dimensional space.

    “In general, it is neither a subcomplex of E n nor does it necessarily behave like an n-dimensional space at all (Figure 2).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisereasoning

    The Rips complex is computationally less expensive than the corresponding Čech complex, even though it generally has more simplices.

    “From a computational point of view, the Rips complex is less expensive that the corresponding Čech complex, even though the Rips complex has more simplices (in general).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The homology of a complex associated to a point cloud at a particular parameter is insufficient, and asking which parameter value is optimal is a mistake.

    “Despite being both computable and insightful, the homology of a complex associated to a point cloud at a particular is insufficient: it is a mistake to ask which value of is optimal.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Close examination of the data point corresponding to the primary circle reveals 3-by-3 patches with one light and one dark region separated by a linear transition, and this nodal curve appears in a circular family parameterized by its angle.

    “A close examination of the data point corresponding to the primary circle reveals a pattern of 3-by-3 patches with one light region and one dark region separated by a linear transition.This nodal curve between light and dark is linear and appears in a circular family parameterized by the angle of the nodal line, as shown in Figure 5.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    Efficient sampling or reduction of the complex with accurate topology is necessary.

    “This necessitates efficient sampling or reduction of the complex with accurate topology.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data

    The persistent generators correspond to meaningful structures in the data and inspire a sensible parametrization of the global structure of the data set.

    “These persistent generators, upon close examination, do correspond to meaningful structures in the data, inspiring a sensible parametrization of the global structure of the data set.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premiseown data

    Each of the two secondary circles intersects the high-k primary circle twice, and the two secondary circles are disjoint from each other.

    “A close examination of these three circles reveals that each intersects the highk primary twice, yet the two secondary circles are disjoint.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation

    Repeating the experiment with the camera held at a constant angle of π/4 yields secondary persistent first homology generators biased toward true vertical and true horizontal, suggesting the axis of pixellation is less relevant than the axis of gravity.

    “[3] reports that a repetition of the experiment with a camera held at a constant angle π/4 yields a data set whose secondary persistent H 1 generators exhibit a bias towards true vertical and true horizontal: the axis of pixellation appears less relevant than the axis of gravity in natural image data.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    For chain filtrations arising from realistic data sets, Rips complexes are of unmanageable size.

    “It is worth noting that for chain filtrations arising from realistic data sets, the Rips complexes are of an unmanageable size.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given

    The computation of persistent relative homology is more subtle because the parameterized chain complex is no longer free as a module over the polynomial ring.

    “The computation of persistent relative homology is more subtle, since the ensuing parameterized chain complex C is no longer free as an F [x]-module.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The first homology primary generator acts as a next term in the expansion of the homotopy type, collating the nodal curve between two contrasting patches, while the secondary circles act as higher-order terms with horizontal and vertical biases.

    “Following this analogy, the H 1 primary generator fills the role of a next term in the expansion of the homotopy type of the data set, collating the nodal curve between two contrasting patches.The secondary circles, interpolating between single and dual nodal curves, act as higher-order terms in the expansion, in which horizontal and vertical biases arise.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    There are fundamental algebraic difficulties in constructing an analogous theory of persistence modules for multi-parameter families.

    “However, there are fundamental algebraic difficulties in constructing an analogous theory of persistence modules in this setting [24].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [24] ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193. source

  • May combine two claims.

    Premisecitation

    Each secondary circle regulates images with three contrasting regions and interpolates between those states and the primary circle; the two secondary circles differ in their bias for horizontal versus vertical stratification.

    “As noted in [3], each secondary circle regulates images with three contrasting regions and interpolates between these states and the primary circle.The difference between the two secondary circles lies in their bias for horizontal and vertical stratification respectively.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

22 of 22

  • The authors state this as an assumption.

    Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.

    “One assumes a finite-type condition that each C i * is finitely generated as an R[x]-module and that the sequence stabilizes in i (in the case of an infinite sequence of chain complexes).”
1 of 1

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Authors
Ghrist, R.
Affiliations
University of Illinois Urbana-Champaign (US)
Corresponding author
Ghrist, Robert
Venue
Bulletin of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Published
2007-10-26
Volume
45
Issue
1
Pages
61-75
ISSN
  • 0002-9904 (ISSN-L)
  • 1088-9485
  • 0273-0979
DOI
10.1090/s0273-0979-07-01191-3
OpenAlex
W2071391326
Semantic Scholar
cad0bff789bb0e626245b21a47791ef76a9cd5c2
Cited by
1,236
References
32
Citations by year
2026: 78 · 2025: 98 · 2024: 81 · 2023: 105 · 2022: 93 · 2021: 111 · 2020: 121 · 2019: 121 · 2018: 86 · 2017: 66 · 2016: 76 · 2015: 46 · 2014: 41 · 2013: 37 · 2012: 18
FWCI
16.70
Type
journal article
Language
en
Open access
Licence
www.ams.org/publications/copyright-and-permissions (am, 366 d)
Locations
  • Bulletin of the American Mathematical Society (publishedVersion)
  • ScholarlyCommons (University of Pennsylvania) (submittedVersion)
  • Scholarly Commons (University of Pennsylvania) (submittedVersion)
Funders
  • Defense Advanced Research Projects Agency (HR0011-07-1-0002)
  • University of Illinois at Urbana-Champaign
Topics
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
  • Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
Keywords
persistent homology, barcodes, computational algebraic topology, feature detection, shape recognition, high-dimensional data, point clouds, natural image classification

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