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

The DOI matches its Crossref record; 12 of the 12 references it cites were found in a registry or an index; none of the 8 sources its claims cite is checked for support yet.

Topological Persistence and Simplification

Open access

Edelsbrunner, Letscher, Zomorodian·2002·Discrete & Computational Geometry

DOI 10.1007/s00454-002-2885-2

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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

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Claims the paper makes

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

A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“While there is considerable topological noise at p = 0, a simplification process which eliminates 1cycles of persistence less than 2688 succeeds in separating the tunnel from the remaining topological attributes detected by measuring homology.”
§ Discrete & Computational Geometry > Feature Detection.
Load-bearingStrength:plain· not yet calibratedCausal:direct· not yet calibratedSingle case
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- Rests on an assumption: 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.

Show all 49 claims

Probably not a substantial claim. 16
It may state two separate claims as one. 24
The quote only partly states this claim. 1

The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Furthermore, if σ ∈ K , then all faces of σ are simplices in K .”
§ Discrete & Computational Geometry > Figure 1 illustrates a two-dimensional example of this construction.
Strength:plain· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passes
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Any two simplices in K are either disjoint or they intersect in a common face, which is a simplex of smaller dimension.”
§ Discrete & Computational Geometry > Figure 1 illustrates a two-dimensional example of this construction.
Strength:plain· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passes
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- 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.
In the paper
“Finally, we note that Robins has independently formulated an alternate but equivalent definition of persistence [10].”
§ Discrete & Computational Geometry > Algebraic Formulation.
What the sources say
- Not matched
  Couldn’t be matched to a paper in Crossref, so it can’t be 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.
Strength:plain· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passes
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Discrete & Computational Geometry > Conflict Resolution.
Strength:hedged· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: fails
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- Rests on an assumption: In alpha complex filtrations, simplices are ordered according to a particular growth model.
In the paper
“The problem of topological simplification was also approached by El-Sana and Varshney [5] using alpha shape inspired ideas of geometric growth.”
§ Discrete & Computational Geometry > Approach and Results.
What the sources say
- Not checked — no full text of the sourcecites [5] 5. J. El-Sana and A. Varshney. Topology simplification for polygonal virtual environments. IEEE Trans. Visual. Comput. Graphics, 4:133-144, 1998. source
Strength:plain· not yet calibratedCausal:correlational· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passes
Probably not a substantial claim. The paper’s conclusions may not depend on it.

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?

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

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Show all 12 references

Research works printed but not linked: 2

[8]8. J. Milnor. Morse Theory. Princeton University Press, Princeton, New Jersey, 1963.
[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.

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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: bronze
doi.org
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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