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arxiv: 2606.30490 · v1 · pith:5PZDCDXXnew · submitted 2026-06-29 · 🧮 math.AT

From Frames to Features: Scalable Zigzag Persistence for Binary Video

David Lanners This is my paper

Pith reviewed 2026-06-30 02:50 UTC · model grok-4.3

classification 🧮 math.AT
keywords zigzag persistencebinary videoconnected componentstopological data analysisbarcode computationreal-time processingH0 homologyH1 homology
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The pith

H0 and H1 barcodes for binary video zigzag persistence are extracted directly from connected-component dynamics encoded in a graph.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the H0 and H1 barcodes in zigzag persistence for binary videos can be obtained directly from the dynamics of connected components rather than from full cubical complexes. Encoding these dynamics in a graph allows use of a near-linear time barcode algorithm, with total runtime scaling linearly in the number of pixels and parallelizable across frames. This makes high-resolution video analysis feasible in real time on consumer hardware. A sympathetic reader cares because it removes the computational bottleneck that previously made topological feature tracking in video impractical at scale.

Core claim

The H0 and H1 barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, cubical complexes are bypassed entirely, and the near-linear time barcode decomposition algorithm by Dey and Hou can be leveraged, leading to significant speedups and real-time performance on 4K video.

What carries the argument

Graph encoding of connected-component dynamics, which carries the zigzag persistence information for H0 and H1.

If this is right

  • Runtime is dominated by graph construction which scales linearly with pixel count.
  • Processing is embarrassingly parallel across frames.
  • Real-time zigzag persistence becomes possible on 4K video on consumer hardware.
  • Significant speedups are achieved compared to methods using cubical complexes and Gaussian elimination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This graph encoding might extend to higher-dimensional or non-binary spatio-temporal data if the information preservation holds.
  • The linear scaling opens the door to processing longer sequences or higher frame rates without proportional cost increases.
  • Integration with existing video pipelines could make topological invariants a standard feature in real-time analysis tools.

Load-bearing premise

That the connected-component dynamics encoded as a graph preserve exactly the same H0 and H1 zigzag persistence information as the full cubical filtration.

What would settle it

Running both the graph method and a standard cubical complex method on the same small binary video sequence and finding differing barcodes would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.30490 by David Lanners.

Figure 1
Figure 1. Figure 1: Pipeline overview. Intermediate frames are generated by taking the union of white regions between original video frames. Connected components in each frame form the vertices of a formigram, while directed edges represent their evolution through intermediate frames. Finally, we compute the corresponding zigzag barcode in near-linear time. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cubical complex constructions induced by binary image I: The T-construction on the foreground TI (1) ⊆ K realises 8-connectivity, while the V -construction VI (1) ⊆ K∗ realises 4-connectivity. Note that TI (1) is homotopy equivalent to a circle, whereas VI (1) consists of four discrete points. Both the T-construction and the V -construction will serve as models for the topology of the foreground and backgr… view at source ↗
Figure 3
Figure 3. Figure 3: Dual topology pairings induced by a binary image I. Subfigures (b) and (c) correspond to the complementary connectivity choices (κI , κ¯I ) = (8, 4) and (4, 8), respectively. In each case, the foreground and background are shown via the geometric realisations of the corresponding subcomplexes of K and K∗ , yielding distinct but consistent topologies. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pointwise logic operations for binary images. The arrows represent inclusions of the foregrounds. Lemma 2.28 shows that unions and intersections of consecutive coloured regions are realised via interpolation frames. This leads us to the following definition. Definition 2.30. Let V = (Ii) n i=1 be a binary video. The insertion of interpolation frames of the form Ii ∨ Ii+1 defines the topological diagrams F … view at source ↗
Figure 5
Figure 5. Figure 5: The curse of resolution: Transition between two frames F1 and F2 of resolution 1000 × 1000 via their union. Pixels in F2 \ F1 (cyan) are inserted going from F1 to F1 ∪ F2, while pixels in F1 \ F2 (red) are deleted going from F1 ∪ F2 to F2. Due to the high resolution, even a small displacement leads to a large symmetric difference; in this example, a total of |F1∆F2| ≈ 4 · 104 pixels change, each inducing c… view at source ↗
Figure 6
Figure 6. Figure 6: Merge graphs of partial formigrams S1, S2 : Pτ → setpar. Note that the edge sets of S1 arise as graphs of total functions, hence S1 is in fact a formigram, not just a partial formigram. For a topological diagram X, the merge graph G(π0(X)) thus encodes the evolution of connected components across X in a graph. Before turning to their efficient construction in the setting of binary videos, we introduce the … view at source ↗
Figure 7
Figure 7. Figure 7: Merge Graph M: Nodes with two colours represent dual-type states. Arrows are consistently oriented with the zigzag type τ , and each vertex is the tail of at most one edge in each direction. Next, we introduce some common constructions. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Disjoint union of two formigrams S1 and S2: G(S1 ⊔ S2) = G(S1) ⊔ G(S2). Moreover, they form the two connected components of this graph. Definition 3.18. Let M be a merge graph. A connected component of M is a connected component of its underlying undirected graph, obtained by forgetting the directions of the edges. Remark 3.19. Each connected component C of a merge graph M determines a subfunctor of F(M) b… view at source ↗
Figure 9
Figure 9. Figure 9: Linearisation k[M] of a Merge graph M. Note that an ordering on the fibres (top to bottom) was chosen to get the matrix representations. The linearisation behaves nicely with respect to disjoint unions. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Connected component labeling (CCL) of a binary image I. The two choices κ ∈ {4, 8} lead to distinct κ-connected components Cκ(I). By Lemma 2.28, these maps are induced by inclusions. Hence each connected component of Ii or Ii+1 is contained in a unique connected component of Ii ∨ Ii+1, so it suffices to track a single representative pixel per component. Let PIi = {p1, . . . , pmIi } and PIi+1 = {q1, . . .… view at source ↗
Figure 11
Figure 11. Figure 11: Merge graph obtained from a binary video V: Interpolation frames Ii ∨ Ii+1 = max(Ii , Ii+1) are inserted, inducing the topological diagram F ∪ V . Vertices correspond to connected components; edges encode their evolution under inclusion. 4. Interval Decomposition Algorithm Given a topological diagram X : Pτ → top, we turn to the problem of efficiently comput￾ing the barcode of its associated H0-zigzag mod… view at source ↗
Figure 12
Figure 12. Figure 12: Left: Merge graph G(π0(X)) with merging and splitting vertices highlighted, obstructing a barcode structure. Right: The goal: a barcode basis of the same zigzag module. Given an elementary formigram S, the algorithm of Dey and Hou constructs a barcode graph Mbar from G(S) such that k[S] ∼= k[Mbar] in near-linear time in the number of vertices. It proceeds by iteratively simplifying the merge graph G(S). W… view at source ↗
Figure 13
Figure 13. Figure 13: 4-Frontier Forest. The truncation M[1, 4] contains no merging vertices and decomposes into bars and rooted trees. To better understand the relation between the vertices of merge graphs, we introduce the following definition. From this point onward, we denote the discrete interval {i ∈ Z | a ≤ i ≤ b} by Ja, bK. Definition 4.6. Let M be a merge graph over τ . Let 1 ≤ tstart ≤ tend ≤ |τ |, a level path from … view at source ↗
Figure 14
Figure 14. Figure 14: H0- and H1-barcode computation. Runtime comparison with the cubical-complex pipeline on synthetic videos consisting of T = 10 frames. 5 runs are performed for each method, and the mean is plotted. The shaded area corresponds to the standard error. For each resolution, we perform five runs of both ImageZigzag and ZigVid, computing the H0- and H1-zigzag barcodes obtained from the union construction. The tim… view at source ↗
Figure 15
Figure 15. Figure 15: Runtime scaling with image resolution. A synthetic video of resolution 3000 × 3000 is downsampled to varying resolutions, and the H0- and H1-zigzag barcodes of the resulting 900-frame videos are computed. The plotted values show the mean over 5 runs. The barcode decomposition stage contributes negligibly to the total runtime and exhibits little dependence on image resolution. through parallelisation. Real… view at source ↗
Figure 16
Figure 16. Figure 16: Parallel scaling on 4K video. Runtime for computing the H0- and H1-zigzag barcodes of a 3840 × 2160, 900-frame video (30 seconds at 30 fps) as a function of thread count. Using 6 threads, the full barcode computation completes in approximately 30 seconds, demonstrating real-time zigzag persistence on 4K video. The dashed curve indicates ideal linear scaling relative to the single-thread runtime. A. Proof … view at source ↗
read the original abstract

Zigzag persistence tracks topological features in spatio-temporal data through combinatorial invariants called barcodes. For binary videos, existing methods are bottlenecked by the construction of prohibitively large cubical complexes and performing Gaussian elimination on large boundary matrices, rendering high-resolution videos out of reach. We show that the $H_0$ and $H_1$ barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, we bypass cubical complexes entirely and are able to leverage the near-linear time barcode decomposition algorithm by Dey and Hou, leading to significant speedups. The total runtime of our pipeline is dominated by the construction of the underlying graph structures, which scales linearly with pixel count and is embarrassingly parallel across frames, ensuring excellent scalability. We demonstrate how this approach enables zigzag persistence on 4k video at real-time rates on consumer hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that H0 and H1 zigzag persistence barcodes for binary video can be extracted directly from connected-component dynamics encoded as a graph. This bypasses construction of large cubical complexes on the space-time grid, allows use of the near-linear Dey-Hou barcode algorithm, and yields linear scaling in pixel count with frame-wise parallelism, enabling real-time 4K video processing.

Significance. If the graph encoding is shown to induce identical H0/H1 zigzag modules, the result would be a substantial practical advance in computational algebraic topology, removing the main bottleneck for spatio-temporal data and opening real-time topological video analysis. The explicit linear-time and parallel claims, together with the avoidance of Gaussian elimination on large boundary matrices, would be the primary strengths.

major comments (2)
  1. [Abstract] Abstract: the central claim that the component-dynamics graph preserves exactly the same H1 zigzag module as the cubical filtration is load-bearing yet unsupported by any proof sketch, boundary-map verification, or numerical check; H1 arises from 2-cycles in the cubical boundary operator, and adjacency recording of components alone does not automatically guarantee preservation of the relevant 1-cycles and their relations across frames.
  2. [Graph construction] Graph-construction section (inferred from abstract): no explicit statement or lemma shows that the inclusion maps or dual cycle information between consecutive frames are faithfully encoded, so it remains possible for the induced persistence diagrams to differ on H1 even when H0 matches.
minor comments (1)
  1. [Abstract] Abstract should cite the Dey-Hou reference explicitly rather than naming the authors only.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit justification of the H1 equivalence. We address the points below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the component-dynamics graph preserves exactly the same H1 zigzag module as the cubical filtration is load-bearing yet unsupported by any proof sketch, boundary-map verification, or numerical check; H1 arises from 2-cycles in the cubical boundary operator, and adjacency recording of components alone does not automatically guarantee preservation of the relevant 1-cycles and their relations across frames.

    Authors: We agree that the abstract states the central claim without a supporting lemma or verification. The graph construction records component adjacencies and their evolution in a manner intended to encode the cycle relations that appear as 1-cycles in the cubical setting. To address the concern directly, the revised manuscript will contain an explicit lemma establishing that the induced H1 zigzag module is identical, together with a short numerical check on a controlled example confirming that the barcodes match. revision: yes

  2. Referee: [Graph construction] Graph-construction section (inferred from abstract): no explicit statement or lemma shows that the inclusion maps or dual cycle information between consecutive frames are faithfully encoded, so it remains possible for the induced persistence diagrams to differ on H1 even when H0 matches.

    Authors: The current text does not contain a formal lemma on the preservation of inclusion maps or cycle information. The graph is built so that edges between component nodes across frames capture the spatial overlaps that determine when 1-cycles form or break; this is why H1 is claimed to be recovered. We will add a precise statement and short proof of this equivalence in the revision, making the encoding of the relevant maps explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; extraction method presented as independent computational reduction

full rationale

The abstract and description present a direct extraction of H0/H1 barcodes from connected-component dynamics encoded as a graph, bypassing cubical complexes and using an external algorithm (Dey and Hou). No equations, fitted parameters, self-citations, or ansatzes appear that would make any claimed result equivalent to its inputs by construction. The equivalence to cubical zigzag is asserted as a derived property of the encoding rather than defined circularly; the derivation chain remains self-contained with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

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Reference graph

Works this paper leans on

61 extracted references · 50 canonical work pages · 2 internal anchors

  1. [1]

    Computing Zigzag Persistence on Graphs in Near- Linear Time

    Tamal K. Dey and Tao Hou. “Computing Zigzag Persistence on Graphs in Near- Linear Time”. In:37th International Symposium on Computational Geometry. Vol. 189. LIPIcs. Leibniz Int. Proc. Inform. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2021, Art. No. 30, 15.isbn: 978-3-95977-184-9

    2021

  2. [2]

    Persistent Homological Cell Tracking Technology

    Haruhisa Oda, Kazuo Tonami, Yoichi Nakata, Naoko Takubo, and Hiroki Kurihara. “Persistent Homological Cell Tracking Technology”. In:Scientific Reports13.1 (July 2023), p. 10882.issn: 2045-2322.doi:10.1038/s41598-023-37760-3. (Visited on 12/02/2025)

    work page doi:10.1038/s41598-023-37760-3 2023

  3. [3]

    Fast Topological Signal Identification and Persistent Cohomological Cycle Matching

    In´ es Garc´ ıa-Redondo, Anthea Monod, and Anna Song. “Fast Topological Signal Identification and Persistent Cohomological Cycle Matching”. In:Journal of Ap- plied and Computational Topology8.3 (Sept. 2024), pp. 695–726.issn: 2367-1734. doi:10.1007/s41468-024-00179-4

    work page doi:10.1007/s41468-024-00179-4 2024

  4. [4]

    Simplicial and Topological Descriptions of Human Brain Dynamics

    Jacob Billings, Manish Saggar, Jaroslav Hlinka, Shella Keilholz, and Giovanni Petri. “Simplicial and Topological Descriptions of Human Brain Dynamics”. In:Network Neuroscience5.2 (June 2021), pp. 549–568.issn: 2472-1751.doi:10.1162/netn_ a_00190

    work page doi:10.1162/netn_ 2021

  5. [5]

    Generalized Morse Theory of Dis- tance Functions to Surfaces for Persistent Homology

    Anna Song, Ka Man Yim, and Anthea Monod. “Generalized Morse Theory of Dis- tance Functions to Surfaces for Persistent Homology”. In:Advances in Applied Mathematics166 (May 2025), p. 102857.issn: 0196-8858.doi:10.1016/j.aam. 2025.102857. (Visited on 12/02/2025)

    work page doi:10.1016/j.aam 2025

  6. [6]

    Anton Fran¸ cois and Rapha¨ el Tinarrage.Train-Free Segmentation in MRI with Cu- bical Persistent Homology. Oct. 2025.doi:10.48550/arXiv.2401.01160. arXiv: 2401.01160 [eess]. (Visited on 12/02/2025)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2401.01160 2025

  7. [7]

    A Homological Approach to a Mathematical Definition of Pulmonary Fibrosis and Emphysema on Computed Tomography

    Naoya Tanabe, Shizuo Kaji, Susumu Sato, Tomoo Yokoyama, Tsuyoshi Oguma, Kiminobu Tanizawa, Tomohiro Handa, Takashi Sakajo, and Toyohiro Hirai. “A Homological Approach to a Mathematical Definition of Pulmonary Fibrosis and Emphysema on Computed Tomography”. In:Journal of Applied Physiology131.2 (2021), pp. 601–612.doi:10.1152/japplphysiol.00150.2021

    work page doi:10.1152/japplphysiol.00150.2021 2021

  8. [8]

    A Semi- Automatic Method for Extracting Mitochondrial Cristae Characteristics from 3D Focused Ion Beam Scanning Electron Microscopy Data

    Chenhao Wang, Leif Østergaard, Stine Hasselholt, and Jon Sporring. “A Semi- Automatic Method for Extracting Mitochondrial Cristae Characteristics from 3D Focused Ion Beam Scanning Electron Microscopy Data”. In:Communications Biol- ogy7.1 (Mar. 2024), p. 377.issn: 2399-3642.doi:10.1038/s42003-024-06045-4

    work page doi:10.1038/s42003-024-06045-4 2024

  9. [9]

    A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems

    Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch, and Joshua R. Tempel- man. “A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems”. In:Nonlinear Dynamics112.6 (Mar. 2024), pp. 4687–4703.issn: 1573-269X.doi:10.1007/s11071-024-09289-1. (Visited on 11/26/2025)

    work page doi:10.1007/s11071-024-09289-1 2024

  10. [10]

    Applications of Persistent Homology to Time Varying Systems

    Elizabeth Munch. “Applications of Persistent Homology to Time Varying Systems”. PhD thesis. 2013. HDL:10161/7180. (Visited on 11/26/2025)

    2013

  11. [11]

    Computing and Visualizing Time-Varying Merge Trees for High-Dimensional Data

    Patrick Oesterling, Christian Heine, Gunther H. Weber, Dmitriy Morozov, and Gerik Scheuermann. “Computing and Visualizing Time-Varying Merge Trees for High-Dimensional Data”. In:Topological Methods in Data Analysis and Visual- ization IV. Ed. by Hamish Carr, Christoph Garth, and Tino Weinkauf. Cham: Springer International Publishing, 2017, pp. 87–101.isbn...

    work page doi:10.1007/978-3-319-44684-4_5 2017

  12. [12]

    Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis

    Jose A. Perea and John Harer. “Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis”. In:Foundations of Computational Math- ematics15.3 (June 2015), pp. 799–838.issn: 1615-3383.doi:10.1007/s10208- 014-9206-z. (Visited on 11/26/2025)

    work page doi:10.1007/s10208- 2015

  13. [14]

    Time-Varying Reeb Graphs for Continuous Space–Time Data

    Herbert Edelsbrunner, John Harer, Ajith Mascarenhas, Valerio Pascucci, and Jack Snoeyink. “Time-Varying Reeb Graphs for Continuous Space–Time Data”. In: Computational Geometry41.3 (Nov. 2008), pp. 149–166.issn: 0925-7721.doi: 10.1016/j.comgeo.2007.11.001. (Visited on 11/26/2025)

    work page doi:10.1016/j.comgeo.2007.11.001 2008

  14. [15]

    Vines and Vineyards by Updating Persistence in Linear Time

    David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. “Vines and Vineyards by Updating Persistence in Linear Time”. In:Proceedings of the Twenty- Second Annual Symposium on Computational Geometry. SCG ’06. New York, NY, USA: Association for Computing Machinery, June 2006, pp. 119–126.isbn: 978-1- 59593-340-9.doi:10.1145/1137856.1137877. (Visited...

    work page doi:10.1145/1137856.1137877 2006

  15. [16]

    Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology

    Mustafa Hajij, Bei Wang, Carlos Scheidegger, and Paul Rosen. “Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology”. In:2018 IEEE Pacific Visualization Symposium (PacificVis)(Apr. 2018), pp. 125–134.doi: 10.1109/PacificVis.2018.00024. (Visited on 11/26/2025)

    work page doi:10.1109/pacificvis.2018.00024 2018

  16. [17]

    Topological Persistence and Simplifica- tion

    Edelsbrunner, Letscher, and Zomorodian. “Topological Persistence and Simplifica- tion”. In:Discrete & Computational Geometry28.4 (Nov. 2002), pp. 511–533.issn: 1432-0444.doi:10.1007/s00454-002-2885-2. (Visited on 11/28/2025)

    work page doi:10.1007/s00454-002-2885-2 2002

  17. [18]

    A Roadmap for the Computation of Persistent Homology

    Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington. “A Roadmap for the Computation of Persistent Homology”. In:EPJ Data Science6.1 (Aug. 2017), p. 17.issn: 2193-1127.doi:10.1140/epjds/s13688- 017-0109-5. (Visited on 11/28/2025)

    work page doi:10.1140/epjds/s13688- 2017

  18. [19]

    Computing Persistent Homology

    Afra Zomorodian and Gunnar Carlsson. “Computing Persistent Homology”. In: Discrete & Computational Geometry33.2 (Feb. 2005), pp. 249–274.issn: 1432- 0444.doi:10.1007/s00454-004-1146-y. (Visited on 11/28/2025)

    work page doi:10.1007/s00454-004-1146-y 2005

  19. [20]

    Zigzag Persistence

    Gunnar Carlsson and Vin de Silva. “Zigzag Persistence”. In:Foundations of Com- putational Mathematics10.4 (Aug. 2010), pp. 367–405.issn: 1615-3383.doi:10. 1007/s10208-010-9066-0. (Visited on 11/26/2025)

    2010

  20. [21]

    Zigzag Persistence for Coral Reef Resilience Using a Stochastic Spatial Model

    R. A. McDonald, R. Neuhausler, M. Robinson, L. G. Larsen, H. A. Harrington, and M. Bruna. “Zigzag Persistence for Coral Reef Resilience Using a Stochastic Spatial Model”. In:Journal of The Royal Society Interface20.205 (Aug. 2023), p. 20230280.doi:10.1098/rsif.2023.0280. (Visited on 11/27/2025)

    work page doi:10.1098/rsif.2023.0280 2023

  21. [22]

    Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

    Woojin Kim, Facundo M´ emoli, and Zane Smith. “Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence”. In:Topological Data Analysis. Ed. by Nils A. Baas, Gunnar E. Carlsson, Gereon Quick, Markus Szymik, and Marius Thaule. Cham: Springer International Publishing, 2020, pp. 371–389.isbn: 978-3- 030-43408-3.doi:10.1007/978-3-030-43408-3_14

    work page doi:10.1007/978-3-030-43408-3_14 2020

  22. [23]

    Tempo- ral Network Analysis Using Zigzag Persistence

    Audun Myers, David Mu˜ noz, Firas A. Khasawneh, and Elizabeth Munch. “Tempo- ral Network Analysis Using Zigzag Persistence”. In:EPJ Data Science12.1 (Mar. 2023), p. 6.issn: 2193-1127.doi:10.1140/epjds/s13688-023-00379-5. (Visited on 11/28/2025). 49

    work page doi:10.1140/epjds/s13688-023-00379-5 2023

  23. [24]

    Topological Analysis of Temporal Hypergraphs

    Audun Myers, Cliff Joslyn, Bill Kay, Emilie Purvine, Gregory Roek, and Made- lyn Shapiro. “Topological Analysis of Temporal Hypergraphs”. In:Algorithms and Models for the Web Graph: 18th International Workshop, WAW 2023, Toronto, ON, Canada, May 23–26, 2023, Proceedings. Berlin, Heidelberg: Springer-Verlag, May 2023, pp. 127–146.isbn: 978-3-031-32295-2.do...

    work page doi:10.1007/978-3-031-32296-9_9 2023

  24. [25]

    Tracking Dy- namical Features via Continuation and Persistence

    Tamal K. Dey, Micha l Lipi´ nski, Marian Mrozek, and Ryan Slechta. “Tracking Dy- namical Features via Continuation and Persistence”. In:38th International Sympo- sium on Computational Geometry (SoCG 2022). Ed. by Xavier Goaoc and Michael Kerber. Vol. 224. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl – Leib...

    work page doi:10.4230/lipics.socg.2022.35 2022

  25. [26]

    Bement.Tracking Temporal Evolution of Topological Features in Image Data

    Susan Glenn, Jessi Cisewski-Kehe, Jun Zhu, and William M. Bement.Tracking Temporal Evolution of Topological Features in Image Data. Aug. 2025.doi:10 . 48550/arXiv.2508.17530. arXiv:2508.17530 [stat]. (Visited on 11/27/2025)

    work page arXiv 2025

  26. [27]

    Topological Classification of Tumour-Immune Interactions and Dynamics

    Jingjie Yang, Heidi Fang, Jagdeep Dhesi, Iris H. R. Yoon, Joshua A. Bull, Helen M. Byrne, Heather A. Harrington, and Gillian Grindstaff. “Topological Classification of Tumour-Immune Interactions and Dynamics”. In:Journal of Mathematical Biology 91.3 (Aug. 2025), p. 25.issn: 1432-1416.doi:10 . 1007 / s00285 - 025 - 02253 - 6. (Visited on 11/26/2025)

    2025

  27. [29]

    Zigzag Persistent Ho- mology in Matrix Multiplication Time

    Nikola Milosavljevi´ c, Dmitriy Morozov, and Primoz Skraba. “Zigzag Persistent Ho- mology in Matrix Multiplication Time”. In:Proceedings of the Twenty-Seventh An- nual Symposium on Computational Geometry. SoCG ’11. New York, NY, USA: Association for Computing Machinery, June 2011, pp. 216–225.isbn: 978-1-4503- 0682-9.doi:10.1145/1998196.1998229. (Visited ...

    work page doi:10.1145/1998196.1998229 2011

  28. [30]

    Ehrhart Polynomials of Matroid Polytopes and Poly- matroids

    Cl´ ement Maria and Hannah Schreiber. “Discrete Morse Theory for Computing Zigzag Persistence”. In:Discrete & Computational Geometry71.2 (Mar. 2024), pp. 708–737.issn: 1432-0444.doi:10.1007/s00454- 023- 00594- x. (Visited on 11/27/2025)

    work page doi:10.1007/s00454- 2024

  29. [31]

    Fast Computation of Zigzag Persistence

    Tamal K. Dey and Tao Hou. “Fast Computation of Zigzag Persistence”. In:30th Annual European Symposium on Algorithms (ESA 2022). Ed. by Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman. Vol. 244. Leibniz Interna- tional Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Informatik, 2022, 43:1–...

    work page doi:10.4230/lipics.esa.2022.43 2022

  30. [32]

    Woojin Kim and Facundo M´ emoli.Extracting Persistent Clusters in Dynamic Data via M¨ obius Inversion. 2022. arXiv:1712.04064 [math.AT]

    work page arXiv 2022

  31. [33]

    Discovering block-structured process models from event logs – A constructive approach,

    Bea Bleile, Ad´ elie Garin, Teresa Heiss, Kelly Maggs, and Vanessa Robins. “The Persistent Homology of Dual Digital Image Constructions”. In:Research in Com- putational Topology 2. Vol. 30. Assoc. Women Math. Ser. Springer, Cham, [2022] ©2022, pp. 1–26.isbn: 978-3-030-95518-2 978-3-030-95519-9.doi:10.1007/978- 3-030-95519-9\_1. 50

    work page doi:10.1007/978- 2022

  32. [34]

    Cotorsion Torsion Triples and the Representation Theory of Filtered Hierarchical Clustering

    Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen. “Cotorsion Torsion Triples and the Representation Theory of Filtered Hierarchical Clustering”. In:Advances in Mathematics369 (2020), p. 107171.issn: 0001-8708.doi:10.1016/ j.aim.2020.107171

    work page arXiv 2020

  33. [35]

    https://arxiv.org/abs/2501.09132v3

    Ulrich Bauer, Magnus Bakke Botnan, Steffen Oppermann, and Johan Steen.On the Additive Image of 0th Persistent Homology. https://arxiv.org/abs/2501.09132v3. Jan. 2025. (Visited on 02/09/2026)

    work page arXiv 2025

  34. [36]

    https://arxiv.org/abs/2411.19319v1

    Riju Bindua, Thomas Br¨ ustle, and Luis Scoccola.Decomposing Zero-Dimensional Persistent Homology over Rooted Tree Quivers. https://arxiv.org/abs/2411.19319v1. Nov. 2024. (Visited on 02/09/2026)

    work page arXiv 2024

  35. [37]

    https://arxiv.org/abs/2008.11532v1

    Jacek Brodzki, Matthew Burfitt, and Mariam Pirashvili.On the Complexity of Zero- Dimensional Multiparameter Persistence. https://arxiv.org/abs/2008.11532v1. Aug

    work page arXiv 2008

  36. [38]

    (Visited on 02/09/2026)

    2026

  37. [39]

    2026.url:https://github.com/Landa233/zigvid(vis- ited on 06/29/2026)

    David Lanners.ZigVid. 2026.url:https://github.com/Landa233/zigvid(vis- ited on 06/29/2026)

    2026

  38. [40]

    Finite Topology as Applied to Image Analysis

    V.A Kovalevsky. “Finite Topology as Applied to Image Analysis”. In:Computer Vision, Graphics, and Image Processing46.2 (1989), pp. 141–161.issn: 0734-189X. doi:10.1016/0734-189X(89)90165-5

    work page doi:10.1016/0734-189x(89)90165-5 1989

  39. [41]

    Digital Topology: Introduction and Survey

    T.Y Kong and A Rosenfeld. “Digital Topology: Introduction and Survey”. In:Com- puter Vision, Graphics, and Image Processing48.3 (1989), pp. 357–393.issn: 0734- 189X.doi:10.1016/0734-189X(89)90147-3

    work page doi:10.1016/0734-189x(89)90147-3 1989

  40. [42]

    Theory and Algo- rithms for Constructing Discrete Morse Complexes from Grayscale Digital Images

    Vanessa Robins, Peter John Wood, and Adrian P. Sheppard. “Theory and Algo- rithms for Constructing Discrete Morse Complexes from Grayscale Digital Images”. In:IEEE Transactions on Pattern Analysis and Machine Intelligence33.8 (2011), pp. 1646–1658.doi:10.1109/TPAMI.2011.95

    work page doi:10.1109/tpami.2011.95 2011

  41. [43]

    May 2020.doi:10.48550/arXiv.2005.04597

    Ad´ elie Garin, Teresa Heiss, Kelly Maggs, Bea Bleile, and Vanessa Robins.Duality in Persistent Homology of Images. May 2020.doi:10.48550/arXiv.2005.04597. arXiv:2005.04597 [math]. (Visited on 03/10/2026)

    work page doi:10.48550/arxiv.2005.04597 2020

  42. [44]

    Unzerlegbare Darstellungen I

    Peter Gabriel. “Unzerlegbare Darstellungen I”. In:manuscripta mathematica6.1 (Mar. 1972), pp. 71–103.issn: 1432-1785.doi:10.1007/BF01298413

    work page doi:10.1007/bf01298413 1972

  43. [45]

    Oudot.Persistence Theory: From Quiver Representations to Data Analy- sis

    Steve Y. Oudot.Persistence Theory: From Quiver Representations to Data Analy- sis. Mathematical Surveys and Monographs 209. American Mathematical Society, 2015

    2015

  44. [46]

    Parametrized Homology via Zigzag Persistence

    Gunnar Carlsson, Vin de Silva, Sara Kaliˇ snik, and Dmitriy Morozov. “Parametrized Homology via Zigzag Persistence”. In:Algebraic and Geometric Topology19.2 (2019), pp. 657–700.issn: 1472-2747,1472-2739.doi:10.2140/agt.2019.19.657

    work page doi:10.2140/agt.2019.19.657 2019

  45. [47]

    The structure and stability of persistence modules

    Fr´ ed´ eric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. “The Structure and Stability of Persistence Modules”. In:ArXivabs/1207.3674 (2012)

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  46. [48]

    Zigzag Persistent Ho- mology and Real-Valued Functions

    Gunnar E. Carlsson, Vin de Silva, and Dmitriy Morozov. “Zigzag Persistent Ho- mology and Real-Valued Functions”. In:Scg ’09. 2009

    2009

  47. [49]

    Cubical Ripser: Software for Com- puting Persistent Homology of Image and Volume Data

    Shizuo Kaji, Takeki Sudo, and Kazushi Ahara. “Cubical Ripser: Software for Com- puting Persistent Homology of Image and Volume Data”. In:ArXivabs/2005.12692 (2020)

    work page arXiv 2005

  48. [50]

    Ripser: Efficient Computation of Vietoris-Rips Persistence Bar- codes

    Ulrich Bauer. “Ripser: Efficient Computation of Vietoris-Rips Persistence Bar- codes”. In:Journal of Applied and Computational Topology5.3 (2021), pp. 391– 423.issn: 2367-1726.doi:10.1007/s41468-021-00071-5. 51

    work page doi:10.1007/s41468-021-00071-5 2021

  49. [52]

    Generalized Persistence Diagrams for Persis- tence Modules over Posets

    Woojin Kim and Facundo M´ emoli. “Generalized Persistence Diagrams for Persis- tence Modules over Posets”. In:Journal of Applied and Computational Topology5.4 (Dec. 2021), pp. 533–581.issn: 2367-1734.doi:10.1007/s41468-021-00075-1

    work page doi:10.1007/s41468-021-00075-1 2021

  50. [53]

    Field Choice Problem in Persistent Homol- ogy

    Ippei Obayashi and Michio Yoshiwaki. “Field Choice Problem in Persistent Homol- ogy”. In:Discrete & Computational Geometry70.3 (Oct. 2023), pp. 645–670.issn: 1432-0444.doi:10.1007/s00454-023-00544-7. (Visited on 04/24/2026)

    work page doi:10.1007/s00454-023-00544-7 2023

  51. [54]

    Sequential Operations in Digital Picture Processing

    Azriel Rosenfeld and John L. Pfaltz. “Sequential Operations in Digital Picture Processing”. In:Journal of The Acm13.4 (Oct. 1966), pp. 471–494.issn: 0004- 5411.doi:10.1145/321356.321357

    work page doi:10.1145/321356.321357 1966

  52. [55]

    Cassola, M

    Lifeng He, Xiwei Ren, Qihang Gao, Xiao Zhao, Bin Yao, and Yuyan Chao. “The Connected-Component Labeling Problem: A Review of State-of-the-Art Algorithms”. In:Pattern Recognition70 (2017), pp. 25–43.issn: 0031-3203.doi:10.1016/j. patcog.2017.04.018

    work page doi:10.1016/j 2017

  53. [56]

    Data Structures for Mergeable Trees

    Loukas Georgiadis, Haim Kaplan, Nira Shafrir, Robert Tarjan, and Renato Wer- neck. “Data Structures for Mergeable Trees”. In:ACM Transactions on Algorithms 7 (Mar. 2011), p. 14.doi:10.1145/1921659.1921660

    work page doi:10.1145/1921659.1921660 2011

  54. [57]

    Zigzag Zoology: Rips Zigzags for Homol- ogy Inference

    Steve Y. Oudot and Donald R. Sheehy. “Zigzag Zoology: Rips Zigzags for Homol- ogy Inference”. In:Proceedings of the Twenty-Ninth Annual Symposium on Com- putational Geometry. SoCG ’13. New York, NY, USA: Association for Computing Machinery, 2013, pp. 387–396.isbn: 978-1-4503-2031-3.doi:10.1145/2462356. 2462371

    work page doi:10.1145/2462356 2013

  55. [58]

    Munkres.Elements of Algebraic Topology

    James R. Munkres.Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984, pp. ix+454.isbn: 0-201-04586-9

    1984

  56. [59]

    Computing Height Persistence and Homology Generators in\mathbb{R}ˆ3\ Efficiently

    Tamal K. Dey. “Computing Height Persistence and Homology Generators in\mathbb{R}ˆ3\ Efficiently”. In:arXiv: Computational Geometry(2018)

    2018

  57. [60]

    Herbert Edelsbrunner and John Harer.Computational Topology: An Introduction. Jan. 2010.isbn: 978-0-8218-4925-5.doi:10.1007/978-3-540-33259-6_7

    work page doi:10.1007/978-3-540-33259-6_7 2010

  58. [61]

    The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs

    David Hartvigsen and Russell Mardon. “The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs”. In:Siam Journal On Discrete Mathematics7 (May 1994), pp. 403–418.doi:10.1137/S0895480190177042

    work page doi:10.1137/s0895480190177042 1994

  59. [62]

    Fabian Lenzen and Leon Renkin.Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology. Dec. 2025.doi:10.48550/ arXiv.2512.09668. arXiv:2512.09668 [math.AT]. (Visited on 06/29/2026)

    work page arXiv 2025

  60. [63]

    Paetzold, and Ulrich Bauer.Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology

    Nico Stucki, Vincent B¨ urgin, Johannes C. Paetzold, and Ulrich Bauer.Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology. July 2024.doi:10.48550/arXiv.2407.04683. arXiv:2407.04683 [math]. (Visited on 12/02/2025)

    work page doi:10.48550/arxiv.2407.04683 2024

  61. [64]

    2024.url:https://github.com/mrzv/dionysus (visited on 06/29/2026)

    Dmitriy Morozov.Dionysus 2. 2024.url:https://github.com/mrzv/dionysus (visited on 06/29/2026). 52

    2024