pith. sign in

arxiv: 2605.22761 · v2 · pith:NEWV6NCTnew · submitted 2026-05-21 · ❄️ cond-mat.soft · cond-mat.mtrl-sci

Topological cell-openness index for porous materials

Micha{\l} Bogdan , Pawe{\l} D{\l}otko This is my paper

Pith reviewed 2026-06-30 15:38 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mtrl-sci
keywords porous materialsBetti numberscell opennesstopologygas pycnometrypore structuretopological index
0
0 comments X

The pith

Betti numbers from the pore network yield an index τ estimating the fraction of open versus closed cells.

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

The paper introduces a method that computes Betti numbers on either the pore space or the solid skeleton of a porous material and uses them to define a cell-openness index τ. This index is offered as a complement to the open-cell volume fraction measured by gas pycnometry, the existing standard. A reader would care because the topological measure can differ from the volume measure in informative ways and because the index correlates with physical properties in both simulated and real samples. The same topological data can also be turned into curves that locate typical pore feature sizes.

Core claim

The authors define a cell-openness index τ from Betti numbers measured on the pore network or solid skeleton; this index estimates the proportion of open versus closed cells and supplies structural information that differs from the volume fraction reported by gas pycnometry, with demonstrated correlations to physical quantities in numerical and experimental cases.

What carries the argument

The cell-openness index τ, obtained by mapping Betti numbers (counts of connected components, loops and voids) to a proportion of open cells.

If this is right

  • τ supplies a complement to the open-celled volume fraction obtained from gas pycnometry.
  • Mismatches between τ and pycnometry results carry additional information about the material structure.
  • τ exhibits significant correlations with measurable physical quantities in both numerical and experimental porous structures.
  • Betti curves derived from the same data can be used to estimate characteristic feature sizes inside the pores.

Where Pith is reading between the lines

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

  • If the mapping holds, τ could guide the design of porous materials whose openness is tuned for specific transport or mechanical roles.
  • Application to image data at varying resolutions would need explicit checks that the same τ values retain the same physical meaning.
  • Systematic comparison of τ across chemically different porous solids might expose topological patterns that are independent of material chemistry.

Load-bearing premise

Betti numbers computed on the pore network or solid skeleton can be translated into a physically meaningful fraction of open versus closed cells that remains consistent across different materials and imaging resolutions.

What would settle it

A collection of porous samples whose open-cell fractions have been measured independently by a non-topological method, yet show no systematic relation to the τ values computed from their Betti numbers.

Figures

Figures reproduced from arXiv: 2605.22761 by Micha{\l} Bogdan, Pawe{\l} D{\l}otko.

Figure 1
Figure 1. Figure 1: Schematic illustration of the synthetic porous-structure generation procedure. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Demonstration of the workflow based on the signed distance transform and persistent [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Correlations between cell-openness indices. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustrative Betti-curve features for an idealised closed-cell system (top) and an [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The relationship between Betti-curve based predictors and characteristic length [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The relationship between Betti-curve based predictors and characteristic length [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relationship between τ and log10(permeability) in the 2D dataset [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original abstract

We propose a method of estimating and parametrising the proportion of open and closed cells in a porous material based on measuring Betti numbers on the structures. We define a cell-openness index {\tau} which can be used to complement the proportion of open-celled volume reported by gas pycnometry, which is the current gold standard for pore type characterization. We discuss in what types of structures mismatches between the two measures can occur and how such mismatches convey additional information about the structure. We demonstrate examples of significant correlations between {\tau} and measurable physical quantities in both numerical and experimental structures. We also discuss how Betti curves can be used to estimate characteristic feature sizes in porous structures.

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 / 0 minor

Summary. The paper proposes a topological method to estimate the proportion of open versus closed cells in porous materials by defining a cell-openness index τ derived from Betti numbers computed on the pore network or solid skeleton. It positions τ as a complement to the open-celled volume fraction from gas pycnometry, discusses structural mismatches between the two measures and their implications, reports correlations between τ and other physical quantities in numerical and experimental cases, and suggests using Betti curves to estimate characteristic feature sizes.

Significance. If the mapping from Betti numbers to a physically meaningful open/closed cell proportion can be shown to be robust, the index would provide a useful topological complement to volume-based characterization methods in porous materials, potentially revealing connectivity information not captured by pycnometry alone. The reported correlations with measurable quantities indicate possible practical relevance, but the absence of direct validation against known ground-truth fractions limits the immediate impact.

major comments (2)
  1. Abstract: the central claim that τ estimates the proportion of open and closed cells requires an explicit mapping rule from Betti numbers, yet none is supplied; without it, the index remains an untested definition whose physical interpretability cannot be assessed.
  2. Abstract (discussion of mismatches and correlations): the reported correlations with physical quantities in numerical/experimental structures do not include direct comparisons to synthetic structures whose true open/closed cell fractions are known by construction (e.g., controlled foams), leaving the required robustness across resolutions, cell-size distributions, and connectivity patterns unverified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: Abstract: the central claim that τ estimates the proportion of open and closed cells requires an explicit mapping rule from Betti numbers, yet none is supplied; without it, the index remains an untested definition whose physical interpretability cannot be assessed.

    Authors: The explicit mapping rule from Betti numbers to the cell-openness index τ is defined in Section 2.2 of the manuscript. To make the central claim immediately interpretable from the abstract alone, we will revise the abstract to include a concise statement of this mapping. revision: yes

  2. Referee: Abstract (discussion of mismatches and correlations): the reported correlations with physical quantities in numerical/experimental structures do not include direct comparisons to synthetic structures whose true open/closed cell fractions are known by construction (e.g., controlled foams), leaving the required robustness across resolutions, cell-size distributions, and connectivity patterns unverified.

    Authors: We agree that direct comparisons against synthetic structures with known ground-truth open/closed fractions by construction would strengthen the validation of τ. Our current numerical examples use controlled generation parameters but do not prescribe exact openness fractions a priori. In the revised manuscript we will add a dedicated subsection presenting simple synthetic test cases (e.g., periodic lattices with deliberately introduced closed cells) to enable direct comparison of τ against the known fractions and to examine robustness to resolution and connectivity variations. revision: yes

Circularity Check

0 steps flagged

Cell-openness index τ introduced as explicit definition from Betti numbers; no fitted predictions or self-referential reductions

full rationale

The paper defines τ directly from Betti numbers computed on pore networks or skeletons and reports empirical correlations in numerical and experimental cases. No equations are presented that fit parameters to data subsets and then relabel the fit as a prediction, nor does any load-bearing step reduce to a self-citation chain or ansatz smuggled from prior work by the same authors. The mapping rule itself is presented as a new definition rather than derived from first principles that presuppose the target proportion, so the derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only; no explicit free parameters, axioms or invented entities beyond the index itself are stated. The index τ is introduced as a new construct without independent evidence supplied.

invented entities (1)
  • cell-openness index τ no independent evidence
    purpose: To estimate and parametrize the proportion of open and closed cells from Betti numbers
    Defined in the paper as a new topological descriptor; no external validation or falsifiable prediction is given in the abstract.

pith-pipeline@v0.9.1-grok · 5646 in / 1216 out tokens · 32774 ms · 2026-06-30T15:38:23.166763+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references

  1. [1]

    Armstrong, James E

    Ryan T. Armstrong, James E. McClure, Vanessa Robins, Zhishang Liu, Christoph H. Arns, Steffen Schlüter, and Steffen Berg. Porous media characterization using Minkowski functionals: Theories, applications and future directions.Transport in Porous Media, 130(1):305–335, 2019

    2019

  2. [2]

    C. H. Arns, M. A. Knackstedt, and K. Mecke. 3D structural analysis: sensitivity of Minkowski functionals.Journal of Microscopy, 240(3):181–196, 2010

    2010

  3. [3]

    A persistence landscapes toolbox for topological statistics.Journal of Symbolic Computation, 78:91–114, 2017

    Peter Bubenik and Paweł Dłotko. A persistence landscapes toolbox for topological statistics.Journal of Symbolic Computation, 78:91–114, 2017

    2017

  4. [4]

    Topology and data.Bulletin of the American Mathematical Society, 46(2):255–308, 2009

    Gunnar Carlsson. Topology and data.Bulletin of the American Mathematical Society, 46(2):255–308, 2009

    2009

  5. [5]

    Cnudde and M

    V. Cnudde and M. N. Boone. High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications.Earth-Science Reviews, 123:1–17, 2013

    2013

  6. [6]

    Computational and applied topology, tutorial, 2018

    Paweł Dłotko. Computational and applied topology, tutorial, 2018. arXiv preprint

    2018

  7. [7]

    Cubical complex

    Paweł Dłotko. Cubical complex. InGUDHI User and Reference Manual. GUDHI Editorial Board, 3.12.0 edition, 2026

    2026

  8. [8]

    Harer.Computational Topology: An Introduction

    Herbert Edelsbrunner and John L. Harer.Computational Topology: An Introduction. American Mathematical Society, Providence, RI, 2010

    2010

  9. [9]

    K. M. Graczyk and M. Matyka. Predicting porosity, permeability, and tortuosity of porous media from images by deep learning.Scientific Reports, 10:21488, 2020

    2020

  10. [10]

    Escolar, Kaname Matsue, and Yasumasa Nishiura

    Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, and Yasumasa Nishiura. Hierarchical structures of amorphous solids char- acterized by persistent homology.Proceedings of the National Academy of Sciences, 113(26):7035–7040, 2016

    2016

  11. [11]

    Lautensack, M

    C. Lautensack, M. Giertzsch, M. Godehardt, and K. Schladitz. Modelling a ceramic foam using locally adaptable morphology.Journal of Microscopy, 230(3):396–404, 2008

    2008

  12. [12]

    Barthel, Paweł Dłotko, S

    Yongjin Lee, Senja D. Barthel, Paweł Dłotko, S. Mohamad Moosavi, Kathryn Hess, and Berend Smit. Quantifying similarity of pore-geometry in nanoporous materials.Nature Communications, 8:15396, 2017

    2017

  13. [13]

    LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations

    Randall J. LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics, 2007

    2007

  14. [14]

    Maire and P

    E. Maire and P. J. Withers. Quantitative X-ray tomography.International Materials Reviews, 59(1):1–43, 2014. 17

    2014

  15. [15]

    Persistent cohomology

    Clément Maria. Persistent cohomology. InGUDHI User and Reference Manual. GUDHI Editorial Board, 3.12.0 edition, 2026

    2026

  16. [16]

    Axel Zeitler

    Daniel Markl, Alexa Strobel, Rüdiger Schlossnikl, Johan Bøtker, Prince Bawuah, Cathy Ridgway, Jukka Rantanen, Thomas Rades, Patrick Gane, Kai-Erik Peiponen, and J. Axel Zeitler. Characterisation of pore structures of pharmaceutical tablets: A review. International Journal of Pharmaceutics, 538:188–214, 3 2018

    2018

  17. [17]

    Klaus R. Mecke. Additivity, convexity, and beyond: Applications of Minkowski func- tionals in statistical physics. In Klaus R. Mecke and Dietrich Stoyan, editors,Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, volume 554 ofLecture Notes in Physics, pages 111–184. Springer, Berlin, Hei...

    2000

  18. [18]

    Nguyen, Jarod C

    Huong Giang T. Nguyen, Jarod C. Horn, Matthew Bleakney, Daniel W. Siderius, and Laura Espinal. Understanding material characteristics through signature traits from helium pycnometry.Langmuir, 35:2115–2122, 2 2019

    2019

  19. [19]

    Robins, M

    V. Robins, M. Saadatfar, O. Delgado-Friedrichs, and A. P. Sheppard. Percolating length scales from topological persistence analysis of micro-CT images of porous materials. Water Resources Research, 52(1):315–329, 2016

    2016

  20. [20]

    Rouquerol, D

    J. Rouquerol, D. Avnir, C. W. Fairbridge, D. H. Everett, J. H. Haynes, N. Pernicone, J. D. F. Ramsay, K. S. W. Sing, and K. K. Unger. Recommendations for the charac- terization of porous solids (IUPAC Technical Report).Pure and Applied Chemistry, 66(8):1739–1758, 1994

    1994

  21. [21]

    Saadatfar, H

    M. Saadatfar, H. Takeuchi, V. Robins, N. Francois, and Y. Hiraoka. Pore configuration landscape of granular crystallization.Nature Communications, 8:15082, 2017

    2017

  22. [22]

    Scholz, F

    C. Scholz, F. Wirner, J. Götz, U. Rüde, G. E. Schröder-Turk, K. Mecke, and C. Bechinger. Permeability of porous materials determined from the Euler characteristic.Physical Review Letters, 109(26):264504, 2012

    2012

  23. [23]

    Sereno, Maria A

    Alberto M. Sereno, Maria A. Silva, and Luis Mayor. Determination of particle density and porosity in foods and porous materials with high moisture content.International Journal of Food Properties, 10:455–469, 7 2007

    2007

  24. [24]

    GUDHI Editorial Board, 3.12.0 edition, 2026

    The GUDHI Project.GUDHI User and Reference Manual. GUDHI Editorial Board, 3.12.0 edition, 2026

    2026

  25. [25]

    Vogel, U

    H.-J. Vogel, U. Weller, and S. Schlüter. Quantification of soil structure based on Minkowski functions.Computers & Geosciences, 36(10):1236–1245, 2010

    2010

  26. [26]

    Paul A. Webb. Volume and density determinations for particle technologies. Technical report, Micromeritics Instrument Corp., Norcross, GA, USA, 2001. 18

    2001