Pith. sign in

REVIEW 3 major objections 5 minor 54 references

Topology of Shape and Data in Material Microstructures

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A scalar bipersistence summary I, integrating Betti-1 counts over a shape-distance and spatial-scale filtration, sharply distinguishes low- from high-strain ice microstructures and detects onset of dynamic recrystallization.

desk verdict Promising TDA+SST framework, but the headline scalar summary is likely confounded by grain density; needs null models and normalization before the claims hold. read the letter →

arxiv 2607.27493 v1 pith:J2NRR6D6 submitted 2026-07-29 cond-mat.mtrl-sci cs.CE

Jeanie Schreiber , Zachary Grey , Adam Creuziger This is my paper · ORCID
classification cond-mat.mtrl-scics.CE MSC 55N31
keywords topologicaldataanalysispersistenthomologymultiparameterpersistenceshapeseparabletensorselectronbackscatterdiffractionnecklacingdynamicrecrystallization
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 sets out to quantify the spatial arrangement of grain shapes in polycrystalline materials, not just their average size or shape. It builds a two-parameter filtration on grain centroids: one axis is the usual spatial scale of a Rips complex, the other is how far each grain's shape sits from the mean shape in a manifold of Separable Shape Tensors. Counting the persistent one-dimensional loops (Betti-1) over this grid yields a surface, and integrating the surface gives a single scalar I. Applied to EBSD scans of ice deformed to increasing axial strain, I rises gently up to 8% strain and then jumps sharply by 12%, matching the strain range where dynamic recrystallization becomes the dominant grain-refinement mechanism. If this holds, the framework gives a quantitative, interpretable measure of 'necklacing' — fine grains arranged in rings around coarse grains — that existing standards describe only qualitatively.

What carries the argument

The central object is the sublevel Rips bifiltration S↑_{ε,ℓ} = φ^{-1}((-∞,ℓ]) ∩ Rips(ε), built on grain centroids, where φ is the distance of each grain's separable shape tensor from the intrinsic mean on the product manifold Gr(d,q) × S^d_{++}. The first parameter ε is the Rips radius controlling spatial connectivity; the second ℓ is a shape-distance threshold selecting grains similar to the mean. Tracking Betti-1 counts β1(ε,ℓ) across the two-parameter grid, and integrating them to define I = ∫∫ β1 dε dℓ, converts the spatial pattern of similar-shaped grains into a single number. The key mechanism is that a ring of fine grains surrounding a coarse grain completes a 1-cycle at relatively s

What would settle it

Take each ice-sample scan, keep the same number of grain centroids and the same image domain, but place the centroids uniformly at random (or as a Poisson process) with the observed shape-distance values assigned to them; compute the same bifiltration and I. If the sharp jump in I between 8% and 12% strain largely persists under random placement, the measure is tracking density or domain effects rather than ring topology; if the jump disappears, the necklace interpretation is supported.

Watch

Extended reading notes

Core claim

The central claim is that a bifiltration combining spatial Rips complexes with shape-distance sublevel sets — using the separable shape-tensor distance from the intrinsic mean as the second parameter — turns the heuristic notion of necklacing into a measurable topological signature: a plateau of elevated Betti-1 counts over a range of spatial scales, visible as an island in the (ε, ℓ) contour map. The scalar I = ∫∫ β1(ε,ℓ) dε dℓ accumulates this persistence and increases monotonically with strain, with a sharp transition between 8% and 12% strain that the authors identify with the onset of dynamic recrystallization as the dominant grain-refinement mechanism. The authors further argue that th

Load-bearing premise

The key load-bearing premise is that the scalar I captures necklace topology rather than merely reflecting the number of grain centroids: the grain count roughly doubles over the same 8%–12% strain range where I jumps, and the paper does not control for point density in its summary.

Editorial extensions

If this is right

  • Necklacing and other duplex grain arrangements can be assigned a quantitative scalar I, or region-specific integrals, providing a basis for ordering or clustering microstructure samples without manual grading.
  • Because the construction uses only grain centroids and shape distances, it can be applied to metals, ceramics, and composites imaged by EBSD or optical microscopy, where standard test methods currently rely on qualitative pattern descriptions.
  • The sharp rise in I between 8% and 12% strain localizes the strain range in which dynamic recrystallization becomes the dominant grain-refinement mechanism, offering a quantitative fingerprint of processing history.
  • Region-restricted integrals of β1 can be tuned to isolate specific topological patterns — necklacing, banding, duplex distributions — allowing standards to be extended with quantitative pattern-specific measures.
  • The scalar I and bifiltration surfaces can serve as reduced-order inputs or calibration targets for phase-field and cellular automata simulations of grain growth and recrystallization.

Reading between the lines

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

  • The paper does not normalize β1 or I by the number of grain centroids; since Rips Betti numbers grow with point count, a comparison against random point configurations matched for count and domain would clarify whether I measures ring topology or simply grain density. (Editorial extension.)
  • If the necklace plateau is the mechanistic signature, the same bifiltration could be used to detect percolation-like transitions in other spatial networks where a second attribute (shape, intensity, orientation) conditions connectivity.
  • The shape-distance axis could be replaced by other grain attributes — orientation, phase, or local misorientation — yielding bifiltrations tuned to different microstructural patterns, such as banding or clustering by crystallographic orientation.
  • A natural testable extension is to normalize I by the integral over a null ensemble of Poisson-distributed centroids; the strain transition's persistence under this normalization would separate topology from density effects.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a dual-parameter topological summary for microstructure images: a Rips bifiltration on grain centroids with a second axis given by sublevel sets of a Separable Shape Tensor (SST) shape distance from an intrinsic mean. It defines a scalar I as the double integral of the Betti-1 count over the two-parameter domain (Eq. 3). The method is demonstrated on four EBSD ice samples strained from 0.03 to 0.20. The central claim (Sections 1.2, 4.1, Fig. 12) is that I reveals a sharp transition in topological texture between 8% and 12% strain, coinciding with the onset of dynamic recrystallization and necklacing as described in ASTM E1181.

Significance. If the empirical claim were established, the framework would be a meaningful contribution: it combines shape and spatial arrangement in one interpretable summary, with no labels or threshold choices. The construction of the two-parameter filtration from SST distances is original and clearly explained, and the paper is generally readable. However, the demonstration is too weak to support the headline result: only four strain levels with one image each are used, no uncertainty quantification is provided, and the key scalar is not separated from a known confound with grain centroid density. As it stands, I is not a validated measure of necklacing topology.

major comments (3)
  1. [Section 4.1, Fig. 12, Eq. (3)] The central claim that I measures topological texture is confounded by grain count. I is a double integral of β1 over a Rips filtration on grain centroids. For any stationary point process, expected β1 is proportional to intensity; doubling the number of centroids roughly doubles I even if the spatial pattern is unchanged. The paper itself cites Fan et al. [42] reporting that grain number density more than doubles between 8% and 12% strain, exactly where I jumps. No normalization by grain count, no comparison with randomized configurations of the same N, and no spatial thinning are provided. The observed rise in I is therefore equally consistent with a density effect, and the claim of a topological transition is unsupported.
  2. [Section 4.1, Fig. 12] The statistical evidence is very limited. Only one EBSD scan is used per strain level, so Figure 12 has four points and no error bars. The 'sharp transition' rests on a comparison of one image at 8% and one at 12%. The 'plateau' between 12% and 20% also rests on two points. To support the monotonic trend and the location of a transition, the authors need replicated scans, bootstrap over subregions, or at least a statement of measurement uncertainty; otherwise the scalar I cannot be distinguished from sample-to-sample variation.
  3. [Section 2.1, Fig. 5; Section 4.1] The identification of β1 cycles with necklacing is not validated. The idealized Figure 5 shows one mechanism by which rings of fine grains create persistent holes, but Rips complexes on random point clouds also produce β1, especially as density increases. The paper provides no synthetic experiments with known necklace versus random arrangements at matched density. Consequently, the mechanistic interpretation in Section 4.1 (core-and-mantle structure, saturation of necklace topology) and the stronger claim in Section 5 that I is a 'direct, quantitative proxy' for creep and ductility are not supported by the evidence presented.
minor comments (5)
  1. [General/typos] Typos: 'difficult' (Section 1), 'recrystalization' and 'bewteen' (Section 4.1). Please proofread.
  2. [Eq. (1), Section 2.2] The dimensions in the product manifold Gr(d,q) × S^d_++ and the role of the low-dimensional submanifold principal directions are not fully defined. Since a,b,d and the number of principal directions are user choices, a sensitivity study or explicit default values are needed for reproducibility.
  3. [Section 5] The authors note that the construction 'may not necessarily reflect standard two-parameter TDA.' This caveat is important and should be moved earlier (e.g., Section 3.1) so that readers do not mistake the scalar I for a multiparameter persistence invariant such as a rank invariant.
  4. [Figure 7] The histograms are not clearly captioned; it is difficult to read which distance is plotted in each panel. Please label directly.
  5. [Reproducibility] Javaplex and public data are cited, but the grid resolution used for the bifiltration, the weights a,b in Eq. (1), and the SST rank d are not reported. For a methods paper, this information should be given so the scalar I can be reproduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: I is a defined summary statistic, not a fitted prediction, and the SST self-citations are not load-bearing.

full rationale

The derivation chain is explicit: SST shape distance (Eq. 1) feeds a sublevel Rips bifiltration (Eq. 2), whose Betti-1 counts are integrated to form I (Eq. 3). I is therefore a direct integral of computed β1 values, not a parameter fitted to the strain trend. The only empirically chosen parameters are the weights a and b in Eq. (1), which the paper states are 'selected empirically to normalize distances for the purposes of combining their distinct value ranges'; they are not tuned to reproduce the strain response or Fan et al.'s observations. The reported monotone increase and sharp rise in I (Fig. 12) is an observed output, not a fitted input renamed as a prediction. The SST machinery is taken from the authors' prior work ([33], [34]), but the paper does not invoke a self-cited uniqueness theorem, and it explicitly says in Section 5 that 'Alternate shape distances, such as those induced by elastic, information-geometric, or kernel-based metrics, may be substituted when required by the application.' Thus the self-citations are not load-bearing. The identification of necklacing with elevated β1 is an externally grounded modeling assumption tied to ASTM E1181's qualitative 'rings of significantly finer grains' definition, not a conclusion defined into existence. The most substantive criticism—that β1 in a Rips complex grows with point count and that Fan et al. report grain number density more than doubling in the 8–12% strain window—is a real statistical confound for the empirical claim, but it is not circularity under the stated definitions: I is not constructed from, or fitted to, Fan et al.'s density values, and no equation makes the claimed transition equal to an input. That concern belongs in a correctness/statistical-validity review, not in the circularity score.

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

The central measurement rests on several data-dependent choices (weights, SST rank, submanifold dimension, grid) and on the modeling choice to use centroids in Euclidean space as the spatial filtration. No new physical entities are introduced.

free parameters (4)
  • a,b (SST product metric weights) = not stated
    Weights in Eq. (1) selected empirically to normalize scale and undulation distances; values not reported.
  • SST rank d = not stated
    Rank of weighted-SVD decomposition of grain boundary curves; d controls the Grassmannian dimension.
  • Submanifold principal-direction count = not stated
    Number of dominant principal directions retained when restricting to a low-dimensional submanifold.
  • Bifiltration grid resolution = not stated
    Number of ε and ℓ grid points and their ranges affect the integral I.
assumptions (4)
  • standard math Rips–Vietoris filtration on Euclidean point clouds gives valid persistence intervals
    Used throughout Section 3; standard TDA.
  • domain assumption SST product manifold Gr(d,q) × S^d_{++} with Fréchet mean and log-map distances is a faithful shape representation
    Depends on [33,34]; the paper inherits this without re-deriving.
  • domain assumption Grain centroids and Euclidean distance adequately represent spatial arrangement for detecting necklace loops
    Section 3.1; ignores actual grain boundary geometry in the spatial filtration; centroids of irregular grains may not form clean rings.
  • standard math The sublevel Rips bifiltration S↑(f) is computable and its Betti counts can be summed over the grid
    Follows [37]; the integral I assumes finite grid approximation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Topology of Shape and Data in Material Microstructures." pith.science (2026). https://pith.science/paper/J2NRR6D6

@misc{pith2026260727493,
  author       = {Pith},
  title        = {Pith review of: Topology of Shape and Data in Material Microstructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J2NRR6D6}},
  note         = {Machine review of arXiv:2607.27493}
}
read the original abstract

One of the challenges in microstructure analysis is the rigorous quantification of the shape, size, and spatial arrangement of the microstructure beyond comparison of average values. We expound on formal principles combining Topological Data Analysis (TDA) and non-Euclidean distances between curves to motivate novel perspectives on the form and nature of pattern and shape in images. Specifically, TDA descriptors extracting persistent topological structures are combined with product submanifold learning of separable shape tensors (SST) to offer unique insights about electron backscatter diffraction (EBSD) images of material microstructures through the lens of a dual-parameter filtration. Beyond standard approaches, our methodology highlights how different choices or permutations of shape distances can lead to distinct notions of topological persistence, thereby broadening the interpretive scope of TDA. The resulting visualizations of feature extraction are designed to be both principled and explanatory, offering novel tools for modern imaging science with applications to material metrology. More broadly, this framework has strong potential to impact domains where precise quantification of topology and shape is critical for uncovering fundamental image patterns and features, and enables additional data-driven tools for microstructure analysis.

Figures

Figures reproduced from arXiv: 2607.27493 by the authors.

Figure 1
Figure 1. An example microstructure (PIL184 from [ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Four example microstructures of ice samples PIL254 (0.03 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Examples of simplices of various dimension (left) and an example 1-chain (right). The [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: For a given -ball distance, (left) the vertices with even indices (middle) form an equivalent cycle to those with odd indices (right). Both cycles are example representations from a single equiv￾alence class constituting the first homology group. That is, both highlig…
Figure 5
Figure 5. Figure 5: Illustration of necklacing and persistent holes using Rips complexes on grain centroids. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Two example microstructures of ice samples PIL184 (left) and PIL185 (right) from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Histograms with fit distributions to visualize the distinction in shape undulation distance [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Increasing sublevels of the product shape distance over the learned product manifold where [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Example bifiltration of PIL 185 as filled contour map. Four selected bifiltration points [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Bifiltration filled contour image created from the example microstructures of ice samples [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Ice microstructures with increasing strain are displayed from left to right as PIL254 (0.03 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Scalar bipersistence summary I (equation 3) as a function of applied axial strain for the four ice microstructure samples PIL254 (0.03 mm mm ), PIL184 (0.08 mm mm ), PIL185 (0.12 mm mm ), and PIL255 (0.20 mm mm ). cycles and overlaying them on EBSD images to identify …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 3 linked inside Pith

  1. [42]

    Temperature and strain controls on ice deformation mechanisms: insights from the microstructures of samples deformed to progressively higher strains at- 10,- 20 and- 30° c

    Sheng Fan, Travis F Hager, David J Prior, Andrew J Cross, David L Goldsby, Chao Qi, Marianne Negrini, and John Wheeler. Temperature and strain controls on ice deformation mechanisms: insights from the microstructures of samples deformed to progressively higher strains at- 10,- 20 and- 30° c. The Cryosphere , 14(11):3875–3905, 2020

    2020

  2. [1]

    Grey and Paul G

    Zachary J. Grey and Paul G. Constantine. Active subspaces of airfoil shape parameterizations. AIAA Journal , 56(5):2003–2017, 2018

    2003

  3. [2]

    Doronina, Zachary J

    Olga A. Doronina, Zachary J. Grey, and Andrew Glaws. G2aero: A python package for separable shape tensors. Journal of Open Source Software , 8(89):5408, 2023

    2023

  4. [3]

    Restoration of noisy orientation maps from electron backscatter diffraction imaging

    Emmanuel Atindama, Peter Lef, Günay Doğan, and Prashant Athavale. Restoration of noisy orientation maps from electron backscatter diffraction imaging. Integrating Materials and Manufacturing Innovation , 12(3):251–266, 2023

    2023

  5. [4]

    Inferential statistics of electron backscatter diffraction data from within individual crystalline grains

    Florian Bachmann, Ralf Hielscher, Peter E Jupp, Wolfgang Pantleon, Helmut Schaeben, and Elias Wegert. Inferential statistics of electron backscatter diffraction data from within individual crystalline grains. Journal of Applied Crystallography , 43(6):1338–1355, 2010. 16 IOP Publishing Modelling and Simulation in Materials Science and Engineering vv (yyyy...

    2010

  6. [5]

    Claire: A distributed-memory solver for constrained large deformation diffeomorphic image registration

    Andreas Mang, Amir Gholami, Christos Davatzikos, and George Biros. Claire: A distributed-memory solver for constrained large deformation diffeomorphic image registration. SIAM Journal on Scientific Computing , 41(5):C548–C584, 2019

    2019

  7. [6]

    Morphometry of anatomical shape complexes with dense deformations and sparse parameters

    Stanley Durrleman, Marcel Prastawa, Nicolas Charon, Julie R Korenberg, Sarang Joshi, Guido Gerig, and Alain Trouvé. Morphometry of anatomical shape complexes with dense deformations and sparse parameters. NeuroImage, 101:35–49, 2014

    2014

  8. [7]

    Generative models for functional data using phase and amplitude separation

    J Derek Tucker, Wei Wu, and Anuj Srivastava. Generative models for functional data using phase and amplitude separation. Computational Statistics & Data Analysis , 61:50–66, 2013

    2013

Show all 54 references
  1. [8]

    Coherence-based underwater target detection from multiple disparate sonar platforms

    J Derek Tucker and Mahmood R Azimi-Sadjadi. Coherence-based underwater target detection from multiple disparate sonar platforms. IEEE Journal of Oceanic Engineering , 36(1):37–51, 2011

    2011

  2. [9]

    Elastic shape analysis of surfaces with second-order sobolev metrics: a comprehensive numerical framework

    Emmanuel Hartman, Yashil Sukurdeep, Eric Klassen, Nicolas Charon, and Martin Bauer. Elastic shape analysis of surfaces with second-order sobolev metrics: a comprehensive numerical framework. International Journal of Computer Vision , 131(5):1183–1209, 2023

    2023

  3. [10]

    Shape analysis of elastic curves in euclidean spaces

    Anuj Srivastava, Eric Klassen, Shantanu H Joshi, and Ian H Jermyn. Shape analysis of elastic curves in euclidean spaces. IEEE transactions on pattern analysis and machine intelligence , 33(7):1415–1428, 2010

    2010

  4. [11]

    Testing equality of cell populations based on shape and geodesic distance

    Charles Hagwood, Javier Bernal, Michael Halter, John Elliott, and Tegan Brennan. Testing equality of cell populations based on shape and geodesic distance. IEEE Transactions on Medical Imaging, 32(12):2230–2237, 2013

    2013

  5. [12]

    Phg-net: Persistent homology guided medical image classification

    Yaopeng Peng, Hongxiao Wang, Milan Sonka, and Danny Z Chen. Phg-net: Persistent homology guided medical image classification. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision , pages 7583–7592, 2024

    2024

  6. [13]

    Train-free segmentation in mri with cubical persistent homology

    Anton François and Raphaël Tinarrage. Train-free segmentation in mri with cubical persistent homology. arXiv preprint arXiv:2401.01160 , 2024

    2024 arXiv

  7. [14]

    The persistent homology mathematical framework provides enhanced genotype-to-phenotype associations for plant morphology

    Mao Li, Margaret H Frank, Viktoriya Coneva, Washington Mio, Daniel H Chitwood, and Christopher N Topp. The persistent homology mathematical framework provides enhanced genotype-to-phenotype associations for plant morphology. Plant physiology , 177(4):1382–1395, 2018

    2018

  8. [15]

    Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features

    Talha Qaiser, Yee-Wah Tsang, Daiki Taniyama, Naoya Sakamoto, Kazuaki Nakane, David Epstein, and Nasir Rajpoot. Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features. Medical image analysis , 55:1–14, 2019

    2019

  9. [16]

    Persistent-homology-based machine learning and its applications–a survey

    Chi Seng Pun, Kelin Xia, and Si Xian Lee. Persistent-homology-based machine learning and its applications–a survey. arXiv preprint arXiv:1811.00252 , 2018

    2018 arXiv

  10. [17]

    A topological” reading” lesson: Classification of mnist using tda

    Adélie Garin and Guillaume Tauzin. A topological” reading” lesson: Classification of mnist using tda. In 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1551–1556. IEEE, 2019

    2019

  11. [18]

    Noise robustness of persistent homology on greyscale images, across filtrations and signatures

    Renata Turkeš, Jannes Nys, Tim Verdonck, and Steven Latré. Noise robustness of persistent homology on greyscale images, across filtrations and signatures. Plos one , 16(9):e0257215, 2021

    2021

  12. [19]

    Electron backscatter diffraction in materials science , volume 2

    Adam J Schwartz, Mukul Kumar, Brent L Adams, and David P Field. Electron backscatter diffraction in materials science , volume 2. Springer, 2009

    2009

  13. [20]

    Saville, Sven C

    Alec I. Saville, Sven C. Vogel, Adam Creuziger, Jake T. Benzing, Adam L. Pilchak, Peeyush Nandwana, Jonah Klemm-Toole, Kester D. Clarke, S. Lee Semiatin, and Amy J. Clarke. Texture evolution as a function of scan strategy and build height in electron beam melted ti-6al-4v. Add...

    2021

  14. [21]

    Location-specific microstructure characterization within in625 additive manufacturing benchmark test artifacts

    Mark R Stoudt, Maureen E Williams, Lyle E Levine, Adam Creuziger, Sandra A Young, Jarred C Heigel, Brandon M Lane, and Thien Q Phan. Location-specific microstructure characterization within in625 additive manufacturing benchmark test artifacts. Integrating Materials and Manufa...

    2020

  15. [22]

    Texture analysis with MTEX–free and open source software toolbox

    F Bachmann, Ralf Hielscher, and Helmut Schaeben. Texture analysis with MTEX–free and open source software toolbox. Solid state phenomena , 160:63–68, 2010

    2010

  16. [23]

    Prior, Andrew J

    Sheng Fan, David J. Prior, Andrew J. Cross, David L. Goldsby, Travis Hager, Marianne Negrini, and Chao Qi. EBSD data for synthetic polycrystalline pure water ice samples deformed at high homologous temperatures. FigShare, 12 2020

    2020

  17. [24]

    Using grain boundary irregularity to quantify dynamic recrystallization in ice

    Sheng Fan, David J Prior, Andrew J Cross, David L Goldsby, Travis F Hager, Marianne Negrini, and Chao Qi. Using grain boundary irregularity to quantify dynamic recrystallization in ice. Acta Materialia , 209:116810, 2021

    2021

  18. [25]

    Hielscher and H

    R. Hielscher and H. Schaeben. A novel pole figure inversion method: specification of the MTEX algorithm. Journal of Applied Crystallography , 41(6):1024–1037, Dec 2008

    2008

  19. [26]

    ASTM E1181-02R23: Test Methods for Characterizing Duplex Grain Sizes, 2023

    ASTM International. ASTM E1181-02R23: Test Methods for Characterizing Duplex Grain Sizes, 2023

    2023

  20. [27]

    ASTM E112-24: Test Methods for Determining A verage Grain Size, 2024

    ASTM International. ASTM E112-24: Test Methods for Determining A verage Grain Size, 2024

    2024

  21. [28]

    ASTM E1382-97R23: Test Methods for Determining A verage Grain Size Using Semiautomatic and Automatic Image Analysis, 2023

    ASTM International. ASTM E1382-97R23: Test Methods for Determining A verage Grain Size Using Semiautomatic and Automatic Image Analysis, 2023

    2023

  22. [29]

    ASTM E45-18a:Test Methods for Determining the Inclusion Content of Steel, 2023

    ASTM International. ASTM E45-18a:Test Methods for Determining the Inclusion Content of Steel, 2023

    2023

  23. [30]

    ASTM E562-19: Test Method for Determining Volume Fraction by Systematic Manual Point Count, 2019

    ASTM International. ASTM E562-19: Test Method for Determining Volume Fraction by Systematic Manual Point Count, 2019

    2019

  24. [31]

    ASTM E1245-03R23: Practice for Determining the Inclusion or Second-Phase Constituent Content of Metals by Automatic Image Analysis, 2023

    ASTM International. ASTM E1245-03R23: Practice for Determining the Inclusion or Second-Phase Constituent Content of Metals by Automatic Image Analysis, 2023

    2023

  25. [32]

    Graphs in statistical analysis

    Francis J Anscombe. Graphs in statistical analysis. The american statistician , 27(1):17–21, 1973

    1973

  26. [33]

    Separable shape tensors for aerodynamic design

    Zachary J Grey, Olga A Doronina, and Andrew Glaws. Separable shape tensors for aerodynamic design. Journal of Computational Design and Engineering , 10(1):468–487, 01 2023

    2023

  27. [34]

    Explainable binary classification of separable shape ensembles

    Zachary Grey, Nicholas Fisher, Andrew Glaws, and Gunay Dogan. Explainable binary classification of separable shape ensembles. arXiv preprint arXiv:2410.12994 , 2024

    2024

  28. [35]

    ASTM E930-18: Test Methods for Estimating the Largest Grain Observed in a Metallographic Section (ALA Grain Size), 2018

    ASTM International. ASTM E930-18: Test Methods for Estimating the Largest Grain Observed in a Metallographic Section (ALA Grain Size), 2018

    2018

  29. [36]

    Topological data analysis in medical imaging: current state of the art

    Yashbir Singh, Colleen M Farrelly, Quincy A Hathaway, Tim Leiner, Jaidip Jagtap, Gunnar E Carlsson, and Bradley J Erickson. Topological data analysis in medical imaging: current state of the art. Insights into Imaging , 14(1):58, 2023

    2023

  30. [37]

    An introduction to multiparameter persistence

    Magnus Bakke Botnan and Michael Lesnick. An introduction to multiparameter persistence. arXiv preprint arXiv:2203.14289 , 2022

    2022 arXiv

  31. [38]

    The theory of multidimensional persistence

    Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. In Proceedings of the twenty-third annual symposium on Computational geometry , pages 184–193, 2007

    2007

  32. [39]

    Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

    Oliver Vipond, Joshua A Bull, Philip S Macklin, Ulrike Tillmann, Christopher W Pugh, Helen M Byrne, and Heather A Harrington. Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors. Proceedings of the National Academy of Sciences, 118(41)...

    2021

  33. [40]

    A multi-parameter persistence framework for mathematical morphology

    Yu-Min Chung, Sarah Day, and Chuan-Shen Hu. A multi-parameter persistence framework for mathematical morphology. Scientific reports, 12(1):6427, 2022

    2022

  34. [41]

    Mix-geneo: A flexible filtration for multiparameter persistent homology detects digital images

    Jiaxing He, Bingzhe Hou, Tieru Wu, and Yue Xin. Mix-geneo: A flexible filtration for multiparameter persistent homology detects digital images. arXiv e-prints , pages arXiv–2401, 2024. 18 IOP Publishing Modelling and Simulation in Materials Science and Engineering vv (yyyy) aa...

    2024

  35. [43]

    Computational topology: an introduction

    Herbert Edelsbrunner and John L Harer. Computational topology: an introduction . American Mathematical Society, 2022

    2022

  36. [44]

    Topology and data

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

    2009

  37. [45]

    Computing persistent homology

    Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. In Proceedings of the twentieth annual symposium on Computational geometry , pages 347–356, 2004

    2004

  38. [46]

    Algebraic topology

    Allen Hatcher. Algebraic topology. Cambridge Univeristy Press, 2002

    2002

  39. [47]

    Randomness of shapes and statistical inference on shapes via the smooth euler characteristic transform

    Kun Meng, Jinyu Wang, Lorin Crawford, and Ani Eloyan. Randomness of shapes and statistical inference on shapes via the smooth euler characteristic transform. Journal of the American Statistical Association, (just-accepted):1–25, 2024

    2024

  40. [48]

    Barcodes: the persistent topology of data

    Robert Ghrist. Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61–75, 2008

    2008

  41. [49]

    Fast construction of the vietoris-rips complex

    Afra Zomorodian. Fast construction of the vietoris-rips complex. Computers & Graphics , 34(3):263–271, 2010

    2010

  42. [50]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds . Princeton University Press, 6 2008

    2008

  43. [51]

    Statistics of shape via principal geodesic analysis on Lie groups

    P Thomas Fletcher, Conglin Lu, and Sarang Joshi. Statistics of shape via principal geodesic analysis on Lie groups. In 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings., volume 1, pages I–I. IEEE, 2003

    2003

  44. [52]

    Javaplex: A research software package for persistent (co) homology

    Henry Adams, Andrew Tausz, and Mikael Vejdemo-Johansson. Javaplex: A research software package for persistent (co) homology. In Mathematical Software–ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 4 , pages 129–136. Springer, 2014

    2014

  45. [53]

    Black, J.C

    Malo Jullien, R.L. Black, J.C. Stinville, Marc Legros, and Damien Texier. Grain size effect on strain localization, slip-grain boundary interaction and damage in the alloy 718 ni-based superalloy at 650 °c. Materials Science and Engineering: A , 912:146927, 2024

    2024

  46. [54]

    Mikhaylovskaya, O.A

    A.V. Mikhaylovskaya, O.A. Yakovtseva, and A.V. Irzhak. The role of grain boundary sliding and intragranular deformation mechanisms for a steady stage of superplastic flow for al–mg-based alloys. Materials Science and Engineering: A , 833:142524, 2022. A TDA F ormalism A.1 Homo...

    2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.