Abstract

Although x-ray tomography is commonly used to characterize the three-dimensional structure of materials, sometimes this is impractical due either to limited time for data collection (such as in rapidly-evolving systems) or the need to limit the radiation exposure of the sample. In such situations, it is desirable to extract as much information as possible from a more limited data set. In this paper, we describe how to extract the size distribution of non-spherical pores (or, equivalently, particles) from single x-ray phase contrast imaging (XPCI). Because the pores overlap in projection, interpreting the images and extracting quantitative information about the size distribution is non-trivial. In this paper we extend a previously-developed Fourier-based framework for interpreting the speckle pattern of XPCI images from materials with spherical pores to the more challenging case of non-spherical pores. We develop an analytical expression for the XPCI image from a distribution of randomly-oriented ellipsoidal pores, and show that we can use this expression to extract quantitative information about the size distribution from single images. We discuss three approaches to evaluating this expression, corresponding to different assumptions about the nature of the size distribution, and validate our results with simulated XPCI images and experimental data from Berea sandstone.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (4)

J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
[Crossref] [PubMed]

J. A. Dijksman, N. Brodu, and R. P. Behringer, “Refractive index matched scanning and detection of soft particle,” Rev. Sci. Instrum. 88, 051807 (2017).
[Crossref]

F. Guillard, B. Marks, and I. Einav, “Dynamic X-ray radiography reveals particle size and shape orientation fields during granular flow,” Sci. Rep. 7, 8155 (2017).
[Crossref] [PubMed]

A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
[Crossref]

2016 (4)

K. Heim, F. Bernier, R. Pelletier, and L. P. Lefebvre, “High resolution pore size analysis in metallic powders by X-ray tomography,” Case Stud. Nondestruct. Test. Eval. 6, 45–52 (2016).
[Crossref]

L. Thwala, “Protamine nanocapsules as carriers for oral peptide delivery,” J. Control. Release 291, 157–168 (2016).
[Crossref]

P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
[Crossref]

D. M. Cooper, C. E. Kawalilak, K. Harrison, B. D. Johnston, and J. D. Johnston, “Cortical bone porosity: What is it, why is it important, and how can we detect it?” Curr. Osteoporos. Rep. 14, 187–198 (2016).
[Crossref] [PubMed]

2015 (2)

M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
[Crossref] [PubMed]

B. Cagnoli and A. Piersanti, “Grain size and flow volume effects on granular flow mobility in numerical simulations: 3-D discrete element modeling of flows of angular rock fragments,” J. Geophys. Res-Sol Ea 120, 2350–2366 (2015).
[Crossref]

2014 (3)

O. Mohnke and B. Hughes, “Jointly deriving NMR surface relaxivity and pore size distributions by NMR relaxation experiments on partially desaturated rocks,” Water Resour. Res. 50, 5309–5321 (2014).
[Crossref]

Y. I. Nesterets and T. E. Gureyev, “Noise propagation in x-ray phase-contrast imaging and computed tomography,” J. Phys. D. Appl. Phys. 47, 105402 (2014).
[Crossref]

A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
[Crossref] [PubMed]

2013 (2)

R. P. Carnibella, M. J. Kitchen, and A. Fouras, “Decoding the structure of granular and porous materials from speckled phase contrast X-ray images,” Opt. Express 21, 19153–19162 (2013).
[Crossref] [PubMed]

A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
[Crossref]

2012 (2)

R. P. Carnibella, M. J. Kitchen, and A. Fouras, “Determining particle size distributions from a single projection image,” Opt. Express 20, 15962–15968 (2012).
[Crossref] [PubMed]

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
[Crossref] [PubMed]

2011 (3)

J. Q. Shi, Z. Xue, and S. Durucan, “Supercritical CO2 core flooding and imbibition in Berea sandstone - CT imaging and numerical simulation,” Energy Procedia 4, 5001–5008 (2011).
[Crossref]

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

M. Sanchez del Rio and R. J. Dejus, “XOP v2.4 : Recent developments of the X-ray optics software toolkit,” Proc. SPIE 8141, 814115 (2011).
[Crossref]

2009 (1)

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
[Crossref]

2008 (1)

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
[Crossref]

2007 (1)

M. Toiya, J. Hettinga, and W. Losert, “3D Imaging of particle motion during penetrometer testing: From microscopic to macroscopic soil mechanics,” Granul. Matter 9, 323–329 (2007).
[Crossref]

2006 (1)

O. Šolcová, L. Matějová, and P. Schneider, “Pore-size distributions from nitrogen adsorption revisited: Models comparison with controlled-pore glasses,” Appl. Catal. A Gen. 313, 167–176 (2006).
[Crossref]

2004 (2)

2003 (1)

P. Fratzl, “Small-angle scattering in materials science - A short review of applications in alloys, ceramics and composite materials,” J. Appl. Crystallogr. 36, 397–404 (2003).
[Crossref]

2002 (2)

S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
[Crossref]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

2001 (1)

K. Sing, “The use of nitrogen adsorption for the characterisation of porous materials,” Colloid Surf. A 187-188, 3–9 (2001).
[Crossref]

2000 (1)

W. B. Lindquist, A. Venkatarangan, J. Dunsmuir, and T. F. Wong, “Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones,” J. Geophys. Res-Sol Ea 105, 21509–21527 (2000).
[Crossref]

1998 (1)

M. Matsumoto and T. Nishimura, “Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
[Crossref]

1997 (1)

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
[Crossref]

1996 (1)

C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
[Crossref]

1986 (1)

W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
[Crossref]

1977 (1)

M. M. Hall Jnr, V. G. Veeraraghavan, H. Rubin, and P. G. Winchell, “The approximation of symmetric x-ray peaks by Pearson type VII distributions,” J. Appl. Crystallogr. 10, 66–68 (1977).
[Crossref]

Allen, A. J.

J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
[Crossref]

Allison, B. J.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

Baby, A. R.

P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
[Crossref]

Balaur, E.

A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
[Crossref]

Baruchel, J.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
[Crossref]

Bazin, J.-E.

J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
[Crossref] [PubMed]

Behringer, R. P.

J. A. Dijksman, N. Brodu, and R. P. Behringer, “Refractive index matched scanning and detection of soft particle,” Rev. Sci. Instrum. 88, 051807 (2017).
[Crossref]

Bernier, F.

K. Heim, F. Bernier, R. Pelletier, and L. P. Lefebvre, “High resolution pore size analysis in metallic powders by X-ray tomography,” Case Stud. Nondestruct. Test. Eval. 6, 45–52 (2016).
[Crossref]

Böck, H.

C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
[Crossref]

Bösecke, P.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
[Crossref]

Brodu, N.

J. A. Dijksman, N. Brodu, and R. P. Behringer, “Refractive index matched scanning and detection of soft particle,” Rev. Sci. Instrum. 88, 051807 (2017).
[Crossref]

Buckley, G. A.

M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
[Crossref] [PubMed]

A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
[Crossref] [PubMed]

Buffiére, J. Y.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
[Crossref]

Cagnoli, B.

B. Cagnoli and A. Piersanti, “Grain size and flow volume effects on granular flow mobility in numerical simulations: 3-D discrete element modeling of flows of angular rock fragments,” J. Geophys. Res-Sol Ea 120, 2350–2366 (2015).
[Crossref]

Carnibella, R. P.

Carroll, A. J.

A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
[Crossref]

Cerbino, R.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
[Crossref]

Cevallos-Candau, J. F.

W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
[Crossref]

Cho, H.

S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
[Crossref]

Churcher, P.

P. Churcher, P. French, J. Shaw, and L. Schramm, “Rock properties of Berea sandstone, Baker Dolomite, and Indiana limestone,” in SPE International Symposium on Oilfield Chemistry, (Society of Petroleum Engineers, 1991).
[Crossref]

Cloetens, P.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
[Crossref]

Conner, W. C.

W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
[Crossref]

Consiglieri, V. O.

P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
[Crossref]

Constantin, J. M.

J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
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D. M. Cooper, C. E. Kawalilak, K. Harrison, B. D. Johnston, and J. D. Johnston, “Cortical bone porosity: What is it, why is it important, and how can we detect it?” Curr. Osteoporos. Rep. 14, 187–198 (2016).
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W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
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P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
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W. B. Lindquist, A. Venkatarangan, J. Dunsmuir, and T. F. Wong, “Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones,” J. Geophys. Res-Sol Ea 105, 21509–21527 (2000).
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J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
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R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
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J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
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Y. I. Nesterets and T. E. Gureyev, “Noise propagation in x-ray phase-contrast imaging and computed tomography,” J. Phys. D. Appl. Phys. 47, 105402 (2014).
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D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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M. M. Hall Jnr, V. G. Veeraraghavan, H. Rubin, and P. G. Winchell, “The approximation of symmetric x-ray peaks by Pearson type VII distributions,” J. Appl. Crystallogr. 10, 66–68 (1977).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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D. M. Cooper, C. E. Kawalilak, K. Harrison, B. D. Johnston, and J. D. Johnston, “Cortical bone porosity: What is it, why is it important, and how can we detect it?” Curr. Osteoporos. Rep. 14, 187–198 (2016).
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Heim, K.

K. Heim, F. Bernier, R. Pelletier, and L. P. Lefebvre, “High resolution pore size analysis in metallic powders by X-ray tomography,” Case Stud. Nondestruct. Test. Eval. 6, 45–52 (2016).
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M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
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A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
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A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
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M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
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O. Mohnke and B. Hughes, “Jointly deriving NMR surface relaxivity and pore size distributions by NMR relaxation experiments on partially desaturated rocks,” Water Resour. Res. 50, 5309–5321 (2014).
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J. M. Constantin, T. Godet, M. Jabaudon, J.-E. Bazin, and E. Futier, “Recruitment maneuvers in acute respiratory distress syndrome,” Ann. Transl. Med. 5, 290 (2017).
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J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
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D. M. Cooper, C. E. Kawalilak, K. Harrison, B. D. Johnston, and J. D. Johnston, “Cortical bone porosity: What is it, why is it important, and how can we detect it?” Curr. Osteoporos. Rep. 14, 187–198 (2016).
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P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
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Kawalilak, C. E.

D. M. Cooper, C. E. Kawalilak, K. Harrison, B. D. Johnston, and J. D. Johnston, “Cortical bone porosity: What is it, why is it important, and how can we detect it?” Curr. Osteoporos. Rep. 14, 187–198 (2016).
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M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
[Crossref] [PubMed]

A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
[Crossref] [PubMed]

A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
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R. P. Carnibella, M. J. Kitchen, and A. Fouras, “Decoding the structure of granular and porous materials from speckled phase contrast X-ray images,” Opt. Express 21, 19153–19162 (2013).
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R. P. Carnibella, M. J. Kitchen, and A. Fouras, “Determining particle size distributions from a single projection image,” Opt. Express 20, 15962–15968 (2012).
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M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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K. Heim, F. Bernier, R. Pelletier, and L. P. Lefebvre, “High resolution pore size analysis in metallic powders by X-ray tomography,” Case Stud. Nondestruct. Test. Eval. 6, 45–52 (2016).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
[Crossref] [PubMed]

A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
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A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
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J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
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L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle X-ray scattering as a contrast mechanism,” J. Appl. Crystallogr. 37, 757–765 (2004).
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M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
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W. B. Lindquist, A. Venkatarangan, J. Dunsmuir, and T. F. Wong, “Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones,” J. Geophys. Res-Sol Ea 105, 21509–21527 (2000).
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J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
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L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle X-ray scattering as a contrast mechanism,” J. Appl. Crystallogr. 37, 757–765 (2004).
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M. Toiya, J. Hettinga, and W. Losert, “3D Imaging of particle motion during penetrometer testing: From microscopic to macroscopic soil mechanics,” Granul. Matter 9, 323–329 (2007).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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Marks, B.

F. Guillard, B. Marks, and I. Einav, “Dynamic X-ray radiography reveals particle size and shape orientation fields during granular flow,” Sci. Rep. 7, 8155 (2017).
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O. Šolcová, L. Matějová, and P. Schneider, “Pore-size distributions from nitrogen adsorption revisited: Models comparison with controlled-pore glasses,” Appl. Catal. A Gen. 313, 167–176 (2006).
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M. Matsumoto and T. Nishimura, “Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
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W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
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D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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Mohnke, O.

O. Mohnke and B. Hughes, “Jointly deriving NMR surface relaxivity and pore size distributions by NMR relaxation experiments on partially desaturated rocks,” Water Resour. Res. 50, 5309–5321 (2014).
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Y. I. Nesterets and T. E. Gureyev, “Noise propagation in x-ray phase-contrast imaging and computed tomography,” J. Phys. D. Appl. Phys. 47, 105402 (2014).
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Nishimura, T.

M. Matsumoto and T. Nishimura, “Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
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Paganin, D.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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A. F. T. Leong, G. A. Buckley, D. M. Paganin, S. B. Hooper, M. J. Wallace, and M. J. Kitchen, “Real-time measurement of alveolar size and population using phase contrast x-ray imaging,” Biomed. Opt. Express 5, 4024–4038 (2014).
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A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
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M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
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W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
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S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
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Pateyron-Salomé, M.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
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Pavlov, K. M.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
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A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
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L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
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P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
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Pelletier, R.

K. Heim, F. Bernier, R. Pelletier, and L. P. Lefebvre, “High resolution pore size analysis in metallic powders by X-ray tomography,” Case Stud. Nondestruct. Test. Eval. 6, 45–52 (2016).
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Peres, D. D.

P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
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R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
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Peyrin, F.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
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Potenza, M. A. C.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing(Cambridge University Press, 1988).

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P. S. Prestes, D. D. Peres, A. Z. de Freitas, V. O. Consiglieri, T. M. Kaneko, M. V. R. Velasco, and A. R. Baby, “Particle size and morphological characterization of cosmetic emulsified systems by optical coherence tomography (OCT),” Brazilian J. Pharm. Sci. 52, 273–280 (2016).
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Robert, A.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008).
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Rubin, H.

M. M. Hall Jnr, V. G. Veeraraghavan, H. Rubin, and P. G. Winchell, “The approximation of symmetric x-ray peaks by Pearson type VII distributions,” J. Appl. Crystallogr. 10, 66–68 (1977).
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M. Sanchez del Rio and R. J. Dejus, “XOP v2.4 : Recent developments of the X-ray optics software toolkit,” Proc. SPIE 8141, 814115 (2011).
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Schlenker, M.

P. Cloetens, M. Pateyron-Salomé, J. Y. Buffiére, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878 (1997).
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Schneider, P.

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Schramm, L.

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[Crossref]

Shaw, J.

P. Churcher, P. French, J. Shaw, and L. Schramm, “Rock properties of Berea sandstone, Baker Dolomite, and Indiana limestone,” in SPE International Symposium on Oilfield Chemistry, (Society of Petroleum Engineers, 1991).
[Crossref]

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J. Q. Shi, Z. Xue, and S. Durucan, “Supercritical CO2 core flooding and imbibition in Berea sandstone - CT imaging and numerical simulation,” Energy Procedia 4, 5001–5008 (2011).
[Crossref]

Shoemake, K.

K. Shoemake, “Uniform random rotations,” in Graphics Gems III, D. Kirk, ed. (Academic Press, 1992), pp. 124–132.

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A. F. T. Leong, D. M. Paganin, S. B. Hooper, M. L. Siew, and M. J. Kitchen, “Measurement of absolute regional lung air volumes from near-field x-ray speckles,” Opt. Express 21, 777–786 (2013).
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O. Šolcová, L. Matějová, and P. Schneider, “Pore-size distributions from nitrogen adsorption revisited: Models comparison with controlled-pore glasses,” Appl. Catal. A Gen. 313, 167–176 (2006).
[Crossref]

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S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
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Svalbe, I.

A. Kingston and I. Svalbe, “Projective transforms on periodic discrete image arrays,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Elsevier Science, 2011), pp. 78–172.

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing(Cambridge University Press, 1988).

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L. Thwala, “Protamine nanocapsules as carriers for oral peptide delivery,” J. Control. Release 291, 157–168 (2016).
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[Crossref]

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M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

Van Riessen, G. A.

A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
[Crossref]

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M. M. Hall Jnr, V. G. Veeraraghavan, H. Rubin, and P. G. Winchell, “The approximation of symmetric x-ray peaks by Pearson type VII distributions,” J. Appl. Crystallogr. 10, 66–68 (1977).
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M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
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J. Q. Shi, Z. Xue, and S. Durucan, “Supercritical CO2 core flooding and imbibition in Berea sandstone - CT imaging and numerical simulation,” Energy Procedia 4, 5001–5008 (2011).
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S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
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[Crossref] [PubMed]

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C. Degueldre, H. Pleinert, P. Maguire, E. Lehman, J. Missimer, J. Hammer, K. Leenders, H. Böck, and D. Townsend, “Porosity and pathway determination in crystalline rock by positron emission tomography and neutron radiography,” Earth Planet. Sci. Lett. 140, 213–225 (1996).
[Crossref]

Energy Procedia (1)

J. Q. Shi, Z. Xue, and S. Durucan, “Supercritical CO2 core flooding and imbibition in Berea sandstone - CT imaging and numerical simulation,” Energy Procedia 4, 5001–5008 (2011).
[Crossref]

Granul. Matter (1)

M. Toiya, J. Hettinga, and W. Losert, “3D Imaging of particle motion during penetrometer testing: From microscopic to macroscopic soil mechanics,” Granul. Matter 9, 323–329 (2007).
[Crossref]

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P. Fratzl, “Small-angle scattering in materials science - A short review of applications in alloys, ceramics and composite materials,” J. Appl. Crystallogr. 36, 397–404 (2003).
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J. Ilavsky, P. R. Jemian, A. J. Allen, F. Zhang, L. E. Levine, and G. G. Long, “Ultra-small-angle X-ray scattering at the Advanced Photon Source,” J. Appl. Crystallogr. 42, 469–479 (2009).
[Crossref]

L. E. Levine and G. G. Long, “X-ray imaging with ultra-small-angle X-ray scattering as a contrast mechanism,” J. Appl. Crystallogr. 37, 757–765 (2004).
[Crossref]

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[Crossref]

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[Crossref]

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D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

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A. J. Carroll, G. A. Van Riessen, E. Balaur, I. P. Dolbnya, G. N. Tran, and A. G. Peele, “An iterative method for near-field Fresnel region polychromatic phase contrast imaging,” J. Opt. 19, 075033 (2017).
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W. C. Conner, J. F. Cevallos-Candau, E. L. Weist, J. Pajares, S. Mendioroz, and A. Cortes, “Characterization of pore structure: porosimetry and sorption,” Langmuir 2, 151–154 (1986).
[Crossref]

Materials (1)

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
[Crossref] [PubMed]

Meas. Sci. Technol. (1)

S. Park, H. Cho, I. Yoon, and K. Min, “Measurement of droplet size distribution of gasoline direct injection spray by droplet generator and planar image technique,” Meas. Sci. Technol. 13, 859–864 (2002).
[Crossref]

Nat. Phys. (1)

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[Crossref]

Opt. Express (4)

Phys. Med. Biol. (2)

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56, 515–534 (2011).
[Crossref] [PubMed]

M. J. Kitchen, G. A. Buckley, A. F. T. Leong, R. P. Carnibella, A. Fouras, M. J. Wallace, and S. B. Hooper, “X-ray specks: low dose in vivo imaging of lung structure and function,” Phys. Med. Biol. 60, 7259–7276 (2015).
[Crossref] [PubMed]

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K. Shoemake, “Uniform random rotations,” in Graphics Gems III, D. Kirk, ed. (Academic Press, 1992), pp. 124–132.

D. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[Crossref]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing(Cambridge University Press, 1988).

P. Churcher, P. French, J. Shaw, and L. Schramm, “Rock properties of Berea sandstone, Baker Dolomite, and Indiana limestone,” in SPE International Symposium on Oilfield Chemistry, (Society of Petroleum Engineers, 1991).
[Crossref]

A. Kingston and I. Svalbe, “Projective transforms on periodic discrete image arrays,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Elsevier Science, 2011), pp. 78–172.

C. T. I. Reviews, Calculus, Single and Multivariable: Business, Mathematics (Cram101, 2016).

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Figures (11)

Fig. 1
Fig. 1 Schematic of the experimental setup for XPCI imaging of a porous/colloidal object, modelled as ellipsoidal-shaped pores/particles embedded in a homogeneous single material.
Fig. 2
Fig. 2 Comparison of simulated XPCI power spectra (solid blue line) with fits to model (dashed red line) for (a) Case 1 (M ≥ 4B), a PSD with a small number of discrete pore shapes, and (b) Case 2 (M < 4B), a PSD that can be described by an analytical function.
Fig. 3
Fig. 3 Representation of an ellipsoid as a distribution of spheres via its XPCI power spectrum. A line profile from the origin of the power spectrum in any direction yields a 1D power spectrum that equivalently represents a population, nsp, of mono-size spheres.
Fig. 4
Fig. 4 The XPCI power spectrum of a single point in the ellipsoid distribution space (left), corresponding to n 1 e l ellipsoids of a particular size (a1, b1, c1), randomly oriented and spatially distributed, is equivalent to a 1D distribution of spherical pores (right), assuming that cibiai.
Fig. 5
Fig. 5 The total number of spheres Nsp in the sphere model normalized to a fixed total number of ellipsoids Nel in the material, for different aspect ratios of ellipsoids b/a and c/a.
Fig. 6
Fig. 6 The median ellipsoid radius mel (left) and median sphere radius msp (right) as functions of ellipsoid aspect ratios (b/a, c/a) with a = 1.
Fig. 7
Fig. 7 Case 3, a sparse PSD. (a) 3D Ellipsoid PSD for the model material. Comparison of the (b) spherical PSD and (c) XPCI power spectrum between that simulated from a model porous material (solid blue line) and that fitted to the simulated XPCI power spectrum using the sphere model (dashed red line).
Fig. 8
Fig. 8 Effectiveness of the sphere model in capturing, from single simulated XPCI images, trends in (a) porosity, (b) total number of pores, and (c) median pore size. In each figure, the solid blue line is an exact calculation based on the true ellipsoid size distribution (E), the dashed red line is a calculation from the equivalent sphere size distribution from the model (SE), and the dash-dotted yellow line is the value obtained from a best-fit of the simulated XPCI power spectrum (SF), assuming a Pearson VII distribution of spherical pores.
Fig. 9
Fig. 9 CT slice of Berea sandstone (a) before and (b) after image thresholding and filtering.
Fig. 10
Fig. 10 Comparison of experimentally recorded XPCI power spectrum (Exp) with that simulated from a CT image of Berea sandstone after (a) binarization and filtering to isolate the pores (BF), then (b) replacing the pores with fitted ellipsoids (El), and then (c) replacing fitted ellipsoids with spheres using the sphere model (Sph).
Fig. 11
Fig. 11 (a) Ellipsoid PSD for Berea sandstone, derived from 3D CT data as described in the text. (b) The corresponding spherical PSD directly converted from the ellipsoidal PSD, the size dimensions of which has been rescaled to match the pore volume, and from fitting to the XPCI power spectrum.

Tables (5)

Tables Icon

Table 1 Parameters used for simulating XPCI images of sandstone.

Tables Icon

Table 2 PSD parameters (pore size in μm) and results of fit for Case 1 (M ≥ 4B). The initial guess and fit result are represented by nominal (range) values across the instantiations.

Tables Icon

Table 3 Parameters and results of fit for Case 2 (M < 4B with the PSD described by an analytical function). The initial guess and fit result are represented by nominal (range) values across the instantiations.

Tables Icon

Table 4 Case 3: Comparison of ellipsoid and sphere PSDs.

Tables Icon

Table 5 Structural properties of Berea sandstone. Watershedded CT refers to values calculated from the watershedded x-ray CT data, in which the irregularly-shaped pores are represented by ellipsoids of equivalent volume. Sphere (calculated) refers to values of a sphere size distribution that is converted from the CT data-computed ellipsoidal size distribution. Sphere (fitted) refers to values obtained from a spherical PSD calculated from fitting its simulated XPCI power spectrum to that experimentally measured.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

| F { I ( r , z = L ) I ( r , z = 0 ) 1 } | 2 = L 2 δ 2 | k | 4 | F { G z ( r ) } | 2 ,
E ( r ) = { 1 if x 2 a 2 + y 2 b 2 + z 2 c 2 1 , 0 otherwise .
G ( r ) = V ( r ) i = 1 B j = 1 n i e l δ ^ ( r r i j ) * E i ( r ) ,
| F { G ( r ) } | 2 = i = 1 B n i e l | F { E i ( r ) } | 2 ,
| F { E i ( r ) } | 2 ( k , θ q , ϕ p ) = | v i e l ( ( R i ( θ p , ϕ q ) k ) 2 [ sin  ( R i ( θ q , ϕ p ) k ) R i ( θ q , ϕ p ) k cos  ( R i ( θ q , ϕ p ) k ) ] | 2 ,
R i ( θ q , ϕ p ) = ( a i cos  θ q sin  ϕ p ) 2 + ( b i sin  θ q sin  ϕ p ) 2 + ( c i cos  ϕ p ) 2 ,
| F { G ( r ) } | 2 = 1 4 π i = 1 B n i e l p = 1 P q = 1 Q sin  ϕ p | v i e l ( R i k ) 2 [ sin  ( R i k ) R i k cos  ( R i k ) ] | 2
| F { I ( r , z = L ) I ( r , z = 0 ) 1 } | 2 = L 2 δ 2 k 4 4 π i = 1 B n i e l p = 1 P q = 1 Q sin  ϕ p | v i e l ( R i k ) 2 [ sin  ( R i k ) R i k cos  ( R i k ) ] | 2 .
h = [ 1 u 1 sin  ( 2 π u 2 ) ,   1 u 1 cos  ( 2 π u 2 ) , u 1 sin  ( 2 π u 3 ) ,   u 1 cos  ( 2 π u 3 ) ] ,
I ( r , z = L ) = | F 1 { P ( k , z = L ) F { ψ ( r , z = 0 ) } } | 2 .
P F ( k ) = p 0 [ 1 + ( k p 1 ) 2 p 2 p 3 2 ] p 2 ,
n e l = 650 h 0 a + h 1 b 1.5 + h 2 c 2 .
S i = { ( r , n s p ) : r = R i , n s p = n i e l v i e l sin  ϕ p 4 π ( v i s p ) 2 q , p | q = [ 1 , Q ] , p = [ 1 , P ] } .
S = { i = 1 B S i , i = 1 , j > i B ( S i S j ) } .
4 3 π ( r , n s p ) S n s p r 3 = 4 3 π i = 1 B n i e l a i b i c i .
N s p = ( r , n s p ) S n s p N e l = i = 1 B n i e l .
r = 0 m e l hist { E } ( r ) r = 0 hist { E } ( r ) = 0.5 ,
E = { i = 1 B E i , i = 1 , j > i B ( E i E j ) }
E i = { ( R i , n i e l ) : θ = { q π Q } , ϕ = { p 2 π P } q , p | q = [ 1 , Q ] , p = [ 1 , P ] } ,
r = 0 m s p hist { S } ( r ) r = 0 hist { S } ( r ) = 0.5 ,
| F { I ( r , z = L ) I ( r , z = 0 ) 1 } | 2 = L 2 δ 2 k 4 4 π i = 1 B [ p = 1 P q = 1 Q n i e l v i e l sin   ϕ p ( v i s p ) 2 × | v i s p ( R i k ) 2 [ sin   ( R i k ) R i k cos   ( R i k ) ] | 2 ]
v i s p = 4 3 π R i .
| F { I ( r , z = L ) I ( r , z = 0 ) 1 } | 2 = L 2 δ 2 k 4 j = 1 W n j s p | v j s p ( r j k ) 2 [ sin  ( r j k ) r j k cos  ( r j k ) ] | 2 ,
n i , p , q = n i e l v i e l sin  ϕ p 4 π ( v i s p ) 2 .
n i , p , q n i , j s p
R i ( θ p , ϕ q ) r i , j .
| F { I ( x , y , z = L ) I ( x , y , z = 0 ) 1 } | 2 = L 2 δ 2 k 4 i = 1 B j = 1 W n i , j s p | v i , j s p ( r i , j k ) 2 [ sin  ( r i , j k ) r i , j k cos  ( r i , j k ) ] | 2 ,
S i = { ( r , n s p ) : r = R i , n s p = n i e l v i e l sin  ϕ p 4 π ( v i s p ) 2 q , p | q = [ 1 , Q ] , p = [ 1 , P ] } ,
S = { i = 1 B S i , i = 1 , j > i B ( S i S j ) } .
4 3 π ( r , n s p ) S n s p r 3 = 4 3 π i = 1 B n i e l a i b i c i ,
lim Q , P 4 3 π ( r , n s p ) S i n s p r 3 = 4 3 π 0 2 π 0 π n i e l v i e l sin  ϕ 4 π ( v i s p ( θ q , ϕ p ) ) 2 R i 3 ( θ , ϕ ) d ϕ d θ .
= 2 ( a i b i ) 2 c i n i e l 3 [ tan 1 ( b i 2 a i 2 tan  θ ) a i b i ] 0 2 π
= 4 π 3 n i e l a i b i c i .

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