pith. sign in

arxiv: 2605.22761 · v1 · 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 2026-05-22 03:17 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mtrl-sci
keywords Betti numbersporous materialscell-openness indexopen cellsclosed cellsgas pycnometrytopological characterizationpore structure
0
0 comments X

The pith

A cell-openness index derived from Betti numbers offers an alternative to gas pycnometry for characterizing open and closed cells in porous materials.

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

The paper proposes computing Betti numbers on the digitized structure of a porous material to estimate the fraction of open versus closed cells. From these numbers it defines a cell-openness index τ that can replace or complement the open-celled volume fraction measured by gas pycnometry. A sympathetic reader would care because mismatches between the two quantities are shown to carry extra information about pore connectivity, and because τ correlates with physical properties in simulated foams. The work also indicates that Betti curves themselves can locate characteristic pore sizes. This matters for materials whose transport and mechanical behavior depend on whether cells are open or sealed.

Core claim

The central claim is that Betti numbers measured on the meshed or voxelized geometry of a porous material directly encode the distinction between open and closed cells, yielding a scalar cell-openness index τ that serves as a topological substitute or complement to the open-cell volume fraction obtained by gas pycnometry; discrepancies between τ and pycnometry results reveal additional structural features, and τ correlates with measurable physical quantities in numerically generated structures.

What carries the argument

The cell-openness index τ, a scalar derived from Betti numbers of the solid and void phases, which quantifies the topological openness of cells without relying on geometric volume measurements.

If this is right

  • Mismatches between the topological index τ and gas-pycnometry open-cell fractions convey additional information about the connectivity of the pore network.
  • In numerically generated structures, τ shows significant correlations with measurable physical quantities such as permeability or stiffness.
  • Betti curves computed across scales can be used to estimate characteristic feature sizes inside the porous structure.
  • The index τ can be applied to any meshed or voxelized geometry, extending pore-type characterization beyond traditional volume-based methods.

Where Pith is reading between the lines

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

  • The method could be applied directly to experimental CT or SEM images, enabling non-destructive topological characterization of real porous specimens.
  • Similar Betti-based indices might be developed for other classes of cellular materials where openness controls fluid flow or insulation performance.
  • Combining τ with existing simulation pipelines could allow rapid screening of virtual material designs for desired open-cell fractions.
  • Betti curves offer a route to multi-scale analysis that simultaneously reports both cell openness and dominant pore diameters.

Load-bearing premise

Betti numbers computed directly on the digitized or meshed structure of a porous material reliably distinguish open from closed cells without material-specific geometric assumptions or extra calibration.

What would settle it

Compute τ from Betti numbers on a collection of foam samples whose open-cell fraction has been measured independently by gas pycnometry; large systematic differences that cannot be attributed to known structural mismatches would falsify the claim that τ provides a valid alternative measure.

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 the proportion of open and closed cells in a porous material based on measuring Betti numbers on the structures. Based on this method, we define a cell-openness index {\tau} which can be used instead of or complementary to 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 also demonstrate initial examples of significant correlations between {\tau} and measurable physical quantities in numerically generated 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

1 major / 2 minor

Summary. The manuscript proposes a cell-openness index τ derived from Betti numbers computed on the digitized structure of porous materials. The index is intended to estimate the proportion of open versus closed cells and to serve as an alternative or complement to the open-cell volume fraction obtained from gas pycnometry. The authors discuss in principle the types of structures where τ and pycnometry may disagree and how such disagreements can provide additional structural information. They report initial correlations between τ and other physical quantities on numerically generated structures and suggest that Betti curves can be used to estimate characteristic feature sizes.

Significance. If the index proves robust, the topological approach could supply a non-destructive, connectivity-sensitive characterization tool that complements volume-fraction methods and yields interpretable discrepancies. The conceptual discussion of mismatches is a constructive contribution. Credit is due for the clear framing of the proposal and the demonstration on simulated data; however, the current evidence base remains limited to numerical examples.

major comments (1)
  1. [Discussion] Discussion of validation and mismatches: the central claim that τ can be used instead of or complementary to gas pycnometry rests on the assumption that Betti numbers computed from 3-D images reliably encode open-cell accessibility. No controlled side-by-side comparison on identical physical specimens is reported, leaving open the possibility that discretization artifacts, imaging resolution, or material-specific connectivity features cause systematic divergence from the gas-accessible openness actually measured by pycnometry.
minor comments (2)
  1. [Abstract] The abstract refers to 'initial examples' of correlations without specifying the number, generation protocol, or statistical details of the numerically generated structures; adding this information would aid reproducibility.
  2. [Methods] Notation for Betti numbers and the precise functional form of τ should be introduced with an equation number in the methods section and used consistently thereafter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review, the positive assessment of the conceptual framing, and the recognition that mismatches between measures can be informative. We address the single major comment below.

read point-by-point responses

  1. Referee: [Discussion] Discussion of validation and mismatches: the central claim that τ can be used instead of or complementary to gas pycnometry rests on the assumption that Betti numbers computed from 3-D images reliably encode open-cell accessibility. No controlled side-by-side comparison on identical physical specimens is reported, leaving open the possibility that discretization artifacts, imaging resolution, or material-specific connectivity features cause systematic divergence from the gas-accessible openness actually measured by pycnometry.

    Authors: We agree that controlled experimental comparisons on physical specimens would provide the strongest test of the index. The present manuscript is framed as a proposal of the topological index τ together with its behavior on numerically generated structures, where the ground-truth cell openness is known exactly; this setting permits direct validation of the index against known topology that is not available in physical samples. The manuscript already contains a section discussing the structural conditions under which τ and pycnometry are expected to diverge and how such divergences can yield additional information. In the revised version we will expand that discussion into a dedicated subsection that explicitly treats discretization artifacts, finite imaging resolution, and material-specific connectivity effects as potential sources of systematic difference from gas pycnometry. We will also add a short paragraph outlining the experimental protocol that would be required for future side-by-side validation. These changes will clarify the current scope of the work while directly addressing the referee’s concern about reliability assumptions. revision: partial

Circularity Check

0 steps flagged

No significant circularity: topological index defined directly from standard Betti numbers

full rationale

The paper defines the cell-openness index τ directly from Betti numbers computed on the digitized porous structure. This is a straightforward application of topological invariants to distinguish open versus closed cells, without any fitting procedures, parameter calibration to the target quantity, or self-referential equations that reduce the output to the input by construction. No load-bearing self-citations, uniqueness theorems from prior author work, or smuggled ansatzes are described. The central claim remains a new definitional proposal that can be evaluated against external benchmarks such as gas pycnometry; the derivation chain is self-contained and does not collapse into tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are described. The approach relies on standard topological invariants (Betti numbers) whose computation is assumed to be well-defined on the input structures.

pith-pipeline@v0.9.0 · 5651 in / 1162 out tokens · 50524 ms · 2026-05-22T03:17:49.712587+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 2013

  6. [6]

    Computational and applied topology, tutorial, 2018

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

    work page 2018

  7. [7]

    Cubical complex

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

    work page 2026

  8. [8]

    Harer.Computational Topology: An Introduction

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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 2014

  15. [15]

    Persistent cohomology

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

    work page 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

    work page 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...

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 2010

  26. [26]

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

    work page 2001