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Morse theory-based segmentation and fabric quantification of granular materials

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Abstract

This article presents a robust Morse theory-based framework for segmenting 3D X-ray computed tomography image (CT) and computing the fabric, relative arrangement of particles, of granular ensembles. The framework includes an algorithm for computing the segmentation, a data structure for storing the segmentation and representing both individual particles and the connectivity network, and visualizations of topological descriptors of the CT image that enable interactive exploration. The Morse theory-based framework produces superior quality segmentation of a granular ensemble as compared to prior approaches based on the watershed transform. The accuracy of the connectivity network also improves. Further, the framework supports the efficient computation of various distribution statistics on the segmentation and the connectivity network. Such a comprehensive characterization and quantification of the fabric of granular ensembles is the first step towards a multiple length scale understanding of the behavior.

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References

  1. Andò, E.: Experimental Investigation of Microstructural Changes in Deforming Granular Media Using X-ray Tomography. Theses, Université de Grenoble (2013)

  2. Andò, E., Hall, S.A., Viggiani, G., Desrues, J., Bésuelle, P.: Experimental micromechanics: grain-scale observation of sand deformation. Géotech. Lett. 2(3), 107–112 (2012). https://doi.org/10.1680/geolett.12.00027

    Article  Google Scholar 

  3. Andò, E., Hall, S.A., Viggiani, G., Desrues, J., Bésuelle, P.: Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotech. 7(1), 1–13 (2012). https://doi.org/10.1007/s11440-011-0151-6

    Article  Google Scholar 

  4. Baule, A., Morone, F., Herrmann, H.J., Makse, H.A.: Edwards statistical mechanics for jammed granular matter. Rev. Mod. Phys. 90(1), 015006 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  5. Bernal, J., Mason, J.: Packing of spheres: co-ordination of randomly packed spheres. Nature 188(4754), 910–911 (1960)

    Article  ADS  Google Scholar 

  6. Bhatia, H., Gyulassy, A.G., Lordi, V., Pask, J.E., Pascucci, V., Bremer, P.T.: Topoms: comprehensive topological exploration for molecular and condensed-matter systems. J. Comput. Chem. 39(16), 936–952 (2018)

    Article  Google Scholar 

  7. Burland, J.B.: On the compressibility and shear strength of natural clays. Géotechnique 40(3), 329–378 (1990). https://doi.org/10.1680/geot.1990.40.3.329

    Article  Google Scholar 

  8. Chan, T., Vese, L.: An active contour model without edges. In: International conference on scale-space theories in computer vision, pp. 141–151. Springer (1999)

  9. Chazal, F., Michel, B.: An introduction to topological data analysis: fundamental and practical aspects for data scientists. arXiv preprint arXiv:1710.04019 (2017)

  10. Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2014)

    Article  Google Scholar 

  11. Desrues, J., Chambon, R., Mokni, M., Mazerolle, F.: Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. Géotechnique 46(3), 529–546 (1996). https://doi.org/10.1680/geot.1996.46.3.529

    Article  Google Scholar 

  12. Donev, A., Connelly, R., Stillinger, F.H., Torquato, S.: Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids. Phys. Rev. E 75(5), 051304 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  13. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Soc (2010)

    MATH  Google Scholar 

  14. Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci., V.: Morse-Smale complexes for piecewise linear 3-manifolds. In: Proc. 19th Ann. symposium on computational geometry, pp. 361–370 (2003)

  15. Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse complexes for piecewise linear 2-manifolds. In: Proceedings of the seventeenth annual symposium on computational geometry, pp. 70–79 (2001)

  16. Fonseca, J., O‘Sullivan, C., Coop, M.R., Lee, P.D.: Non-invasive characterization of particle morphology of natural sands. Soils Found. 52(4), 712–722 (2012). https://doi.org/10.1016/j.sandf.2012.07.011

    Article  Google Scholar 

  17. Forman, R.: A user‘s guide to discrete morse theory. Sém. Lothar. Combin. 48, 35 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Günther, D., Boto, R.A., Contreras-Garcia, J., Piquemal, J.P., Tierny, J.: Characterizing molecular interactions in chemical systems. IEEE Trans. Vis. Comput. Graphics 20(12), 2476–2485 (2014)

    Article  Google Scholar 

  19. Gyulassy, A., Duchaineau, M., Natarajan, V., Pascucci, V., Bringa, E., Higginbotham, A., Hamann, B.: Topologically clean distance fields. IEEE Trans. Vis. Comput. Graphics 13(6), 1432–1439 (2007)

    Article  Google Scholar 

  20. Hall, S.A., Bornert, M., Desrues, J., Pannier, Y., Lenoir, N., Viggiani, G., Bésuelle, P.: Discrete and continuum analysis of localised deformation in sand using x-ray micro-CT and volumetric digital image correlation. Géotechnique 60(5), 315–322 (2010). https://doi.org/10.1680/geot.2010.60.5.315

    Article  Google Scholar 

  21. Hasan, A., Alshibli, K.: Three dimensional fabric evolution of sheared sand. Granul. Matter 14(4), 469–482 (2012). https://doi.org/10.1007/s10035-012-0353-0

    Article  Google Scholar 

  22. Herring, A., Robins, V., Sheppard, A.: Topological persistence for relating microstructure and capillary fluid trapping in sandstones. Water Resour. Res. 55(1), 555–573 (2019)

    Article  ADS  Google Scholar 

  23. Hsieh, J.: Computed Tomography: Principles, Design, Artifacts And Recent Advances. SPIE press (2003)

    Google Scholar 

  24. Imseeh, W.H., Druckrey, A.M., Alshibli, K.A.: 3d experimental quantification of fabric and fabric evolution of sheared granular materials using synchrotron micro-computed tomography. Granul. Matter (2018). https://doi.org/10.1007/s10035-018-0798-x

    Article  Google Scholar 

  25. Kader, M., Brown, A., Hazell, P., Robins, V., Escobedo, J., Saadatfar, M.: Geometrical and topological evolution of a closed-cell aluminium foam subject to drop-weight impact: an x-ray tomography study. Int. J. Impact Eng 139, 103510 (2020)

    Article  Google Scholar 

  26. Kong, D., Fonseca, J.: Quantification of the morphology of shelly carbonate sands using 3d images. Géotechnique 68(3), 249–261 (2018). https://doi.org/10.1680/jgeot.16.P.278

    Article  Google Scholar 

  27. Krissian, K., Westin, C.F.: Fast and accurate redistancing for level set methods. Comput. Aided Syst. Theory (EUROCAST’03) pp. 48–51 (2003)

  28. Matsumoto, Y.: An Introduction to Morse Theory. Amer. Math. Soc.: Translated from Japanese by K. Hudson and M, Saito (2002)

  29. McCormick, M.M., Liu, X., Ibanez, L., Jomier, J., Marion, C.: Itk: enabling reproducible research and open science. Front. Neuroinform. 8, 13 (2014)

    Article  Google Scholar 

  30. Meyer, F., Beucher, S.: Morphological segmentation. J. Vis. Commun. Image Represent. 1(1), 21–46 (1990)

    Article  Google Scholar 

  31. Oda, M.: Initial fabrics and their relations to mechanical properties of granular material. Soils Found. 12(1), 17–36 (1972)

    Article  Google Scholar 

  32. Oda, M.: Fabric tensor for discontinuous geological materials. Soils Found. 22(4), 96–108 (1982). https://doi.org/10.3208/sandf1972.22.4_96

    Article  Google Scholar 

  33. Oda, M., Konishi, J., Nemat-Nasser, S.: Experimental micromechanical evaluation of strength of granular materials: Effects of particle rolling. Mech. Mater. 1(4), 269–283 (1982). https://doi.org/10.1016/0167-6636(82)90027-8

    Article  Google Scholar 

  34. Oda, M., Nemat-Nasser, S., Konishi, J.: Stress-induced anisotropy in granular masses. Soils Found. 25(3), 85–97 (1985). https://doi.org/10.3208/sandf1972.25.3_85

    Article  Google Scholar 

  35. Otsu, N.: A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979)

    Article  Google Scholar 

  36. Robins, V., Wood, P.J., Sheppard, A.P.: Theory and algorithms for constructing discrete morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)

    Article  Google Scholar 

  37. Roerdink, J.B., Meijster, A.: The watershed transform: Definitions, algorithms and parallelization strategies. Fundamenta Informa. 41(1, 2), 187–228 (2000)

    Article  MathSciNet  Google Scholar 

  38. Saadatfar, M., Takeuchi, H., Robins, V., Francois, N., Hiraoka, Y.: Pore configuration landscape of granular crystallization. Nat. Commun. 8(1), 1–11 (2017)

    Article  Google Scholar 

  39. Schroeder, W., Martin, K., Lorensen, B.: The Visualization Toolkit, 4th edn. kitware, New York (2006)

  40. Shivashankar, N., Natarajan, V.: Parallel computation of 3D Morse-Smale complexes. Comput. Graph. Forum 31(3), 965–974 (2012)

    Article  Google Scholar 

  41. Shivashankar, N., Natarajan, V.: Efficient software for programmable visual analysis using Morse-Smale complexes. Topol. Methods Data Anal. Visual. (2017). https://doi.org/10.1007/978-3-319-44684-4_19

    Article  MATH  Google Scholar 

  42. Shivashankar, N., Pranav, P., Natarajan, V., van de Weygaert, R., Bos, E.P., Rieder, S.: Felix: a topology based framework for visual exploration of cosmic filaments. IEEE Trans. Visual Comput. Graphics 22(6), 1745–1759 (2015)

    Article  Google Scholar 

  43. Singh, S.: Weakly cemented granular materials: study at multiple length scales. Ph.D. thesis, Indian Institute of Science, Bengaluru, Department of Civil Engineering (2020)

  44. Singh, S., Miers, J., Saldana, C., Murthy, T.: Quantification of fabric in cemented granular materials. Comput. Geotech. 125, 103644 (2020). https://doi.org/10.1016/j.compgeo.2020.103644

    Article  Google Scholar 

  45. Soille, P.: Morphological Image Analysis: Principles and Applications. Springer (1999)

    Book  Google Scholar 

  46. Sowers, G.B., Sowers, G.F.: Introductory soil mechanics and foundations. LWW (1951). https://doi.org/10.1097/00010694-195111000-00014

    Article  Google Scholar 

  47. Van der Walt, S., Schönberger, J.L., Nunez-Iglesias, J., Boulogne, F., Warner, J.D., Yager, N., Gouillart, E., Yu, T.: scikit-image: image processing in python. PeerJ 2, e453 (2014)

    Article  Google Scholar 

  48. Wiebicke, M., Andò, E., Šmilauer, V., Herle, I., Viggiani, G.: A benchmark strategy for the experimental measurement of contact fabric. Granul. Matter 21(3), 1–13 (2019). https://doi.org/10.1007/s10035-019-0902-x

    Article  Google Scholar 

  49. Yang, Z.X., Li, X.S., Yang, J.: Quantifying and modelling fabric anisotropy of granular soils. Géotechnique 58(4), 237–248 (2008). https://doi.org/10.1680/geot.2008.58.4.237

    Article  Google Scholar 

  50. Zheng, J., Hryciw, R.D.: Segmentation of contacting soil particles in images by modified watershed analysis. Comput. Geotech. 73, 142–152 (2016). https://doi.org/10.1016/j.compgeo.2015.11.025

    Article  Google Scholar 

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Acknowledgements

This work is supported by an Indo-Swedish joint network Project (DST/ INT/ SWD/VR/P-02/2019) and Swedish Research Council (VR) Grant 2018-07085 and the VR Grant 2019-05487. KP and VN are partially supported by a Swarnajayanti Fellowship from the Department of Science and Technology, India (DST/SJF/ETA-02/2015-16) and a Mindtree Chair research grant.

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Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

    Karran Pandey & Vijay Natarajan

  2. Department of Science and Technology (ITN), Linköping University, Norrköping, Sweden

    Talha Bin Masood & Ingrid Hotz

  3. Department of Civil Engineering, Indian Institute of Science, Bangalore, India

    Saurabh Singh & Tejas G. Murthy

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  1. Karran Pandey
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Correspondence to Tejas G. Murthy.

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Pandey, K., Bin Masood, T., Singh, S. et al. Morse theory-based segmentation and fabric quantification of granular materials. Granular Matter 24, 27 (2022). https://doi.org/10.1007/s10035-021-01182-7

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