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REVIEW 3 major objections 6 minor 31 references

Topology across Scales on Heterogeneous Cell Data

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that plotting the death simplex back onto the tissue image, coloured by birth and weighted by persistence, visually and statistically separates healthy from diseased cell patterns.

desk verdict PWDS is a cheap, interpretable visualization and the persistence-image weightings are worth having, but the death-simplex localization claim is unproved and the biological claims are oversold; still, it deserves a serious referee. read the letter →

arxiv 2505.02717 v1 pith:55MN3HIU submitted 2025-05-05 q-bio.QM math.AT

Maria Torras-Pérez , Iris H.R. Yoon , Praveen Weeratunga , Ling-Pei Ho , Helen M. Byrne , Ulrike Tillmann , Heather A. Harrington This is my paper · ORCID
classification q-bio.QMmath.AT MSC 55N3162R40
keywords persistenthomologymultiplexedimagingspatialbiologypersistenceweighteddeathsimplicesimagesBetticurveslupusspleenCOVID-19lungs
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

Persistent homology summarises the multiscale shape of a cell cloud, but its output does not say where in the tissue a detected loop or void sits. This paper argues that the simplex which kills a feature—its death simplex—can be plotted back onto the original image, coloured by feature birth and weighted by persistence, to give a visualisation (PWDS) that locates biological structures and separates low-density from high-density features at no extra computational cost. The paper also tunes persistence-image weightings so that vectorised topological summaries emphasise either small-scale dense-tissue patterns or large-scale noisy loops. Applied to a lupus mouse spleen CODEX dataset and a COVID-19 human lung IMC dataset, the pipeline finds that 10 spleen cell types plus red and white pulp cells form spatial patterns that differ between healthy and diseased samples, and that endothelial-cell topology separates COVID-19 disease stages. If the claims hold, the payoff is an interpretable and computationally cheap route from multiplexed tissue images to multiscale spatial biomarkers.

What carries the argument

The death simplex is the simplex that kills a homological feature when it enters the filtration (for a void in degree 1, the last empty triangle that fills it in). It is unstable under small perturbations of the points, but the paper assumes it lies inside the feature, so it gives an approximate location. PWDS draws these simplices in the original coordinates and assigns colour by birth thresholds $b_{\mathrm{prox}}=\langle P_{90}\rangle$, $b_{\mathrm{dist}}=\langle P_{98}\rangle$ and intensity by persistence $d-b$. The other machinery is the persistence-image vectorisation: a persistence diagram is converted to birth–persistence coordinates, convolved with Gaussians weighted by $w_{\mathrm{flat}}$, $w_1=p$, $w_2=p^2$, $w_3=bp$, $w_4=bp^2$, and integrated on a grid; together with normalised Betti curves and percentile statistics these feed k-means clustering.

What would settle it

Take a synthetic loop with several nearly empty triangles of similar circumradius inside it and perturb the points slightly so the death simplex jumps between triangles; if the chosen triangle can land far from the feature's interior, the localisation claim fails. A direct check is to compute exact feature boundaries on synthetic loops and measure the Hausdorff distance between those boundaries and the PWDS triangles; large distances would falsify the claim that the triangles indicate feature positions.

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Extended reading notes

Core claim

The central discovery is that the death simplex—the simplex whose addition to the filtration destroys a persistent feature—can serve as a proxy for the feature's location in the original point cloud. By plotting these simplices on the tissue image, coloured red for features with small birth values (formed by proximal cells), blue for features with large birth values (formed by distal cells), and purple for intermediate births, with colour intensity proportional to persistence, PWDS turns a persistence diagram into a spatial map. The paper claims this reproduces the main advantage of multi-parameter persistent homology—distinguishing features at different densities—while staying within ordinary one-parameter persistent homology and hence at no additional computational cost. On the vectorisation side, the paper proposes persistence-image weightings $w_1=p$, $w_2=p^2$, $w_3=bp$, $w_4=bp^2$ alongside a flat weighting, so that summaries can emphasise large-scale structure or noisy loops; for dense tissues like the red pulp a stronger weight such as $w_2$ is needed before large-scale features become visible. The application claims are that PWDS and normalised Betti curves reveal red-pulp infiltration into white pulp in lupus and the loss of alveolar cavities with dense cellular infiltrate in COVID-19 lungs, and that topological descriptors separate healthy from diseased samples for 10 spleen cell types plus the two pulp groups, with endothelial cells giving the best separation of COVID-19 disease stages.

Load-bearing premise

The load-bearing premise is that the death simplex always lies inside the topological feature it kills, so the coloured triangle on the tissue image really marks where the loop or void is; the paper asserts this but does not prove it, and the death simplex itself is unstable under small data perturbations.

Editorial extensions

If this is right

  • PWDS gives a spatial readout of persistent homology for any 2D point cloud, so cell-type-specific tissue architecture can be inspected directly instead of only through summary diagrams.
  • The birth-threshold colouring distinguishes features formed by proximal versus distal cells, so infiltration or exclusion phenomena become visible as blue triangles appearing inside large red voids.
  • Weighted persistence images can be tuned to the scale of interest; the paper shows that for dense populations a strong weight like $p^2$ is needed to expose large-scale features that a flat or linear weight masks.
  • Normalised Betti curves detect changes in the density of dense cell regions even when overall cell density does not change significantly, as the paper demonstrates for CD4+ T cells in lupus spleen (t-test p-value ≈ 0.16).
  • The same pipeline applied to COVID-19 lungs identifies endothelial-cell topology as the descriptor that best separates alveolitis from later disease stages, and this separation is not simply density thresholding.

Reading between the lines

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

  • The paper does not test whether the PWDS red/blue counts exceed what a null model with randomly relabelled cell types would produce; such a permutation test would strengthen the claim that the observed infiltration signal is spatial rather than compositional.
  • Because death simplices are unstable, quantitative use of PWDS may need aggregation over many features or a smoothed version; the paper presents PWDS as a visualisation, so an aggregate PWDS statistic is a natural next step.
  • The birth-value thresholds are set by averaging the 90th and 98th percentiles across samples; an adaptive per-sample or scale-free threshold could remove this heuristic when PWDS is applied to new tissues.
  • The 'no additional computational cost' claim could be tested directly by running PWDS and a multi-parameter persistent-homology pipeline on the same dataset and comparing runtime and density-distinguishing power.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a pipeline for topological analysis of multiplexed imaging data. It introduces a visualisation, persistence-weighted death simplices (PWDS), which plots the simplex that kills each persistent homology feature, coloured by birth-value thresholds (proximal/distal) and weighted by persistence. It also studies persistence-image weightings and combines nine vectorisations with k-means clustering to compare healthy and diseased tissue in two datasets (lupus spleen CODEX and COVID-19 lung IMC). The authors report that 10 spleen cell types plus red/white pulp show different spatial patterns between health and disease, and that endothelial-cell descriptors separate COVID-19 disease stages.

Significance. If the central visualisation claim were established, PWDS would be a useful, interpretable and computationally cheap complement to persistence diagrams for spatial biology, localising multiscale features in the original tissue while avoiding the cost of multiparameter persistence. The paper is clearly written, uses standard persistent homology computations, and provides code and data at an accessible repository. The comparison of persistence-image weightings is systematic, and the authors are transparent about the instability of death simplices. However, the method's key assumption that the death simplex lies inside the detected feature is not proved, and the biological conclusions rest on in-sample screening of hundreds of descriptors without correction.

major comments (3)
  1. [Visualisation of PH] The assertion 'the death simplex will always correspond to a simplex within the interior of the topological feature' is load-bearing for the entire PWDS method and is stated without proof or reference. In an alpha filtration, a degree-1 feature is killed when a 2-simplex completes a 2-chain homologous to the feature; the geometric location of that simplex is not constrained to lie inside the void, and it can fail to do so for non-convex or non-simple cycles. For witness filtrations with random landmarks, the coupling between a landmark simplex and the Euclidean location of the feature is even weaker. Since the biological readings of Figures 5 and 7 (triangles as infiltrating cells or cavities) depend directly on this localisation property, the central methodological claim is currently unsupported. The authors should either prove the interior-location claim for the planar alpha case under suitable genericity/convexity assumptions, or replace the death-simplex visualisation with computed representative or optimal cycles, or validate PWDS on synthetic non-convex examples and state the failure modes.
  2. [Results: Topological vectorisations of 10 cell types distinguish lupus samples] The screening procedure selects descriptors that yield correct k-means clusterings from 383 descriptors for the spleen and 548 for the lung, with no correction for multiple comparisons and no hold-out validation. The phrase '10 cell types ... form significantly different spatial patterns' is therefore not supported as a significance claim: under a null model in which descriptors are independent of the disease label, the maximum silhouette or Rand score over hundreds of descriptors will be high by chance. The problem is compounded by the fact that bprox, bdist, p0, and the persistence-image variance and pixel size are estimated from the same samples. The authors should add a permutation null (e.g., shuffling sample labels), nested cross-validation, or an explicit false-discovery control, and should present the distribution of scores across all descriptors rather than only the selected ones.
  3. [Introduction] The introductory claim that PWDS 'replicat[es] the main advantage of MPH over PH at no additional computational cost' is not tested anywhere in the paper. Multiparameter persistent homology is never computed on these data or on a synthetic example, so the equivalence is unsupported. The phrase 'features in point clouds at different densities' is also not formalised; the colour thresholds are quantile-based and data-dependent. The authors should either remove the MPH-replication claim or substantiate it with an explicit comparison (for example, a synthetic two-density example where MPH and PWDS are both computed), and they should define 'different densities' precisely.
minor comments (6)
  1. [Visualisation of PH] The displayed definition of blue features reads '{(b,d )∈D| b > bdist⟩}' with a stray angle bracket; also, bprox and bdist are introduced before their quantile-based definitions, which makes the threshold rule harder to follow.
  2. [Vectorisations] The description of wflat is clear, but the nearest-neighbour scale s used to set p0 is estimated from the same samples under analysis; the sensitivity of the reported clusterings to s and p0 should be reported.
  3. [Methods] The witness filtration is described without stating the number and selection rule for landmarks beyond 'chosen randomly'; since witness results appear in Tables 2-4, these parameters should be specified.
  4. [Normalised Betti curves] The normalisation by the total number of features is not unit-area normalisation; although the text notes this, the term 'normalised' may still mislead readers and could be renamed.
  5. [Figure 5] The caption states that thresholds are computed per cell type, while the main text defines them as averages across samples for each point cloud; the dependence of bprox and bdist on the point cloud should be stated consistently.
  6. [Table 2] Several listed descriptors have very low silhouette scores (e.g., 0.173) and are still presented as 'correct'; a reporting threshold or a clarification of what 'correct' means in these cases would strengthen the table.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PH-to-vectorisation-to-clustering pipeline is self-contained, with only in-sample parameter fitting and an unproved PWDS localisation assertion.

full rationale

The paper's central pipeline is not circular. Persistence diagrams are computed by standard alpha/witness filtrations via gudhi, then deterministic vectorisations (normalised Betti curves, percentile statistics, persistence images with stated weight functions) are applied, and k-means clustering is run separately on each descriptor; no descriptor is defined in terms of the health/disease labels it is later compared with. PWDS colours death simplices by birth-value thresholds (bprox = (P90), bdist = (P98)), so its proximal/distal distinction is transparently definitional rather than a fitted quantity disguised as a discovery; the biological interpretation in Figures 5 and 7 depends on the unproved assertion that the death simplex lies inside the feature, but that is an unsupported correctness assumption, not a circular reduction. The persistence-image variances, p0, and threshold percentiles are chosen from the same samples later clustered, which is an in-sample selection/overfitting concern, but the vectorisations themselves are still computed from the persistence diagrams rather than from the labels. The only author-overlapping citation, [21], supplies the COVID-19 imaging data set, not a load-bearing theorem, so it does not constitute circular support. Consequently no load-bearing step reduces by construction to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method rests on standard TDA machinery plus several data-dependent choices. The main additions are PWDS and the weighting functions; both require parameters estimated from the same samples, and the death-simplex localizer is asserted rather than proven. No new physical entities are introduced.

free parameters (4)
  • bprox and bdist birth thresholds = average 90th and 98th percentile of birth values per point cloud, data-dependent
    Chosen from the same samples being analyzed; they define the red, blue, and purple regions in PWDS and directly affect the biological interpretation shown in Figures 5 and 7.
  • nearest-neighbour scale s and wflat threshold p0 = average nearest-neighbour distance across samples for all cell types; p0 = s/2 for H0 and (1/sqrt(3)-1/2)s for H1
    Estimated from the data being analyzed; the 'natural scale' assumption sets the minimum persistence considered relevant in the flat weighting and is not independently validated.
  • persistence image variance and pixel size = Table 1: spleen H0 10/10, H1 25/25, witness H1 50/50; lung H0 5/5, H1 10/10, witness H1 25/25
    Chosen separately for each dataset and homology degree, with no robustness analysis for these particular datasets.
  • witness filtration landmark count and selection = unspecified
    Landmarks are chosen randomly but neither the number nor the random seed is reported, which affects the reproducibility of witness-complex results.
assumptions (5)
  • ad hoc to paper Unique simplex death: only one simplex enters the filtration at a time, so death simplices are well-defined.
    Assumed in the PWDS section for generic position; the paper notes that gudhi still outputs a death simplex under ties, but with an arbitrary internal ordering.
  • ad hoc to paper Death simplex interior localizer: the death simplex of a feature lies in the interior of the feature and roughly indicates its position.
    Stated without proof in the PWDS section; the paper itself admits the death simplex is unstable under small perturbations, making the localization claim load-bearing.
  • domain assumption Red and white pulp cell-type grouping from Goltsev et al. [20] is valid for the spleen samples.
    The paper assigns individual cell types to red or white pulp based on the prior CODEX study's two large association clusters, and all pulp-level analyses rely on that grouping.
  • domain assumption Cell type annotations and disease staging in the two source datasets are correct.
    The analysis starts from processed and annotated data in [20] and [21]; segmentation or annotation errors would propagate directly into the point clouds and persistence diagrams.
  • standard math Stability theorem of persistence diagrams and standard TDA background results.
    Used to justify PH as stable and vectorisations as valid summaries; the paper does not prove new stability results, so this is inherited from the cited literature.

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Cite this review

Pith. "Pith review of Topology across Scales on Heterogeneous Cell Data." pith.science (2026). https://pith.science/paper/55MN3HIU

@misc{pith2026250502717,
  author       = {Pith},
  title        = {Pith review of: Topology across Scales on Heterogeneous Cell Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55MN3HIU}},
  note         = {Machine review of arXiv:2505.02717}
}
read the original abstract

Multiplexed imaging allows multiple cell types to be simultaneously visualised in a single tissue sample, generating unprecedented amounts of spatially-resolved, biological data. In topological data analysis, persistent homology provides multiscale descriptors of ``shape" suitable for the analysis of such spatial data. Here we propose a novel visualisation of persistence homology (PH) and fine-tune vectorisations thereof (exploring the effect of different weightings for persistence images, a prominent vectorisation of PH). These approaches offer new biological interpretations and promising avenues for improving the analysis of complex spatial biological data especially in multiple cell type data. To illustrate our methods, we apply them to a lung data set from fatal cases of COVID-19 and a data set from lupus murine spleen.

Figures

Figures reproduced from arXiv: 2505.02717 by the authors.

Figure 1
Figure 1. Topological data analysis pipeline. Starting from point cloud data—e.g., cell centroid locations in tissue—we construct filtered simplicial complexes to approximate the underlying geometric structure. Persistent homology is then computed to extract topological features such as connected components, loops, and voids across multiple scales. These features are vectorised, and we explore weightings that emphasise struct… view at source ↗
Figure 2
Figure 2. Lupus murine spleen cell centroid data. A) Cell centroid spatial distribution of the 25 cell types in lupus murine spleen CODEX data set. The data set of consists of 3 healthy samples (BALBc-1, BALBc-2, BALBc-3), 3 samples in an early stage of the disease (MRL-4, MRL-5, MRL-6), 2 samples in an intermediate stage of the disease (MRL-7, MRL-8) and 1 sample in a late stage of the disease (MRL-9). B) Cell counts of the … view at source ↗
Figure 3
Figure 3. Examples of persistent homology. A) Example of the alpha filtration for a point cloud forming a loop. Corresponding persistence diagram with one persistent feature in degree 1 corresponding to the loop. B) Example of persistent homology for a point cloud consisting of four loops of various sizes and densities. For each loop, we indicate the corresponding feature in the persistence diagram. Less dense loops have larg… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Persistence weighted death simplices (PWDS) visualisation. A) Colouring and weighting of the persistence diagram. The colouring in the diagram distinguishes features based on their birth values, with red representing features formed by proximal points, blue for those f…
Figure 5
Figure 5. Figure 5: PWDS of red and white pulp cell populations in the murine spleen tissue. Visualisation of PWDS for the red and white pulp in a healthy sample (BALBc-1) and a diseased sample (MRL-4). Triangles correspond to loops detected by PH with alpha complexes. Red triangles corre…
Figure 6
Figure 6. Figure 6: Betti curves of red pulp, white pulp, and B cells populations in the murine spleen tissue. A) Red pulp cell locations in the lupus murine spleen CODEX data. B) Red pulp normalised Betti-1 curves, which count the proportion of loops that are alive at a certain parameter…
Figure 7
Figure 7. Figure 7: PWDS and Betti curves for all cell types together in the human lung tissue. A) Visualisation of PWDS for all cell types together with a choice of 6 representative samples out of 32. The visualisation corresponds to loops in PH alpha, with threshold parameters bprox = ⟨…
Figure 8
Figure 8. Figure 8: Clustering based on the topology of Endothelial cells in the human lung tissue. A) Clustering based on the topological descriptor that attains the best Rand score when compared to the biological classification: percentiles of persistence values of degree 1 PH (alpha) o…
Figure 9
Figure 9. Figure 9: Witness filtration. A) Example of the witness filtration for a point cloud forming a loop. Landmarks points, in black, are chosen randomly. The persistence diagram contains only one point in each degree, corresponding to a single connected component and the loop, both …
Figure 10
Figure 10. Figure 10: Betti curves of CD4(+) T cells, 220(+) DN T cells and megakaryocytes populations in the murine spleen tissue. A) CD4(+) T cells locations in the lupus murine spleen CODEX data. B) CD4(+) T cells normalised Betti-1 curves. The differences in the location of the peak in…
Figure 11
Figure 11. Figure 11: Persistence images with 5 different weightings for the red pulp in the murine spleen tissue. Examples of persistence images with 5 different weighting functions for the red pulp with alpha filtration (BALBc-1, MRL-4). Here we observe how weights that depend on persist…

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Reference graph

Works this paper leans on

31 extracted references · 14 canonical work pages

  1. [1]

    ‘Spatial analysis with SPIAT and spaSim to characterize and simulate tissue microenvironments’

    Yuzhou Feng et al. ‘Spatial analysis with SPIAT and spaSim to characterize and simulate tissue microenvironments’. In: Nature Communications 14.1 (May 2023). issn: 2041-1723. doi: 10.1038/ s41467-023-37822-0 . url: http://dx.doi.org/10.1038/s41467-023-37822-0

    doi:10.1038/s41467-023-37822-0 2023

  2. [2]

    Mah et al

    Clarence K. Mah et al. ‘Bento: a toolkit for subcellular analysis of spatial transcriptomics data’. In: Genome Biology 25.1 (Apr. 2024). issn: 1474-760X. doi: 10.1186/s13059-024-03217-7 . url: http://dx.doi.org/10.1186/s13059-024-03217-7

    work page doi:10.1186/s13059-024-03217-7 2024

  3. [3]

    Nirmal and Peter K

    Ajit J. Nirmal and Peter K. Sorger. ‘SCIMAP: A Python Toolkit for Integrated Spatial Analysis of Multiplexed Imaging Data’. In: Journal of Open Source Software 9.97 (May 2024), p. 6604. issn: 2475-9066. doi: 10.21105/joss.06604. url: http://dx.doi.org/10.21105/joss.06604

    doi:10.21105/joss.06604 2024

  4. [4]

    Stoltzfus et al

    Caleb R. Stoltzfus et al. ‘CytoMAP: A Spatial Analysis Toolbox Reveals Features of Myeloid Cell Organization in Lymphoid Tissues’. In: Cell Reports 31.3 (Apr. 2020), p. 107523. issn: 2211-1247. doi: 10.1016/j.celrep.2020.107523 . url: http://dx.doi.org/10.1016/j.celrep.2020. 107523

    arXiv 2020

  5. [5]

    ‘GraphCompass: spatial metrics for differential analyses of cell organization across conditions’

    Mayar Ali et al. ‘GraphCompass: spatial metrics for differential analyses of cell organization across conditions’. In: Bioinformatics 40.Supplement 1 (June 2024), pp. i548–i557. issn: 1367-4811. doi: 10.1093/bioinformatics/btae242 . url: http://dx.doi.org/10.1093/bioinformatics/ btae242

    doi:10.1093/bioinformatics/btae242 2024

  6. [6]

    ‘Stability of Persistence Diagrams’

    David Cohen-Steiner, Herbert Edelsbrunner and John Harer. ‘Stability of Persistence Diagrams’. In: Discrete & Computational Geometry 37.1 (Dec. 2006), pp. 103–120. doi: 10.1007/s00454-006- 1276-5. url: https://doi.org/10.1007/s00454-006-1276-5

    doi:10.1007/s00454-006- 2006

  7. [7]

    ‘Induced matchings and the algebraic stability of persistence barcodes’

    Ulrich Bauer and Michael Lesnick. ‘Induced matchings and the algebraic stability of persistence barcodes’. en. In: Journal of Computational Geometry (2015), Vol. 6 No. 2 (2015): Special issue of Selected Papers from SoCG 2014. doi: 10.20382/JOCG.V6I2A9. url: https://jocg.org/index. php/jocg/article/view/2983

    doi:10.20382/jocg.v6i2a9 2015

  8. [8]

    Ali et al

    D. Ali et al. ‘A Survey of Vectorization Methods in Topological Data Analysis’. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 45.12 (2023), pp. 14069–14080. issn: 1939-3539

    work page 2023

Show all 31 references
  1. [9]

    Topaz, Lori Ziegelmeier and Tom Halverson

    Chad M. Topaz, Lori Ziegelmeier and Tom Halverson. ‘Topological Data Analysis of Biological Aggreg- ation Models’. In: PLOS ONE 10.5 (May 2015). Ed. by Bard Ermentrout, e0126383. issn: 1932-6203. doi: 10.1371/journal.pone.0126383 . url: http://dx.doi.org/10.1371/journal.pone. 0126383

    2015 doi

  2. [10]

    Nardini et al

    John T. Nardini et al. ‘Topological data analysis distinguishes parameter regimes in the Anderson- Chaplain model of angiogenesis’. In: PLOS Computational Biology 17.6 (June 2021). Ed. by Mark Chaplain, e1009094. issn: 1553-7358. doi: 10.1371/journal.pcbi.1009094 . url: http:/...

    2021 doi

  3. [11]

    McGuirl, Alexandria Volkening and Bj¨ orn Sandstede

    Melissa R. McGuirl, Alexandria Volkening and Bj¨ orn Sandstede. ‘Topological data analysis of zebrafish patterns’. In: Proceedings of the National Academy of Sciences 117.10 (Feb. 2020), pp. 5113–5124. issn: 1091-6490. doi: 10.1073/pnas.1917763117 . url: http://dx.doi.org/10.1...

    2020 doi

  4. [12]

    Gardner et al

    Richard J. Gardner et al. ‘Toroidal topology of population activity in grid cells’. In: Nature 602.7895 (Jan. 2022), pp. 123–128. issn: 1476-4687. doi: 10.1038/s41586- 021- 04268- 7 . url: http: //dx.doi.org/10.1038/s41586-021-04268-7

    2022 doi

  5. [13]

    ‘A tool for mapping microglial morphology, morphOMICs, reveals brain-region and sex-dependent phenotypes’

    Gloria Colombo et al. ‘A tool for mapping microglial morphology, morphOMICs, reveals brain-region and sex-dependent phenotypes’. In: Nature Neuroscience 25.10 (Sept. 2022), pp. 1379–1393. issn: 1546-1726. doi: 10.1038/s41593-022-01167-6 . url: http://dx.doi.org/10.1038/s41593-...

    2022 doi

  6. [14]

    ‘Topological data analysis of contagion maps for examining spreading processes on networks’

    Dane Taylor et al. ‘Topological data analysis of contagion maps for examining spreading processes on networks’. In: Nature Communications 6.1 (July 2015). issn: 2041-1723. doi: 10.1038/ncomms8723. url: http://dx.doi.org/10.1038/ncomms8723

    2015 doi

  7. [15]

    ‘Topological data analysis of task-based fMRI data from experiments on schizophrenia’

    Bernadette J Stolz et al. ‘Topological data analysis of task-based fMRI data from experiments on schizophrenia’. In: Journal of Physics: Complexity 2.3 (May 2021), p. 035006. issn: 2632-072X. doi: 10.1088/2632-072x/abb4c6. url: http://dx.doi.org/10.1088/2632-072X/abb4c6

    2021 doi

  8. [16]

    ‘Analyzing collective motion with machine learning and topology’

    Dhananjay Bhaskar et al. ‘Analyzing collective motion with machine learning and topology’. In: Chaos: An Interdisciplinary Journal of Nonlinear Science 29.12 (Dec. 2019). issn: 1089-7682. doi: 10.1063/1.5125493. url: http://dx.doi.org/10.1063/1.5125493

    2019 doi

  9. [17]

    ‘The Theory of Multidimensional Persistence’

    Gunnar Carlsson and Afra Zomorodian. ‘The Theory of Multidimensional Persistence’. In: Discrete & Computational Geometry 42.1 (Apr. 2009), pp. 71–93. issn: 1432-0444. doi: 10.1007/s00454- 009-9176-0. url: http://dx.doi.org/10.1007/s00454-009-9176-0

    2009 doi

  10. [18]

    ‘Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors’

    Oliver Vipond et al. ‘Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors’. In: Proceedings of the National Academy of Sciences 118.41 (Oct. 2021). issn: 1091-6490. doi: 10 . 1073 / pnas . 2102166118. url: http : / / dx . doi . org / 1...

    2021

  11. [19]

    ‘Multiscale topology classifies cells in subcellular spatial transcriptomics’

    Katherine Benjamin et al. ‘Multiscale topology classifies cells in subcellular spatial transcriptomics’. In: Nature 630.8018 (June 2024), pp. 943–949. issn: 1476-4687. doi: 10.1038/s41586-024-07563-1 . url: http://dx.doi.org/10.1038/s41586-024-07563-1

    2024 doi

  12. [20]

    ‘Deep Profiling of Mouse Splenic Architecture with CODEX Multiplexed Imaging’

    Yury Goltsev et al. ‘Deep Profiling of Mouse Splenic Architecture with CODEX Multiplexed Imaging’. In: Cell 174.4 (Aug. 2018), 968–981.e15. doi: 10 . 1016 / j . cell . 2018 . 07 . 010. url: https : //doi.org/10.1016/j.cell.2018.07.010

    2018 doi

  13. [21]

    ‘Single cell spatial analysis reveals inflammatory foci of immature neutrophil and CD8 T cells in COVID-19 lungs’

    Praveen Weeratunga et al. ‘Single cell spatial analysis reveals inflammatory foci of immature neutrophil and CD8 T cells in COVID-19 lungs’. In: Nature Communications 14.1 (Nov. 2023). issn: 2041-1723. doi: 10.1038/s41467-023-42421-0 . url: http://dx.doi.org/10.1038/s41467-023...

    2023 doi

  14. [22]

    Edelsbrunner, D

    H. Edelsbrunner, D. Kirkpatrick and R. Seidel. ‘On the shape of a set of points in the plane’. In: IEEE Transactions on Information Theory 29.4 (1983), pp. 551–559. doi: 10.1109/TIT.1983.1056714

    1983

  15. [23]

    Edelsbrunner

    H. Edelsbrunner. ‘The union of balls and its dual shape’. In: Discrete & Computational Geometry 13.3-4 (June 1995), pp. 415–440. doi: 10.1007/bf02574053. url: https://doi.org/10.1007/ bf02574053

    1995 doi

  16. [24]

    ‘A weak characterisation of the Delaunay triangulation’

    Vin de Silva. ‘A weak characterisation of the Delaunay triangulation’. In: Geometriae Dedicata 135.1 (Apr. 2008), pp. 39–64. doi: 10.1007/s10711-008-9261-1 . url: https://doi.org/10.1007/ s10711-008-9261-1

    2008 doi

  17. [25]

    ‘Topological estimation using witness complexes’

    Vin de Silva and Gunnar Carlsson. ‘Topological estimation using witness complexes’. In: SPBG’04 Symposium on Point - Based Graphics 2004 . Ed. by Markus Gross et al. The Eurographics Association,

    2004

  18. [26]

    GUDHI: Witness Complex Documentation

    GUDHI Project. GUDHI: Witness Complex Documentation . Accessed: 2025-04-15. 2025. url: https://gudhi.inria.fr/doc/latest/group__witness__complex.html

    2025

  19. [27]

    ‘Persistence Images: A Stable Vector Representation of Persistent Homology’

    Henry Adams et al. ‘Persistence Images: A Stable Vector Representation of Persistent Homology’. In: Journal of Machine Learning Research 18.8 (2017), pp. 1–35. url: http://jmlr.org/papers/ v18/16-337.html

    2017

  20. [28]

    ‘Clique topology reveals intrinsic geometric structure in neural correlations’

    Chad Giusti et al. ‘Clique topology reveals intrinsic geometric structure in neural correlations’. In: Proceedings of the National Academy of Sciences 112.44 (Oct. 2015), pp. 13455–13460. issn: 1091-

    2015

  21. [29]

    ‘A Topological Representation of Branching Neuronal Morphologies’

    Lida Kanari et al. ‘A Topological Representation of Branching Neuronal Morphologies’. In: Neuroin- formatics 16.1 (Oct. 2017), pp. 3–13. issn: 1559-0089. doi: 10.1007/s12021-017-9341-1 . url: http://dx.doi.org/10.1007/s12021-017-9341-1 . 31

    2017 doi

  22. [2004]

    doi: 10.2312/SPBG/SPBG04/157-166

    isbn: 3-905673-09-6. doi: 10.2312/SPBG/SPBG04/157-166

    doi

  23. [6490]

    url: http://dx.doi.org/10.1073/pnas.1506407112

    doi: 10.1073/pnas.1506407112. url: http://dx.doi.org/10.1073/pnas.1506407112. 30

    doi

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.