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Cited by 1,811Topological and Geometric Data AnalysisDigital Image Processing TechniquesGeometric and Algebraic Topology

Topological Persistence and Simplification

Open access
Edelsbrunner, Letscher, Zomorodian·2002·Discrete & Computational Geometry
DOI 10.1007/s00454-002-2885-2
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AI snapshot

Written from the paper’s full text.

TL;DR

This paper formalizes topological persistence in a filtration, using the lifetime of homology classes to distinguish features from noise, and gives algorithms to compute persistence and simplify filtrations. It demonstrates the approach on alpha-complex filtrations of molecular and surface data, including separation of a tunnel in Gramicidin A from shorter-lived cycles.

Key findings

  • Persistence is defined for Betti numbers and non-bounding cycles by factoring a cycle group at one filtration stage by the boundary group at a later stage; cycles with short lifetimes are treated as topological noise, while longer-lived ones are features.
  • The pairing algorithm matches positive \(k\)-simplices with negative \((k+1)\)-simplices; index-based persistence is \(j-i-1\), and the worst-case running time is \(O(m^3)\), with union-find acceleration for \(k=0\) and \(k=2\) giving amortized \(O(A^{-1}(m))\) per operation.
  • On five datasets, total persistence timings ranged from 0.06 s for Gramicidin A to 19.93 s for the Buddha dataset with union-find, versus 30.52 s without it; average collision counts and list lengths were small, and conflicts were rare (basic conflicts 0–1, recursive conflicts 0–212 across datasets).
  • For Gramicidin A, removing 1-cycles with persistence below 2688 isolated the molecular tunnel from other topological attributes.

Methods

  • The work studies finite simplicial complexes in \(\mathbb{R}^3\), especially alpha complexes of Delaunay triangulations, ordered by an age filter induced by growing weighted balls.
  • Persistence is formulated algebraically as \(H^{\ell,p}_k = Z^\ell_k / (B^{\ell+p}_k \cap Z^\ell_k)\), and an abstract pairing algorithm uses a hash table and cycle search to pair simplex births with deaths; simplification is performed by reordering simplices via migration, with conflict resolution or diminution.
  • Experiments implemented the algorithms with alpha-shapes software on five datasets: three weighted molecular structures and two unweighted macroscopic surfaces; timings were collected on a 266 MHz Pentium II with 128 MB RAM under Solaris 8.

Limitations

  • Stated: the detailed analysis of the 1-cycle computation is left open, and the authors ask for a different, more efficient algorithm; worst-case cubic behavior contrasts with observed near-linear timing, and conflicts, though rare, require handling.
  • Stated: conflict resolution by subdivision may not correspond to the alpha-complex growth model, while conflict diminution changes when cycles affect Betti numbers.
  • Evident from the text: the experimental evidence covers only five datasets and a prototype implementation; time-based persistence is not developed further, and the setting is restricted to \(\mathbb{R}^3\) complexes with \(\mathbb{Z}_2\) coefficients.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1007/s00454-002-2885-2

Cites

12 of 12 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 8 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The primary topological feature of the Gramicidin A data is a tunnel that runs through the molecule.

    “The primary topological feature of this data is a tunnel that runs through the molecule, as shown in Fig. 17.”
    Quote in the paperSubstantial claim
  • Findingnone given

    The image of the homomorphism from ordinary homology to later homology is isomorphic to the p-persistent homology group.

    “The image of the homomorphism is isomorphic to the p-persistent homology group of K ℓ , im η ℓ, p k H ℓ, p k .”
    Quote in the paperSubstantial claim
  • Findingreasoning

    In Case 2, if there is no k-cycle in K_ell homologous to d, then beta_{ell,p}^k = beta_{ell,p-1}^k.

    “Then Z ℓ k ∩ B ℓ+ p-1 k = Z ℓ k ∩ B ℓ+ p k , and hence β ℓ, p k = β ℓ, p-1 k”
    Quote in the paperSubstantial claim
  • Findingown data

    After accelerating with union-find, the slowest portion of the algorithm is adding 1-cycles.

    “After accelerating with union-find, the slowest portion of the algorithm is adding 1-cycles.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Increasing p by one shortens the persistence of all nonbounding cycles by one.

    “In other words, increasing p by one shortens the persistence of all nonbounding cycles by one.”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 16
  • May combine two claims. 24
  • The quote only partly states this claim. 1

  • Classified as probably not a substantial claim.

    Findingreasoning

    The persistence of a k-cycle is one short of the difference between the indices of its paired simplices, j-i-1.

    “The persistence of that k-cycle is one short of the difference between indices, ji -1.”
    Quote in the paperSubstantial claim

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

    Findingreasoning

    The first Betti number beta_1 is the size of a basis of tunnels.

    “We can write each tunnel as a sum of tunnels in a basis, and β 1 is the size of the basis.”
    Quote in the paperNot a substantial claim

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

    Findingreasoning

    The second Betti number beta_2 is the size of a basis of voids, which equals the number of voids.

    “As before, β 2 is the size of a basis of voids, which is equal to the number of voids.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    All Betti numbers are non-negative, so pos_k >= neg_{k+1} for all levels in the filtration.

    “All Betti numbers are non-negative so pos k ≥ neg k+1 for all ℓ.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The data structure guarantees that d is homologous to the sum of cycles represented by the simplices in the list, and that the list contains the youngest simplex in (d).

    “All we guarantee is that d is homologous to the sum of cycles represented by the simplices in the list, and that the list contains the youngest simplex in (d), which is σ i as above.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Basic and recursive conflicts exist in practice, but are rather rare, as shown in Section 6.

    “Basic and recursive conflicts exist in practice, but are rather rare, as shown in Section 6.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Findingnone given

    For each simplex sigma in the Delaunay triangulation D, there is a unique birth time alpha(sigma) such that sigma belongs to K(alpha) if and only if alpha >= alpha(sigma).

    “For each simplex σ ∈ D, there is a unique birth time α(σ ) defined such that σ ∈ K (α) iff α ≥ α(σ ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The pairs (su, suv) and (tw, stu) in Fig. 6 constitute a conflict of type (1,2) and show that conflicts do occur.

    “The pairs (su, suv) and (tw, stu) in Fig. 6 constitute a conflict of type (1,2) and show that conflicts do occur, although our experiments in Section 6 suggest that they are rather rare.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The results suggest a possible linear dependence of the running time on the size of the data, which is substantially faster than the cubic dependence proved in Section 4.

    “The results suggest a possible linear dependence of the running time on the size of the data, which is substantially faster than the cubic dependence proved in Section 4.”
    Quote in the paperNot a substantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    Tables 3 and 4 provide evidence that the average number of collisions and the average list length are constant.

    “Tables 3 and 4 provide evidence that this might indeed be the case.”
    Quote in the paperSubstantial claim

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

    Findingnone given

    If a simplex belongs to K, then all its faces are simplices in K.

    “Furthermore, if σ ∈ K , then all faces of σ are simplices in K .”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The point set is in general position, so at most four (three in R^2) Voronoi regions can have a non-empty common intersection.

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

    Findingnone given

    The incremental algorithm in Section 2 asserts that beta_k at level ell is the number of k-intervals that contain index ell on the axis.

    “The incremental algorithm in Section 2 asserts that β ℓ k is the number of k-intervals that contain index ℓ on the axis.”
    Quote in the paperNot a substantial claim

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

    Findingnone given

    Every 0-cycle is represented by a list of length 2.

    “We do not show statistics for 0-cycles in Table 4 as every 0-cycle is represented by a list of length 2.”
    Quote in the paperNot a substantial claim

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

    Findingnone given

    Any two simplices in the dual complex K are either disjoint or intersect in a common face, which is a simplex of smaller dimension.

    “Any two simplices in K are either disjoint or they intersect in a common face, which is a simplex of smaller dimension.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The point set is in general position, so at most four (three in R^2) Voronoi regions can have a non-empty common intersection.

  • May combine two claims.

    Findingown data

    As substantiated in Table 2, the union-find improvement subsumes adding 0- and 2-cycles in the marking process, shrinking the time for these steps to essentially nothing.

    “As substantiated in Table 2, this improvement subsumes adding 0-and 2-cycles in the marking process, shrinking the time for these to steps to essentially nothing.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Table 5 shows that the average persistence is considerably larger than the average number of collisions and list length.

    “Table 5 shows that the average persistence is considerably larger than the average number of collisions and list length.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The total algorithm runs in time at most O(p_i^2), which is at most O(m^3).

    “The total algorithm runs in time at most O( p 2 i ), which is at most O(m 3 ).”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given

    Our motivation for formulating persistent homology in (3) was to eliminate cycles with low persistence.

    “Our motivation for formulating persistent homology in (3) was to eliminate cycles with low persistence.”
    Quote in the paperSubstantial claim

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

    Interpretationreasoning

    There are 6 possible types of conflicts, each identified by the pair (dim σ_i, dim σ_h) of the dimensions of the main participants.

    “There are 4 2 = 6 possible types of conflicts, each identified by the pair (dim σ i , dim σ h ) of the dimensions of the main participants.”
    Quote in the paperNot a substantial claim

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

    Premisenone given

    The sequence of iso-surfaces obtained by increasing alpha represents a growth process similar to that represented by a filtration.

    “The sequence of iso-surfaces obtained by increasing α represents a growth process similar to that represented by a filtration.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    As cx is not a face of abx, we have removed the conflict and preserved the Betti numbers of a refined filtration.

    “As cx is not a face of abx, we have removed the conflict and preserved the Betti numbers of a refined filtration.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    By complementarity and duality, there are no conflicts of types (1, 3) and (2, 3).

    “By complementarity and duality, there are no conflicts of types (1, 3) and (2, 3).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data

    We can explain the apparently linear running time documented in Table 2 by showing that the average number of collisions and the average list length are both constant.

    “We can explain the apparently linear running time documented in Table 2 by showing that the average number of collisions and the average list length are both constant.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The cycle search continues until it finds a collision-free cycle homologous to d, and that cycle has the correct youngest positive simplex.

    “The cycle search continues until it finds a collision-free cycle homologous to d, and the Collision Lemma implies that that cycle has the correct youngest positive simplex.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    A single collision takes time at most O(p_i), and the entire search for σ_i takes time at most O(p_i^2).

    “A single collision takes time at most O( p i ), and the entire search for σ i takes time at most O( p 2 i ).”
    Quote in the paperSubstantial claim

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

    Premisecitation

    Robins has independently formulated an alternate but equivalent definition of persistence.

    “Finally, we note that Robins has independently formulated an alternate but equivalent definition of persistence [10].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [10] 10. V. Robins. Computing topology at multiple resolutions: foundations and applications to fractals and dynamics. Ph.D. thesis, Department of Applied Mathematics, University of Colorado, June 2000.

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

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

    Interpretationnone given

    The ordering of the new simplices specified by subdivision in Fig. 13 might not have a corresponding set of weighted balls that would generate the filtration under the growth model.

    “The ordering of the new simplices specified by subdivision in Fig. 13 might not have a corresponding set of weighted balls that would generate the filtration under the growth model.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: In alpha complex filtrations, simplices are ordered according to a particular growth model.

  • May combine two claims.

    Interpretationnone given

    The number of collisions and the length of lists is bounded from above by the persistence of cycles.

    “Recall that the number of collisions and the length of lists is bounded from above by the persistence of cycles.”
    Quote in the paperNot a substantial claim

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

    Premisenone given

    Simplices in a filter correspond to critical points of a density function.

    “Specifically, simplices in a filter correspond to critical points of a density function.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The list for i contains fewer than (k+2)p_i k-simplices, and the list for g contains fewer than (k+2)p_g < (k+2)p_i k-simplices.

    “Hence, i contains fewer than (k + 2) p i k-simplices, and similarly g contains fewer than (k + 2) p g < (k + 2) p i k-simplices.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    For k = 0, each set is the vertex set of a connected component, and each set has exactly one yet unpaired vertex, namely the oldest one in the component.

    “For k = 0, each set is the vertex set of a connected component.Each set has exactly one yet unpaired vertex, namely the oldest one in the component.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    As a consequence of the formulation, the lifetime of every cycle is reduced regardless of its persistence, leading to the creation of k-triangles.

    “As a consequence of the formulation, the lifetime of every cycle is reduced regardless of its persistence, leading to the creation of k-triangles.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    The problem of topological simplification was also approached by El-Sana and Varshney using alpha shape inspired ideas of geometric growth.

    “The problem of topological simplification was also approached by El-Sana and Varshney [5] using alpha shape inspired ideas of geometric growth.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [5] 5. J. El-Sana and A. Varshney. Topology simplification for polygonal virtual environments. IEEE Trans. Visual. Comput. Graphics, 4:133-144, 1998. source

  • May combine two claims.

    Interpretationreasoning

    We cannot achieve the same acceleration for 1-cycles using this method, as there can be multiple unpaired positive edges at any time.

    “We cannot achieve the same acceleration for 1-cycles using this method, however, as there can be multiple unpaired positive edges at any time.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The number of collisions in cycle search is at most the number of positive k-simplices strictly between σ_i and σ_j, which is less than p_i.

    “The number of collisions is at most the number of positive k-simplices strictly between σ i and σ j , which is less than p i .”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    We interpret these intervals as k-intervals of a simplified version of the original filtration.

    “We interpret these intervals as k-intervals of a simplified version of the original filtration.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    We may resolve a conflict by subdivision, achieving correct Betti numbers for a refined complex and its corresponding filtrations.

    “We may resolve a conflict by subdivision.Here, we achieve the correct Betti numbers for a refined complex and its corresponding filtrations.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    At the time we move it, the face is already collocated with its matching negative simplex, and the two cancel each other's contributions.

    “At the time we move it, the face is already collocated with its matching negative simplex, and the two cancel each other's contributions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    A collision happens at T[g] only if σ_g already forms a pair, which implies its k-interval [g, h) is contained inside [i, j).

    “A collision happens at T[g] only if σ g already forms a pair, which implies its k-interval [g, h) is contained inside [i, j).”
    Quote in the paperSubstantial claim
39 of 39

  • The authors state this as an assumption.

    The point set is in general position, so at most four (three in R^2) Voronoi regions can have a non-empty common intersection.

    “We assume general position, where at most four (three in R 2 ) Voronoi regions can have a non-empty common intersection.”

  • The paper relies on this without support.

    Complexes embedded in three-dimensional Euclidean space have no torsion.

    “There is no torsion in complexes embedded in three-dimensional Euclidean space, so the homology groups are vector spaces.”

  • The paper relies on this without support.

    In alpha complex filtrations, simplices are ordered according to a particular growth model.

    “For example, in alpha complex filtrations, simplices are ordered according to a particular growth model.”

  • The paper relies on this without support.

    The five data sets used in the experiments are representative of their respective classes of molecular structures and macroscopic shapes.

    “We have applied the software to a variety of datasets and show the results for five representative sets in this section.”

  • The paper relies on this without support.

    The Gramicidin A dataset contains the time-averaged molecular dynamics structure of Gramicidin A, a peptide that forms a channel for ion and water movement across lipid membranes.

    “This dataset contains the time-averaged molecular dynamics structure of Gramicidin A, a peptide that forms a channel for ion and water movement across lipid membranes.”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Persistent homology groups of subcomplexes of S^3 over Z2 coefficients have a simple description in terms of a set of intervals.

    “In other words, the authors prove constructively that persistent homology groups of subcomplexes of S 3 , if computed over Z 2 coefficients, have a simple description in terms of a set of intervals.”

    From Computing Persistent Homology (2005) · cites this paper as [11]

    The Abstract Algorithm section describes a constructive pairing of simplices and represents each pair by a half-open interval; the paper defines the corresponding persistent homology groups using Z2 coefficients. But the Approach and Results section restricts the work to finite simplicial complexes in R3, not all subcomplexes of S3. The Homology Groups section also says its R3 complexes have no 3-cycles, so the cited result does not establish the claim at the stated scope. (LLM, conf=0.86)

    Experimental — accuracy not yet measured

  • The previously known algorithm's intervals allowed the correct computation of the rank of persistent homology groups.

    “Surprisingly, the authors show that these intervals allowed the correct computation of the rank of persistent homology groups.”

    From Computing Persistent Homology (2005) · cites this paper as [11]

    The Abstract Algorithm section (page unavailable) says the paper’s pairing algorithm represents simplex pairs as intervals and proves that the corresponding triangles count persistent Betti numbers—the ranks of persistent homology groups. The Approach and Results section (page unavailable) says this algorithm was based on an earlier incremental Betti-number algorithm, but does not describe the intervals as products of that earlier algorithm. (LLM, conf=0.84)

    Experimental — accuracy not yet measured

  • The previously known persistent homology algorithm worked only for spaces that were subcomplexes of S^3 over Z2 coefficients.

    “The authors also provide an algorithm that worked only for spaces that were subcomplexes of S 3 over Z 2 coefficients.”

    From Computing Persistent Homology (2005) · cites this paper as [11]

    The Approach and Results section says the authors provide a persistence algorithm and restrict their attention to finite simplicial complexes in R³, not subcomplexes of S³. The Background and Chains, Cycles, Boundaries sections use Z₂ coefficients. Thus the coefficient restriction is supported, but the stated space is not. (LLM, conf=0.88)

    Experimental — accuracy not yet measured

  • The P-intervals are precisely the k-intervals utilized in [11] to describe persistent Z2-homology.

    “Therefore, the P -intervals discussed here are precisely the so-called k -intervals utilized in [11] to describe persistent Z 2-homology.”

    From Computing Persistent Homology (2005) · cites this paper as [11]

  • The rank β_{l,p}^k of H_{l,p}^k is the number of triangles in T containing the point (l,p), where T is the set of triangles defined by P-intervals for the k-dimensional persistence module.

    “Lemma 3.1. Let T be the set of triangles defined by P -intervals for the k-dimensional persistence module . The rank β l , p k of H l , p k is the number of triangles in T containing the point ( l , p ) .”

    From Computing Persistent Homology (2005) · cites this paper as [11]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Induced homology maps are described as injective despite cycle deaths

    Minor

    An injective map from an earlier homology group cannot send one of its nonzero classes to zero. The paper also says that later negative simplices cancel classes created earlier, so the homology maps it describes cannot generally be injective.

  • One reported average list length exceeds average persistence

    Minor

    For G's 2-cycles, Table 4 gives an average list length of 4.86, while Table 5 gives an average persistence of 4.47. Thus the stated comparison does not hold for that dataset and cycle dimension.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. [8]8. J. Milnor. Morse Theory. Princeton University Press, Princeton, New Jersey, 1963.
  2. [11]11. J. A. Sethian. Level Set Methods. Cambridge University Press, Cambridge, 1996. Received March 19, 2001, and in revised form October 4, 2001. Online publication October 29, 2002.

Cited by

Sorted by citations

Related

Authors
Edelsbrunner, Letscher, & Zomorodian
Affiliations
  • Duke University (US)
  • Oklahoma State University (US)
  • University of Illinois Urbana-Champaign (US)
Venue
Discrete & Computational Geometry
Publisher
Springer Science and Business Media LLC
Published
2002-11
Volume
28
Issue
4
Pages
511-533
ISSN
  • 0179-5376 (ISSN-L)
  • 1432-0444
DOI
10.1007/s00454-002-2885-2
OpenAlex
W3013843370
Cited by
1,779
References
12
Citations by year
2026: 130 · 2025: 152 · 2024: 125 · 2023: 94 · 2022: 91 · 2021: 139 · 2020: 106 · 2019: 131 · 2018: 101 · 2017: 72 · 2016: 89 · 2015: 70 · 2014: 74 · 2013: 47 · 2012: 63
FWCI
13.65
Type
journal article
Language
en
Open access
Licence
www.springer.com/tdm (tdm)
Locations
Discrete & Computational Geometry (publishedVersion)
Funders
  • National Science Foundation (9712088)
  • Army Research Office
Topics
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
  • Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Digital Image Processing Techniques
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
Keywords
topological persistence, topological simplification, simplicial complexes, filtration, computational topology

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