Pith. sign in

REVIEW 4 major objections 8 minor 43 references

Tracking Temporal Evolution of Topological Features in Image Data

T0 review · 4 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Maximum Void method treats time as a spatial dimension in an image stack, uses a permutation test to find one statistically significant cavity, and tracks the evolution of the corresponding loops on a zigzag persistence diagram…

desk verdict A plausible new TDA pipeline for tracking image features over time, but the statistical threshold is built on an unproven null model and the simulations dodge the multi-cavity case that matters most. read the letter →

arxiv 2508.17530 v1 pith:DNN3W47V submitted 2025-08-24 stat.ME stat.AP

Susan Glenn , Jessi Cisewski-Kehe , Jun Zhu , William M Bement This is my paper · ORCID
classification stat.MEstat.AP MSC 62R4062M1062M40
keywords ImageProcessingTimeSeriesTopologicalDataAnalysisZigzagPersistencePersistentHomologyHypothesisTestingMaximumTest
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a method, the Maximum Void (MV) method, for detecting and tracking holes (loops or connected components) in a time series of grayscale images. The central idea is to stack the images so that time becomes an extra spatial dimension; a loop that persists and evolves over time then appears as a two-dimensional cavity (an H2 feature) in the stack. A permutation-based hypothesis test finds the most persistent such cavity, and its birth time supplies a data-driven threshold that isolates the relevant loops at each time point. These loops are then connected across time with zigzag persistence, producing a summary of how features appear, split, merge, and disappear. In simulations mimicking cell wound healing, the method detects the true loops 95% of the time, compared with 50% for the point-cloud Vietoris-Rips comparison method.

What carries the argument

The load-bearing mechanism is the Maximum Persistence Test run on the time-stacked array: each pixel intensity is permuted across the whole stack, the maximum persistence of the H_{M-1} features is recomputed, and the observed maximum is compared to this null distribution. The birth time of the single significant H2 cavity is used as the global threshold (Equation 19), and zigzag persistence over unions of consecutive upper-level-set simplicial complexes (Equation 21) links the H1 and H0 features into a time-evolving summary. The method is called the Maximum Void method because it uses this statistically significant higher-dimensional void, or cavity, to define which lower-dimensional loops belong to the feature of interest at every time point.

What would settle it

Run the Maximum Void method on a simulated image sequence containing two separate loops of equal intensity that both persist across the whole time series; if the method's threshold is the birth time of only one of the two cavities, the zigzag diagram should either drop the second loop or jump between loops. If the method still tracks both loops correctly, the single-dominant-cavity assumption is not necessary.

Watch

Extended reading notes

Core claim

The paper claims that topological features in a time series of grayscale images can be identified and tracked statistically without a user-chosen threshold, by treating time as an intrinsic dimension. Under the assumption that one cavity in the image stack is the most persistent and is well separated from background features, the birth time of that cavity is a valid threshold for the whole sequence. Applying the threshold at each time point yields simplicial complexes whose lower-dimensional homology (loops and connected components) is then linked across consecutive times via zigzag persistence; the resulting zigzag persistence diagram records the temporal evolution of each feature. The paper argues that this approach outperforms existing point-cloud-based tracking, and demonstrates it on simulated data and on two wounded-cell image sequences.

Load-bearing premise

The method assumes that in the stack of images there is one cavity that stands out from background noise more than any other, and that its pixel values at one time are closer to its own values at other times than to any other structure.

Editorial extensions

If this is right

  • Tracking loops without predefined thresholds becomes possible for any image sequence that has a dominant cavity in the space-time stack.
  • The zigzag persistence diagram provides a summary that distinguishes splitting, merging, appearance, and disappearance of loops, which persistence-only summaries cannot show.
  • The method can be used to quantify wound healing dynamics, comparing cells with different treatments by the number and persistence of loops making up the wound over time.
  • Extending the same stacking logic to higher-dimensional image sequences, such as three-dimensional volumes over time, would track H2 voids through H3 features.
  • The comparison with the point-cloud Vietoris-Rips method shows that intensity-based upper-level set filtrations are better suited than distance-based Vietoris-Rips filtrations for detecting small-area loops in images.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The method's reliance on one dominant H2 cavity suggests a testable diagnostic: if a time series yields two H2 features of comparable persistence, the zigzag summary will likely flip between them, and an automatic check could flag such ambiguity.
  • The permutation null assumes exchangeability of pixel intensities, which ignores spatial correlation; smoothing the images likely changes the test's operating characteristics in ways the paper does not quantify.
  • Because the threshold is the birth time of a single cavity, a natural extension would be a time-varying threshold derived from the same idea, which could handle features whose contrast changes over time.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 8 minor

Summary. The paper proposes the Maximum Void (MV) method for detecting and tracking topological features (mainly loops, H1) in a time series of grayscale images. The method stacks the image sequence into a spatiotemporal array, computes persistent homology of the upper-level set filtration of that array, and applies a permutation-based Maximum Persistence Test to identify the most persistent higher-dimensional feature (H2 for 2D images). The birth time of this feature is then used as a global threshold to binarize each time slice, and zigzag persistence is computed on the resulting simplicial complexes to connect H0 and H1 features across time. The method is evaluated on simulated image sequences designed to mimic wound healing and applied to two wounded-cell datasets (C3 and Control). The paper claims the MV method outperforms a point-cloud Vietoris-Rips approach in the simulations and provides a useful representation of the temporal evolution of wound loops.

Significance. If the identification step were fully valid, the paper would address a real gap: it offers a data-driven, statistically motivated threshold for TDA on image time series and uses space-time cavities to label feature trajectories across time, overcoming the feature-correspondence problem in persistence summaries. The algorithm is clearly described, the simulation setup is relevant to the application, and the wounded-cell example is a persuasive motivation. The paper also gives a concrete algorithmic pipeline (Algorithm 1) that could be built upon. However, the central claims rest on statistical and identification assumptions that are not adequately supported: the null distribution for birth/death times is stated without justification, the C3 application appears to violate the single-dominant-feature assumption, and the simulation comparison is not replicated. These issues affect the manuscript's main claims rather than its presentation.

major comments (4)
  1. [Section 5, Figure 7c; Assumptions 2-3, Eq. (19)] The C3 application is presented as a demonstration of the method, but the manuscript itself states in Section 5 that the C3 array has 'several persistent H2 features' on its persistence diagram, while the Control has only one prominent H2. The MV method's Step 3 uses the birth time of the single most persistent H2 feature as the global threshold (Eq. 19), which is justified only under Assumption 2 that the target gamma_1 dominates all background features in persistence at every time. The Maximum Persistence Test considers only the maximum-persistence feature and therefore cannot determine which of several significant H2 features corresponds to the wound. If the selected maximum is a background cavity, the threshold in Eq. (19) binarizes the wrong voxels and the zigzag diagram in Figure 9 tracks loops that are not the wound. Moreover, the simulation study in Section 4 constructs every scenario with exactly one cylindrical H2 feature, so it does not exercise the regime in which the C3 data fall. The authors should either restrict the method's scope to the single-dominant-feature setting or develop and validate a selection procedure that identifies gamma_1 when multiple persistent H2 features are present, including simulations with competing H2 features.
  2. [Section 3.2.1, Eq. (12)] Equation (12) asserts that under the null hypothesis the birth and death times of all homology features are i.i.d. draws from the pixel intensity distribution F(mu_0, sigma_0^2). This is not established and is generally false for upper-level set filtrations: birth and death values of persistent homology features are functionals of the order relations among pixel values and are not independent draws from the pixel distribution. The permutation test in Algorithm 1 is a reasonable test of spatial exchangeability conditional on the observed pixel values, but the null distribution it generates is not the distribution stated in Eq. (12). Because the abstract and Section 3 describe the threshold as statistically grounded, the authors should either replace Eq. (12) with a correct statement based on exchangeability and the permutation distribution, or prove the distributional claim under explicit assumptions on the noise and filtration.
  3. [Section 4, Figure 6c] The paper's claim that the MV method 'outperforms' the PCVR method, detecting true loops 95% of the time versus 50%, appears to rest on a single simulated realization for each of the four scenarios in Figure 6. No number of independent noise replicates is reported, no matching criterion between detected loops and true loops is defined, and the PCVR threshold used to binarize the images is not specified. A single realization cannot support a general claim of superiority. The authors should add replicated simulations, define the detection and tracking criteria precisely, and report standard errors or confidence intervals for the detection rates. They should also report the PCVR threshold value or state how it was chosen, so the comparison is not sensitive to an unfavorable default.
  4. [Section 3.3, Eq. (19) and Eq. (20)] Step 3 applies the single threshold theta_1^* = min f over G_1(t*) to every time slice. Assumption 3 only says that values of the target partition are closer to each other than to other partitions; it does not ensure that the same level set isolates G_1(t_o) at each time when the feature intensity varies systematically over time. The manuscript should state explicit conditions under which the birth time of the space-time H_{M-1} feature yields the correct partition at every time slice, or it should examine the sensitivity of the zigzag summary to the threshold choice. Without this, the binarized slices in Figure 8 may include different amounts of background at different times, and the resulting zigzag connections could reflect threshold artifacts rather than biological dynamics.
minor comments (8)
  1. [Abstract] The final sentence contains a grammatical error: 'Since, the wounded cell images exhibit nonlinear, dynamic, spatial, and temporal structures...' should read 'Since the wounded cell images exhibit...' without the comma after 'Since'.
  2. [Figure 4 and Section 3.2] The text near Figure 4 describes the smoothed images as t7 and t12, while the caption refers to t1 and t12; these should be reconciled.
  3. [Section 4, notation] The text uses A_sigma_1, ..., A_sigma_4 for the simulated arrays, while Figure 6b labels them A0_2, A0_3, A0_4; the notation should be unified.
  4. [Table 1] The p-value is reported as 0, but the number of permutations B is not stated anywhere; report the p-value as < 1/B and give B for both the simulation and the application.
  5. [Figures 2 and 9] The axes of the persistence diagrams are inconsistent: Figure 2d has birth on the y-axis and death on the x-axis, while the caption of Figure 9 states that birth is on the x-axis and death on the y-axis; make the conventions uniform and clearly labeled.
  6. [References] Reference [4] and reference [9] are the same Benink and Bement (2005) paper, and references [5] and [6] are the same Abreu-Blanco et al. (2014) paper; these duplicates should be consolidated.
  7. [Equation (19)] Equation (19) writes the partition as f^{-1}(theta), but f is not an invertible map; express the upper-level set directly in terms of pixel coordinates and intensity values.
  8. [Section 4, PCVR comparison] The PCVR comparison does not state the predetermined threshold used to binarize images; without this detail, readers cannot assess whether the comparison is fair.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation is self-contained: the threshold is estimated from data via a permutation test, and the simulations and cell application are evaluated against externally defined ground truth; the self-citations supply details and smoothing choices but are not load-bearing for the central claim.

full rationale

The central pipeline (Steps 1-4) estimates a data-driven threshold using the Maximum Persistence Test, whose test statistic and permutation null distribution are fully specified in the paper (Eqs. 11-18). The threshold is then applied to identify lower-dimensional features, and the output is validated against simulated image sequences with known true loops and qualitatively against the wounded-cell application. The claim to 'outperform a current approach' is an empirical comparison with an external benchmark (PCVR), not a consequence of fitting a parameter to the same data used for evaluation. The self-citations to [43] and [12] are used for implementation details, smoothing, and prior context; they do not supply an unverified uniqueness theorem or otherwise force the paper's conclusions. The skeptical concern about multiple persistent H2 features in the C3 cell is a real limitation of Assumptions 2 and 3, but it is explicitly stated as an assumption in the paper and is a correctness/robustness risk rather than a circular reduction of the derivation to its inputs. No equation is defined in terms of the result it is used to predict, and no fitted value is renamed as a prediction. Therefore no significant circularity is present.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claim rests on four explicit domain assumptions (Assumptions 1 through 5 in Section 3) plus standard triangulability; the method introduces no new physical or mathematical entities. There are no fitted model parameters in the usual sense, but the smoothing span, resolution reduction, and permutation count are user-chosen values that directly influence the detected features and are not audited.

free parameters (5)
  • Smoothing span = 0.1
    Local polynomial regression span of 0.1 is used for cell images (Section 2.2) and reduces the number of detected loops from 39,895 to 10; no sensitivity analysis is provided.
  • Smoothing degree = 2
    Local polynomial degree 2 is selected for image smoothing in Section 2.2; it affects which loops survive the filtration.
  • Image resolution reduction = unspecified
    Section 5 states resolution was reduced to make computations feasible, but the reduction factor is not given; it changes the persistence output.
  • Number of permutations B = unspecified
    Algorithm 1 takes B as input for the Maximum Persistence Test; no value is reported, so the p-values in Table 1 cannot be reproduced.
  • Significance level alpha = unspecified
    Algorithm 1 uses alpha to decide rejection; no explicit level is reported in the application, though Table 1 reports the 95th percentile of the null distribution.
assumptions (7)
  • domain assumption Assumption 1: each image partitions into k contiguous regions with constant function values mu_k.
    Used in Section 3.1, Eq. (10), to define partitions G_k(t_o) that represent topological features; real images satisfy this only after smoothing and approximation.
  • domain assumption Assumption 2: the target H_{M-1} feature has persistence higher than every background feature.
    Section 3.2; it justifies selecting the maximum-persistence H2 feature as the true temporal feature. If background features are more persistent, the method selects the wrong threshold.
  • domain assumption Assumption 3: the target partition's functional value at different times is closer than to any other partition.
    Section 3.2; needed so that a single global threshold from the H2 birth time identifies the same wound-related partition at every time.
  • domain assumption Assumption 4: under the null, the noise-free array is constant mu0, so no true H_m features exist for m>=1.
    Section 3.2.1; defines the null hypothesis for the Maximum Persistence Test.
  • domain assumption Assumption 5: under the null, noise is homoskedastic with distribution F(0,sigma0^2).
    Section 3.2.1; permits the permutation test to treat pixel values as exchangeable; violated by spatially varying or autocorrelated image noise.
  • standard math The underlying space is triangulable so simplicial complexes can approximate image homology.
    Section 2.2; standard assumption to apply simplicial zigzag persistence to image data.
  • standard math A stack of loops at middle times capped by connected components forms an H2 feature.
    Section 4 simulation design states this cylindrical structure is 'consistent with an H2 feature'; this geometric fact justifies Step 1 for 2D image time series.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tracking Temporal Evolution of Topological Features in Image Data." pith.science (2026). https://pith.science/paper/DNN3W47V

@misc{pith2026250817530,
  author       = {Pith},
  title        = {Pith review of: Tracking Temporal Evolution of Topological Features in Image Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNN3W47V}},
  note         = {Machine review of arXiv:2508.17530}
}
read the original abstract

Topological Data Analysis (TDA) can be used to detect and characterize holes in an image, such as zero-dimensional holes (connected components) or one-dimensional holes (loops). However, there is currently no widely accepted statistical framework for modeling spatiotemporal dependence in the evolution of topological features, such as holes, within a time series of images. We propose a hypothesis testing framework to identify statistically significant topological features of images in space and time, simultaneously. This addition of time may induce higher-dimensional topological features which can be used to establish temporal connections between the lower-dimensional features at each point in time. The temporal evolution of these lower-dimensional features is then represented on a zigzag persistence diagram, as a topological summary statistic focused on time dynamics. We demonstrate that the method effectively captures the emergence and progression of topological features in a study of a series of images of a wounded cell as it repairs. The proposed method outperforms a current approach in a simulation study that includes features of the wound healing process. Since, the wounded cell images exhibit nonlinear, dynamic, spatial, and temporal structures during single-cell repair, they provide a good application for this method.

Figures

Figures reproduced from arXiv: 2508.17530 by the authors.

Figure 1
Figure 1. Examples of cell wounds at different points of time in the healing process. Each row [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (a) Smoothed C3 cell at time 𝑡15, denoted as A˜C3 𝑡15 . (b) The upper-level set (A˜C3 𝑡15 ) [𝑏𝑗 ,∞) at the birth of the most persistent loop with the birth time 𝑏 𝑗 = 2007. (c) The upper-level set at the death of the most persistent loop (A˜C3 𝑡15 ) [𝑑𝑗 ,∞) with the death time 𝑑 𝑗 = 1404. (d) The persistence diagram of the upper-level set filtration of A˜C3 𝑡15 where the black dots represent connected components and… view at source ↗
Figure 3
Figure 3. Each row is a partition of a topological feature in an image set changing across time. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Pipeline for the Maximum Persistence Test for [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: (a) Slices of the partition G1 (𝑡2), G1 (𝑡3), G1 (𝑡4). The nesting structure of the homology of the wound throughout times 𝑡2, 𝑡3, 𝑡4 is unpredictable. (b) The top row of simplicial complexes are those representing the features at time points 𝑡2, 𝑡3, 𝑡4 and bottom row …
Figure 6
Figure 6. Figure 6: Simulation setup and results. (a) An example of an [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: (a) Time series of images of the C3 cell where each row is a sequence of ten consecutive [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: (a) Slices of the cell wound GˆC3 1 (𝑡11), GˆC3 1 (𝑡12), GˆC3 1 (𝑡13) where color is the intensity value of the pixels which are above the threshold 𝜃ˆC3 1 . (b) The black simplicial complexes are the simplicial complexes representing the wound at time points 𝑡11, 𝑡12,…
Figure 9
Figure 9. Figure 9: Zigzag diagrams for the array of the C3 cell (a) and the array of the Control cell. The [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 41 canonical work pages

  1. [12]

    Confidence regions for a persistence diagram of a single image with one or more loops

    GLENN , S., C ISEWSKI -K EHE , J., Z HU, J. AND BEMENT , W.M. (2024). Confi- dence regions for a persistence diagram of a single image with one or more loops. , arXiv:2405.01651

    work page Pith review arXiv 2024

  2. [1]

    L., O RTIGÃO -FARIAS , J

    JOYNER , A. L., O RTIGÃO -FARIAS , J. R., and KORNBERG , T. (2024). Conserved roles of engrailed: patterning tissues and specifying cell types. Development, 151(24), dev204250

    work page 2024

  3. [2]

    M., M ILLER , A

    BEMENT , W. M., M ILLER , A. L., and VON DASSOW , G. (2006). Rho GTPase activity zones and transient contractile arrays. Bioessays, 28(10), 983-993

    work page 2006

  4. [3]

    M., G ORYACHEV , A

    BEMENT , W. M., G ORYACHEV , A. B., M ILLER , A. L., and VON DASSOW , G. (2024). Patterning of the cell cortex by Rho GTPases. Nature Reviews Molecular Cell Biology, 25(4), 290-308

    work page 2024

  5. [4]

    A., and BEMENT , W

    BENINK , H. A., and BEMENT , W. M. (2005). Concentric zones of active RhoA and Cdc42 around single cell wounds. The Journal of cell biology, 168(3), 429-439. 22 Glenn et al

    work page 2005

  6. [6]

    T., V ERBOON , J

    ABREU -B LANCO , M. T., V ERBOON , J. M., and PARKHURST , S. M. (2014). Coordi- nation of Rho family GTPase activities to orchestrate cytoskeleton responses during cell wound repair. Current biology, 24(2), 144-155

    work page 2014

  7. [7]

    A., and BEMENT , W

    MANDATO , C. A., and BEMENT , W. M. (2001). Contraction and polymerization co- operate to assemble and close actomyosin rings around Xenopus oocyte wounds. The Journal of cell biology, 154(4), 785-798

    work page 2001

  8. [8]

    T., V ERBOON , J

    ABREU -B LANCO , M. T., V ERBOON , J. M., and PARKHURST , S. M. (2011). Cell wound repair in Drosophila occurs through three distinct phases of membrane and cytoskeletal remodeling. Journal of Cell Biology, 193(3), 455-464

    work page 2011

Show all 43 references
  1. [9]

    A., and BEMENT , W

    BENINK , H. A., and BEMENT , W. M. (2005). Concentric zones of active RhoA and Cdc42 around single cell wounds. The Journal of cell biology, 168(3), 429-439

    2005

  2. [10]

    E., Z OLA , J., S WIDER , Z

    MOE, A., H OLMES , W., G OLDING , A. E., Z OLA , J., S WIDER , Z. T., E DELSTEIN - KESHET , L., and BEMENT , W. (2021). Cross-talk-dependent cortical patterning of Rho GTPases during cell repair. Molecular biology of the cell, 32(16), 1417-1432

    2021

  3. [11]

    CRESSIE , N., and WIKLE , C. K. (2011). Statistics for spatio-temporal data. John Wiley and Sons

    2011

  4. [13]

    R., and CIOCANEL , M

    DAWSON , M., D UDLEY , C., O MOMA , S., T UNG , H. R., and CIOCANEL , M. V. (2023). Characterizing emerging features in cell dynamics using topological data analysis methods. Mathematical Biosciences and Engineering, 20(2), 3023-3046

    2023

  5. [14]

    A., T ILLMANN , U., G RINDROD , P., and HARRINGTON , H

    OTTER , N., P ORTER , M. A., T ILLMANN , U., G RINDROD , P., and HARRINGTON , H. A. (2017). A roadmap for the computation of persistent homology. . EPJ Data Science, 6, 1-38

    2017

  6. [15]

    UMEDA , Y. (2017). Time series classification via topological data analysis.Information and Media Technologies, 12, 228-239

    2017

  7. [16]

    LIM, B., S ON, S., K IM, H., N AH, S., and LEE, K. M. (2017). Proceedings of the IEEE conference on computer vision and pattern recognition workshops. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition Workshops (Vol. 9, p. 136)

    2017

  8. [17]

    A., and HARER , J

    PEREA , J. A., and HARER , J. (2015). Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of computational mathematics, 15, 799-838

    2015

  9. [18]

    J., and BERGER , M

    TRALIE , C. J., and BERGER , M. (2018). Topological eulerian synthesis of slow motion periodic videos. 2018 25th IEEE international conference on image processing (ICIP) (pp. 3573-3577). IEEE

    2018

  10. [19]

    TYMOCHKO , S., M UNCH , E., K HASAWNEH , F. A. (2020). Using zigzag persistent homology to detect Hopf bifurcations in dynamical systems. Algorithms, 13(11), 278

    2020

  11. [20]

    DAKURAH , S., and CISEWSKI -K EHE , J. (2024). A Subsequence Approach to Topolog- ical Data Analysis for Irregularly-Spaced Time Series. arXiv preprint arXiv:2410.13723. Temporal Evolution of Topological Features 23

    2024 arXiv

  12. [21]

    M., H ATHAWAY, Q

    SINGH , Y., F ARRELLY , C. M., H ATHAWAY, Q. A., L EINER , T., J AGTAP, J., C ARLS - SON , G. E., and ERICKSON , B. J. (2023). Topological data analysis in medical imaging: current state of the art. Insights into Imaging , 14(1), 58

    2023

  13. [22]

    M., Z IEGELMEIER , L., and HALVERSON , T

    TOPAZ , C. M., Z IEGELMEIER , L., and HALVERSON , T. (2015). Topological data analysis of biological aggregation models. PloS one, 10(5), e0126383

    2015

  14. [23]

    COHEN -S TEINER , D., E DELSBRUNNER , H., M OROZOV , D. (2006). Vines and vine- yards by updating persistence in linear time. In Proceedings of the twenty-second annual symposium on Computational geometry (pp. 119-126)

    2006

  15. [24]

    V., J UENEMANN , R., D AWES , A

    CIOCANEL , M. V., J UENEMANN , R., D AWES , A. T., and MCKINLEY , S. A. (2021). Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks. Bulletin of Mathematical Biology, 83, 1-25

    2021

  16. [25]

    CARLSSON , G., and DE SILVA, V. (2010). Zigzag persistence. Foundations of compu- tational mathematics, 10, 367-405

    2010

  17. [26]

    MATA, G., M ORALES , M., R OMERO , A., and RUBIO , J. (2015). Zigzag persistent homology for processing neuronal images. Pattern Recognition Letters, 62, 55-60

    2015

  18. [27]

    TAUSZ , A., and CARLSSON , G. (2011). Applications of zigzag persistence to topological data analysis. arXiv:1108.3545

    2011 arXiv

  19. [28]

    CARLSSON , G., D E SILVA, V., and MOROZOV , D. (2009). Zigzag persistent homol- ogy and real-valued functions. In Proceedings of the twenty-fifth annual symposium on Computational geometry (pp. 247-256)

    2009

  20. [29]

    T., L ECCI , F., R INALDO , A., W ASSERMAN , L., B ALAKRISHNAN , S., and SINGH , A

    FASY, B. T., L ECCI , F., R INALDO , A., W ASSERMAN , L., B ALAKRISHNAN , S., and SINGH , A. (2014). Confidence sets for persistence diagrams. The Annals of Statistics, 24, 2301-2339

    2014

  21. [30]

    DIVASÓN , J., R OMERO , A., S ANTOLARIA , P., and YÁNIZ , J. L. (2024). Zigzag persis- tence for image processing: New software and applications. Pattern Recognition Letters, 184, 111-118

    2024

  22. [31]

    TYMOCHKO , S., M UNCH , E., D UNION , J., C ORBOSIERO , K., and TORN , R. (2020). Using persistent homology to quantify a diurnal cycle in hurricanes. Pattern Recognition Letters, 133, 137-143

    2020

  23. [32]

    B., C HUNG , M

    EL-YAAGOUBI , A. B., C HUNG , M. K., O MBAO , H. (2023). Topological data analysis for multivariate time series data. Entropy, 25(11), 1509

    2023

  24. [33]

    A., T ILLMANN , U., G RINDROD , P., and HARRINGTON , H

    OTTER , N., P ORTER , M. A., T ILLMANN , U., G RINDROD , P., and HARRINGTON , H. A. (2017). A roadmap for the computation of persistent homology. EPJ Data Science, 6, 1-38

    2017

  25. [34]

    (2007, June)

    CARLSSON , G., and ZOMORODIAN , A. (2007, June). The theory of multidimensional persistence. In Proceedings of the twenty-third annual symposium on Computational geometry (pp. 184-193)

    2007

  26. [35]

    GHAHREMANI , Y., and METSIS , V. (2025). Time Series Embedding Methods for Clas- sification Tasks: A Review. arXiv preprint arXiv:2501.13392

    2025 arXiv

  27. [36]

    (2018, April)

    HAJIJ , M., W ANG , B., S CHEIDEGGER , C., and ROSEN , P. (2018, April). Visual de- tection of structural changes in time-varying graphs using persistent homology. In 2018 ieee pacific visualization symposium (pacificvis) (pp. 125-134). IEEE

    2018

  28. [37]

    KRAMÁR , M., L EVANGER , R., T ITHOF , J., S URI , B., X U, M., P AUL, M., and MIS- CHAIKOW , K. (2016). Analysis of Kolmogorov flow and Rayleigh–Bénard convection using persistent homology. Physica D: Nonlinear Phenomena, 334, 82-98. 24 Glenn et al

    2016

  29. [38]

    (2013).Applications of persistent homology to time varying systems (Doc- toral dissertation, Duke University)

    MUNCH , E. (2013).Applications of persistent homology to time varying systems (Doc- toral dissertation, Duke University)

    2013

  30. [39]

    D., K RINGELBACH , M

    EXPERT , P., L ORD , L. D., K RINGELBACH , M. L., P ETRI , G. (2019). Topological neuroscience. Network Neuroscience, 3(3), 653-655

    2019

  31. [40]

    BUKKURI , A., A NDOR , N., and DARCY, I. K. (2021). Applications of topological data analysis in oncology. Frontiers in artificial intelligence, 4, 659037

    2021

  32. [41]

    CARLSSON , G., and VEJDEMO -J OHANSSON , M. (2021). Topological data analysis with applications. Cambridge University Press

    2021

  33. [42]

    K., B UBENIK , P., and KIM, P

    CHUNG , M. K., B UBENIK , P., and KIM, P. T . (2009, July). Persistence diagrams of cortical surface data. In International Conference on Information Processing in Medical Imaging (pp. 386-397). Berlin, Heidelberg: Springer Berlin Heidelberg

    2009

  34. [43]

    GLENN , S. (2024). Inference on a Time Series of Images Using Topological Data Analysis. The University of Wisconsin-Madison

    2024

  35. [44]

    TRALIE , C. (2016). HIGH -DIMENSIONAL GEOMETRY OF SLIDING WINDOW EMBED - DINGS OF PERIODIC VIDEOS . In 32nd international symposium on computational geom- etry (socg 2016) (pp. 71-1). Schloss Dagstuhl–Leibniz-Zentrum für Informatik

    2016

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.