Abstract

Differential X-ray phase-contrast tomography (DPCT) refers to a class of promising methods for reconstructing the X-ray refractive index distribution of materials that present weak X-ray absorption contrast. The tomographic projection data in DPCT, from which an estimate of the refractive index distribution is reconstructed, correspond to one-dimensional (1D) derivatives of the two-dimensional (2D) Radon transform of the refractive index distribution. There is an important need for the development of iterative image reconstruction methods for DPCT that can yield useful images from few-view projection data, thereby mitigating the long data-acquisition times and large radiation doses associated with use of analytic reconstruction methods. In this work, we analyze the numerical and statistical properties of two classes of discrete imaging models that form the basis for iterative image reconstruction in DPCT. We also investigate the use of one of the models with a modern image reconstruction algorithm for performing few-view image reconstruction of a tissue specimen.

© 2012 OSA

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2011

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

T. Köhler, K. Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38, S106–S116 (2011).
[CrossRef]

X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
[CrossRef]

2010

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[CrossRef] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[CrossRef] [PubMed]

2009

Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79, 013815 (2009).
[CrossRef]

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
[CrossRef] [PubMed]

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in “Proc. SPIE,”  7258A1–A8 (2009)

J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. 28, 1781–1790 (2009).
[CrossRef]

2008

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

R. Fatehi, M. Fayazbakhsh, and M. Manzari, “On discretization of second-order derivatives in smoothed particle hydrodynamics,” Proceedings of World Academy of Science, Engineering and Technology.  30 pp. 243–246 (2008).

2007

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, 89–91 (2007).
[CrossRef]

M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett. 32, 3167–3169 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[CrossRef]

J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
[CrossRef] [PubMed]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology 14, 119–139 (2006).

2005

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105–124105-3 (2005).
[CrossRef]

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. 44, 8219–8221 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys. 68, 1703–1760 (2005).
[CrossRef]

2004

A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. 197, 253–274 (2004).
[CrossRef]

S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
[CrossRef] [PubMed]

2003

M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, 866–868 (2003).
[CrossRef]

A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
[CrossRef]

2000

F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33, 1596–1605 (2000).
[CrossRef]

T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. 19, 474–484 (2000).
[CrossRef]

1998

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

1997

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

1996

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase–contrast X–ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

S. Matej and R. Lewitt, “Practical considerations for 3-D image reconstruction using spherically symmetric volume elements,” IEEE Trans. Med. Imag. 15, 68–78 (1996).
[CrossRef]

1995

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Meth. A 352, 622–628 (1995).
[CrossRef]

1994

J. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag. 13, 290–300 (1994).
[CrossRef]

1992

R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. 37, 705–716 (1992).
[CrossRef] [PubMed]

1990

1988

S. Lo, “Strip and line path integrals with a square pixel matrix: A unified theory for computational CT projections,” IEEE Trans. Med. Imag. 7, 355–363 (1988).
[CrossRef]

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
[CrossRef] [PubMed]

1985

R. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[CrossRef] [PubMed]

1984

Abramovitz, M.

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Anastasio, M. A.

J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. 28, 1781–1790 (2009).
[CrossRef]

M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett. 32, 3167–3169 (2007).
[CrossRef] [PubMed]

J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
[CrossRef] [PubMed]

M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

Ando, M.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105–124105-3 (2005).
[CrossRef]

Aoki, S.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, 89–91 (2007).
[CrossRef]

Arfelli, F.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Barrett, H.

H. Barrett, K. Myers, and S. Dhurjaty, Foundations of Image Science (Wiley-Interscience, 2003), 2nd ed.

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Bertero, M.

M. Bertero, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[CrossRef]

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in “Proc. SPIE,”  7258A1–A8 (2009)

Bian, J.

X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
[CrossRef]

Brankov, J.

J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
[CrossRef] [PubMed]

M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Bravin, A.

S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
[CrossRef] [PubMed]

Brendel, B.

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

Bunk, O.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[CrossRef]

Byer, R.

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[CrossRef]

Chaniotis, A.

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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
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E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology 14, 119–139 (2006).

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S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
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K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33, 1596–1605 (2000).
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S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
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X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
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T. Köhler, K. Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38, S106–S116 (2011).
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M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
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I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. 44, 8219–8221 (2005).
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A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, 866–868 (2003).
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T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. 19, 474–484 (2000).
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J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
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A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105–124105-3 (2005).
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R. Fatehi, M. Fayazbakhsh, and M. Manzari, “On discretization of second-order derivatives in smoothed particle hydrodynamics,” Proceedings of World Academy of Science, Engineering and Technology.  30 pp. 243–246 (2008).

Marone, F.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
[CrossRef] [PubMed]

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T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. 19, 474–484 (2000).
[CrossRef]

S. Matej and R. Lewitt, “Practical considerations for 3-D image reconstruction using spherically symmetric volume elements,” IEEE Trans. Med. Imag. 15, 68–78 (1996).
[CrossRef]

S. Matej and R. Lewitt, “Image representation and tomographic reconstruction using spherically-symmetric volume elements,” in Nuclear Science Symposium and Medical Imaging Conference, 1992., Conference Record of the 1992 IEEE, (IEEE, 1992), pp. 1191–1193.

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P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
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D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
[CrossRef] [PubMed]

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Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, 89–91 (2007).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

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A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
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A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, 866–868 (2003).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase–contrast X–ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Meth. A 352, 622–628 (1995).
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K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33, 1596–1605 (2000).
[CrossRef]

Muehleman, C.

J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
[CrossRef] [PubMed]

M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

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H. Barrett, K. Myers, and S. Dhurjaty, Foundations of Image Science (Wiley-Interscience, 2003), 2nd ed.

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T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. 19, 474–484 (2000).
[CrossRef]

Oltulu, O.

M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Orion, I.

F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
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X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
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E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
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M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett. 32, 3167–3169 (2007).
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E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology 14, 119–139 (2006).

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

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K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33, 1596–1605 (2000).
[CrossRef]

Pfeiffer, F.

S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

Pisano, E.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Poulikakos, D.

A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. 197, 253–274 (2004).
[CrossRef]

Qi, Z.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in “Proc. SPIE,”  7258A1–A8 (2009)

Ren, B.

F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
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X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
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T. Köhler, K. Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. 38, S106–S116 (2011).
[CrossRef]

T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
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E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
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E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
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D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
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M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
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X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. 30 pp. 606–620 (2011).
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E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
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E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology 14, 119–139 (2006).

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P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010).
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S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16, 562–572 (2009).
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T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
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V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
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I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. 44, 8219–8221 (2005).
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Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, 89–91 (2007).
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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
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E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
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E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
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S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
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S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
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F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
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S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. 49, 175–188 (2004).
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F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
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Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, 89–91 (2007).
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J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. 28, 1781–1790 (2009).
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Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A 79, 013815 (2009).
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J. Brankov, M. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. 33, 278–289 (2006).
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Figures (18)

Fig. 1
Fig. 1

A schematic of differential phase-contrast imaging tomography. The black box represents the system of optical elements that is specific to the implementation.

Fig. 2
Fig. 2

Singular value spectra associated with the system matrices Hpixel with pixel size 50 μm.

Fig. 3
Fig. 3

Singular value spectra associated with the system matrices Hblob with m = 2, relative radius a = 1.5 (physical size 75μm).

Fig. 4
Fig. 4

Singular value spectra associated with the system matrices Hblob with m = 2, relative radius a = 2 (physical size 100μm).

Fig. 5
Fig. 5

Profiles of the differential projection value of one blob with m = 2, relative radius a = 2.

Fig. 6
Fig. 6

The three highest singular value spectra are replotted for comparison. Two of the spectra correspond to Hblob with α = 10.4, relative radius a = 1.5 and a = 2. The third spectra corresponds to Hpixel based on the linear weighting kernel.

Fig. 7
Fig. 7

The numerical phantom employed in our simulation studies with an ROI indicated

Fig. 8
Fig. 8

Examples of reconstructed images by use of PLS algorithms based on pixel system matrix Hpixel and blob system matrix Hblob. Regularization parameter γ = 10 for both cases. (a) An reconstructed image produced by pixel system matrix with linear interpolation. (b) An reconstructed image produced by blobs with relative radius a = 2 (physical size 100μm), m = 2 and α = 10.4.

Fig. 9
Fig. 9

(Color online) Profiles through the center row of the reconstructed images in Fig. 8. The solid blue line corresponds to Fig. 8(a). The dashed red and dashdotted black lines correspond to Fig. 8(b) and the true phantom.

Fig. 10
Fig. 10

Variance versus resolution curves corresponding to use of the system matrices Hpixel.

Fig. 11
Fig. 11

Variance versus resolution curves corresponding to use of the system matrices Hblob. (a) Curves are produced by blobs with relative radius a = 1.5 (physical size 75μm), m = 2 and varying α. (b) Curves are produced by blobs with relative radius a = 2 (physical size 100μm), m = 2 and varying α.

Fig. 12
Fig. 12

The three best variance-resolution curves picked from the pixel and blob cases.

Fig. 13
Fig. 13

Images reconstructed from 90 projections by use of the (a) FBP (b) ASD-POCS algorithm. The dashed boxes indicate two ROIs chosen for comparison. All images are displayed in the same grey scale window [0 1].

Fig. 14
Fig. 14

Images reconstructed from 180 projections by use of the (a) FBP (b) ASD-POCS algorithm Two dashed boxes indicate two ROIs chosen for comparison. All images are displayed in the same grey scale window [0 1].

Fig. 15
Fig. 15

Zoomed-in images of the smaller ROIs denoted in Figs. 13(a) and (b), reconstructed from 90 view angles, are displayed in subfigures (a) and (b). Subfigure (c) displays the corresponding reference ROI corresponding to an image reconstructed from 720 projections by use of a DPCT FBP algorithm. All images are displayed in the same grey scale window [0 1].

Fig. 16
Fig. 16

Zoomed-in images of the smaller ROIs denoted in Figs. 14(a) and (b), reconstructed from 180 view angles, are displayed in subfigures (a) and (b). Subfigure (c) displays the corresponding reference ROI corresponding to an image reconstructed from 720 projections by use of a DPCT FBP algorithm. All images are displayed in the same grey scale window [0 1].

Fig. 17
Fig. 17

Zoomed-in images of the larger ROIs denoted in Figs. 13(a) and (b), reconstructed from 90 view angles, are displayed in subfigures (a) and (b). Subfigure (c) displays the corresponding reference ROI corresponding to an image reconstructed from 720 projections by use of a DPCT FBP algorithm. All images are displayed in the same grey scale window [0 1].

Fig. 18
Fig. 18

Zoomed-in images of the larger ROIs denoted in Figs. 14-(a) and (b), reconstructed from 180 view angles, are displayed in subfigures (a) and (b). Subfigure (c) displays the corresponding reference ROI corresponding to an image reconstructed from 720 projections by use of a DPCT FBP algorithm. All images are displayed in the same grey scale window [0 1].

Equations (37)

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g(xr,θ;z)=xrdyrδ(r2;z)xrRδ(r2;z).
g[s,t;h]g(xr,θ;z)|xr=sΔd,θ=tΔθ,z=hΔd,
δa(r2;z=hΔd)=n=0N1bnhϕn(r2),
g[s,t;h]xrRδa(r2;z=hΔd)|xr=sΔd,θ=tΔθ,
g[s,t;h]xrSRδa(r2;z=hΔd)|xr=sΔd,θ=tΔθ,
g[s,t;h]n=0N1bnhxr(Rϕn(r2))(xr,θ)|xr=sΔd,θ=tΔθ.
g=Hb,
ϕnpixel(r2)=rect(xxnɛ)rect(yynɛ),
p[s,t](Rδa(r2))[s,t]=(Rδa(r2))(xr,θ)|xr=sΔd,θ=tΔθj=0N1wstjbj,
p=HRb,
[HR]m=t×S+s,n=wstn,
pk'=1ρki=kK/2i=k+K/2(pipk)W(didk,h)xr,
ρk=i=kK/2i=k+K/2W(didk,h).
p=HDp,
gp=Hpixelb,
HpixelHDHR.
ϕnblob(r2;m,a,α)={[1(rb/a)2]mIm[α1(rb/a)2]Im(α),rba0,otherwise,
Rϕnblob(r2;m,a,α)=aIm(α)(2πα)1/2[1(ξ/a)2]m+1/2Im+1/2(α1(ξ/a)2),
(Rϕnblob(r2;m,a,α))xr=(2πα)1/2Im(α)ξa(1(ξ/a)2)m1/2Im1/2(α1(ξ/a)2).
g[s,t](2πα)1/2Im(α)×n=0N1bnξa(1(ξ/a)2)m1/2Im1/2(α1(ξ/a)2)|ξ=sΔdxncos(tΔθ)ynsin(tΔθ),
gHblobb,
[Hblob]m,n=[Hblob]m=t×S+s,n=(2πα)1/2Im(α)×ξa(1(ξ/a)2)m1/2Im1/2(α1(ξ/a)2)|ξ=sΔdxncos(tΔθ)ynsin(tΔθ),
b^=argminbgHb+γL(b),
L(b)=n=0N1k𝒩n([b]n[b]k)2,
G(x)=I1+I2I12(1+erf(xμσ2)),
b^=argminbbTVs.t.|gHblobb|ɛ,
HD=(H110000H2200000Htt000000HTT),
W1(d,h)=nd{1s0s<1,s=|d|h,0s1,
Htt=(boundarycondition1/201/20001/201/2001/201/2boundarycondition)S×S,
W2(d,h)=nd{34s20s<12,s=|d|h,9832s+s2212s<32,0s32,
Htt=(boundarycondition1/81/401/41/8001/81/401/41/80001/81/401/41/8boundarycondition)S×S.
W3(d,h)=nd{23s2+s320s<1,s=|d|h,432s+s2s361s<2,0s2,
Htt=(boundarycondition1/321/85/3205/321/81/32001/321/85/3205/321/81/320001/321/85/3205/321/81/32boundarycondition)S×S.
Rϕnblob(r2;m,a,α)=aIm(α)(2πα)1/2[1(ξ/a)2]m+1/2Im+1/2(α1(ξ/a)2),
ddz{z±mIm(z)}=z±mIm1(z),
Rϕnblob(r2;m,a,α)=aIm(α)(2πα)1/2(1α)m+1/2zm+1/2Im+1/2(z).
(Rϕnblob(m,a,α,r))xr=(Rϕnblob(m,a,α,r))zzξξxr=aIm[α](2πα)1/2(1α)m+1/2zm+1/2Im1/2(z)×(αa)2(ξz)=(2πα)1/2Im(α)ξa(1(ξ/a)2)m1/2Im1/2(α1(ξ/a)2).

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