Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements

Cheng-Ying Chou; Yin Huang; Daxin Shi; Mark A. Anastasio

doi:10.1364/OE.15.010002

Optics Express
Vol. 15,
Issue 16,
pp. 10002-10025
(2007)
•https://doi.org/10.1364/OE.15.010002

Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements

Cheng-Ying Chou, Yin Huang, Daxin Shi, and Mark A. Anastasio

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Cheng-Ying Chou, Yin Huang, Daxin Shi, and Mark A. Anastasio, "Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements," Opt. Express 15, 10002-10025 (2007)

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Related Topics
Table of Contents Category
- Imaging Systems
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Phase retrieval (100.5070)
- X-ray imaging (110.7440)
- Image reconstruction techniques (170.3010)

About this Article
History
- Original Manuscript: February 5, 2007
- Revised Manuscript: May 7, 2007
- Manuscript Accepted: May 20, 2007
- Published: July 25, 2007
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 2, Iss. 9

Abstract

X-ray phase-contrast imaging is a technique that aims to reconstruct the projected absorption and refractive index distributions of an object. One common feature of reconstruction formulas for phase-contrast imaging is the presence of isolated Fourier domain singularities, which can greatly amplify the noise levels in the estimated Fourier domain and lead to noisy and/or distorted images in spatial domain. In this article, we develop a statistically optimal reconstruction method that employs multiple (>2) measurement states to mitigate the noise amplification effects due to singularities in the reconstruction formula. Computer-simulation studies are carried out to quantitatively and systematically investigate the developed method, within the context of propagation-based X-ray phase-contrast imaging. The reconstructed images are shown to possess dramatically reduced noise levels and greatly enhanced imaging contrast.

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References

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Year
Author
Publication

A. Krol, A. Ikhlef, J.-C. Kieffer, D. Bassano, C. C. Chamberlain, Z. Jiang, H. Pepin, and S. C. Prasad, “Laser-based microfocused X-ray source for mammography: feasibility study,” Med. Phys. 24, 725–732 (1997).
[Crossref] [PubMed]
R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. 6, 1465–1469 (2000).
[Crossref]
H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods Phys. Res. B 199, 509–516 (2003).
R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[Crossref] [PubMed]
W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).
T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).
K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]
D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[Crossref] [PubMed]
C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[Crossref] [PubMed]
X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[Crossref] [PubMed]
M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[Crossref]
E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[Crossref] [PubMed]
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]
B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]
F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43, 2845–2852 (1998).
[Crossref] [PubMed]
F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Mammography with Synchrotron Radiation: Phase-Detection Techniques,” Radiology 215, 286–293 (2000).
M. Z. Kiss, D. E. Sayers, Z. Zhong, C. Parham, and E. D. Pisano, “Improved image contrast of calcifications in breast tissue specimens using diffraction enhanced imaging,” Phys. Med. Biol. 49, 3427–3439 (2004).
[Crossref] [PubMed]
E. Pisano, R. Johnston, D. Chapman, J. Geradts, M. Iacocca, C. Livasy, D. Washburn, D. Sayers, Z. Zhong, M. Kiss, and W. Thomlinson, “Human Breast Cancer Specimens: Diffraction-enhanced Imaging with Histologic Correlation-Improved Conspicuity of Lesion Detail Compared with Digital Radiography,” Radiology 214, 895–901 (2000).
[PubMed]
T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[Crossref] [PubMed]
C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]
J. G. Brankov, M. N. 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. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. D 36, A24–A29 (2003).
P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]
P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999).
A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]
M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[Crossref]
P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. lett. 75, 2912–2914 (1999).
[Crossref]
A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
[Crossref]
T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]
M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]
Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).
Y. I. Nesterets, T. E. Gureyev, K. M. Pavlov, D. M. Paganin, and S. W. Wilkins, “Combined analyser-based and propagation-based phase-contrast imaging of weak objects,” Opt. Commun. 259, 19–31 (2006).
[Crossref]
T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]
D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]
Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626–642 (2007).
[Crossref]
D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[Crossref]
T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]
Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).
K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to X-ray phase-contrast imaging,” J. Phys. D 37, 2746–2750 (2004).
J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).
J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).
T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).
S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]
W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

2007 (1)

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626–642 (2007).
[Crossref]

2006 (7)

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

Y. I. Nesterets, T. E. Gureyev, K. M. Pavlov, D. M. Paganin, and S. W. Wilkins, “Combined analyser-based and propagation-based phase-contrast imaging of weak objects,” Opt. Commun. 259, 19–31 (2006).
[Crossref]

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

J. G. Brankov, M. N. 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]

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. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

2005 (3)

W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).

T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta, S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada, M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. 40, 385–396 (2005).
[Crossref] [PubMed]

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).

2004 (7)

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to X-ray phase-contrast imaging,” J. Phys. D 37, 2746–2750 (2004).

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[Crossref]

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[Crossref] [PubMed]

M. Z. Kiss, D. E. Sayers, Z. Zhong, C. Parham, and E. D. Pisano, “Improved image contrast of calcifications in breast tissue specimens using diffraction enhanced imaging,” Phys. Med. Biol. 49, 3427–3439 (2004).
[Crossref] [PubMed]

2003 (6)

X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[Crossref] [PubMed]

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

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
[Crossref] [PubMed]

H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods Phys. Res. B 199, 509–516 (2003).

A. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. D 36, A24–A29 (2003).

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

2002 (1)

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
[Crossref]

2000 (3)

R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. 6, 1465–1469 (2000).
[Crossref]

E. Pisano, R. Johnston, D. Chapman, J. Geradts, M. Iacocca, C. Livasy, D. Washburn, D. Sayers, Z. Zhong, M. Kiss, and W. Thomlinson, “Human Breast Cancer Specimens: Diffraction-enhanced Imaging with Histologic Correlation-Improved Conspicuity of Lesion Detail Compared with Digital Radiography,” Radiology 214, 895–901 (2000).
[PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Mammography with Synchrotron Radiation: Phase-Detection Techniques,” Radiology 215, 286–293 (2000).

1999 (3)

C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[Crossref] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. lett. 75, 2912–2914 (1999).
[Crossref]

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

1998 (1)

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43, 2845–2852 (1998).
[Crossref] [PubMed]

1997 (3)

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

A. Krol, A. Ikhlef, J.-C. Kieffer, D. Bassano, C. C. Chamberlain, Z. Jiang, H. Pepin, and S. C. Prasad, “Laser-based microfocused X-ray source for mammography: feasibility study,” Med. Phys. 24, 725–732 (1997).
[Crossref] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

1996 (2)

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

1993 (1)

B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

1977 (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1970 (1)

S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

Anastasio, M. A.

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626–642 (2007).
[Crossref]

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Arfelli, F.

Arsenault, H.

S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

Assante, M.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

Baruchel, J.

Bassano, D.

Birch, I. P.

C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[Crossref] [PubMed]

Bonvicini, V.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Brankov, J. G.

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

Bravin, A.

A. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. D 36, A24–A29 (2003).

Bronnikov, A. V.

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
[Crossref]

Bunk, O.

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]

Cantatore, G.

Carlo, F. D.

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Castelli, E.

Chamberlain, C. C.

Chapman, D.

W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).

Cloetens, P.

P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999).

Cookson, D.

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

David, C.

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]

Davis, J.C.

Davis, T.

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Donnelly, E. F.

Dougherty, G. R.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

Dougherty, R.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

Dyck, D.

Fabrizioli, M.

Furukawa, A.

Galatsanos, N. P.

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Geradts, J.

Gmur, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).

Guigay, J. P.

Guigay, J.-P.

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Gullikson, E.M.

Gureyev, T. E.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

Henke, B.L.

Honda, C.

Huang, Y.

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626–642 (2007).
[Crossref]

Iacocca, M.

Ikhlef, A.

Jiang, Z.

Johnston, R.

Johnston, R. E.

Kieffer, J.-C.

Kiss, M.

Kiss, M. Z.

Kitchen, M.

Kotre, C. J.

C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[Crossref] [PubMed]

Krol, A.

Landuyt, J.

Lewis, R.

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[Crossref] [PubMed]

Lewis, R. A.

Li, J.

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

Liu, H.

X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[Crossref] [PubMed]

Livasy, C.

Longo, R.

Lowenthal, S.

S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

Ludwig, W.

Matsuo, S.

Mcmahon, P. J.

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

Menk, R.

Menk, R. H.

Michiel, M. D.

Morgan, M. J.

Muehleman, C.

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

Murata, K.

Myers, G. R.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

Nesterest, Y. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

Nesterets, Y.

Nesterets, Y. I.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

Nitta, N.

Noma, K.

Nugent, K. A.

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

Ohara, H.

Olivo, A.

Oltulu, O.

Ota, S.

Paganin, D.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

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

Paganin, D.M.

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Palma, L. D.

Pan, X.

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Pani, S.

Parham, C.

Pavlov, K. M.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

Pepin, H.

Pfeiffer, F.

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]

Pickens, D. R.

Pisano, E.

Pisano, E. D.

Pogany, A.

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Pontoni, D.

Poropat, P.

Prasad, S. C.

Prest, M.

Price, R. R.

Rashevsky, A.

Ratti, M.

Raven, C.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

Rigon, L.

Sakashita, Y.

Sayers, D.

Sayers, D. E.

Schlenker, M.

Shi, D.

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Snigirev, A.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

Snigireva, I.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

Spanne, P.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

Stanley, W. D.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

Stevenson, A.

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Suortti, P.

W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).

Takahashi, M.

Tanaka, T.

Thomlinson, W.

W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).

Tromba, G.

Tsuchiya, K.

Vacchi, A.

Vallazza, E.

Washburn, D.

Waynant, R.

R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. 6, 1465–1469 (2000).
[Crossref]

Weitkamp, T.

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]

Wernick, M. N.

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

Wilkins, S.

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Wilkins, S. W.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Wilkins, S.W.

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Wirjadi, O.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Wu, X.

X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[Crossref] [PubMed]

Yamada, A.

Yamada, H.

H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods Phys. Res. B 199, 509–516 (2003).

Yamasaki, M.

Yang, Y.

Zanconati, F.

Zhong, Z.

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

Appl. Phys. lett. (2)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[Crossref]

At. Data Nucl. Data Tables (1)

IEEE J. Quantum Electron. (1)

R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. 6, 1465–1469 (2000).
[Crossref]

Invest. Radiol. (1)

J. Anat. (1)

C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. 208, 115–124 (2006).
[Crossref] [PubMed]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[Crossref]

J. Opt. Soc. Am. (3)

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24, 626–642 (2007).
[Crossref]

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
[Crossref]

S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
[Crossref]

J. Phys. (4)

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D 38, 4259–4271 (2005).

A. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. D 36, A24–A29 (2003).

Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D 37, 1262–1274 (2004).

Med. Phys. (4)

X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[Crossref] [PubMed]

Nature (London) (1)

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Nature Phys. (1)

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]

Nucl. Instrum. Methods Phys. Res. (2)

W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. A 543, 288–296 (2005).

H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods Phys. Res. B 199, 509–516 (2003).

Opt. Commun. (4)

T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Optik (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Med. Biol. (8)

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–749 (1999).
[Crossref] [PubMed]

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[Crossref] [PubMed]

C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref] [PubMed]

Proc. SPIE (1)

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Stability and locality of amplitude and phase contrast tomographies,” Proc. SPIE 6318, 63180V (2006).

Radiology (2)

Rev. Sci. Instrum. (1)

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

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

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

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Fig. 1. A schematic of a generic X-ray phase-contrast imaging system.

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Fig. 2. The measurement geometry of propagation-based X-ray phase-contrast imaging employing multiple detector-planes.

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Fig. 3. Images of the true object properties (a) A(x,y) and (b) ϕ(x,y).

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Fig. 4. Images of theoretical and empirical estimates of Var{Ã(u, v)} measured in Geometry ‘A’ are displayed logarithmically in subfigures (a)–(b), respectively. The corresponding variance maps of ϕ̃(u, v) are contained in subfigures (c)–(d), respectively.

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Fig. 5. Variance profiles of images in Fig. 4. Subfigure (a) contains the theoretically and empirically determined variance profiles of Ã(u, v), which are depicted by solid and dashed curves, respectively. The corresponding variance profiles of ϕ̃(u, v) are shown in subfigure (b).

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Fig. 6. Images of theoretical and empirical estimates of Var{Ã(u, v)} measured in Geometry ‘B’ are displayed logarithmically in subfigures (a)–(b), respectively. The corresponding variance maps of ϕ̃(u, v) are contained in subfigures (c)–(d), respectively.

Download Full Size | PPT Slide | PDF

Fig. 7. Variance profiles of images in Fig. 6. Subfigure (a) contains the theoretically and empirically determined variance profiles of Ã(u, v), which are depicted by solid and dashed curves, respectively. The corresponding variance profiles of ϕ̃(u, v) are shown in subfigure (b).

Download Full Size | PPT Slide | PDF

Fig. 8. Estimates of A(x, y) reconstructed from noisy intensity data measured in Geometry ‘A’ by use of detector-planes (a)(1,2), (b)(1,3), (c)(2,3), and (d) an optimally-weighted combination of all three detector-planes. The corresponding estimates of ϕ(x, y) are shown in subfigures (e)–(h).

Download Full Size | PPT Slide | PDF

Fig. 9. Empirical variance profiles of Ã(u, v) and ϕ̃(u, v) measured in Geometry ‘A’ are displayed in subfigures (a)–(b), respectively. Each subfigure contains profiles corresponding to the Fourier variances estimated by use of detector-planes (1,2) (dashed curve), detector-planes (1,3) (dashdotted curve), detector-planes (2,3) (dotted curve), and the optimal one (solid curve). Subfigures (c) and (d) display empirical variance profiles of the corresponding images in the spatial domain.

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Fig. 10. Estimates of A(x, y) reconstructed from noisy intensity data measured in Geometry ‘B’ by use of detector-planes (a)(1,2), (b)(1,3), (c)(2,3), and (d) an optimally-weighted combination of all three detector-planes. The corresponding estimates of ϕ(x, y) are shown in subfigures (e)–(h).

Download Full Size | PPT Slide | PDF

Fig. 11. Empirical variance profiles of Ã(u, v) and ϕ̃(u, v) measured in Geometry ‘B’ are displayed in subfigures (a)–(b), respectively. Each subfigure contains profiles corresponding to the Fourier variances estimated by use of detector-planes (1,2) (dashed curve), detectorplanes (1,3) (dashdotted curve), detector-planes (2,3) (dotted curve), and the optimal one (solid curve). Subfigures (c) and (d) display empirical variance profiles of the corresponding images in the spatial domain.

Download Full Size | PPT Slide | PDF

Fig. 12. The optimally determined estimates of A(x, y) are reconstructed from noisy intensity data measured in Geometry ‘A’ with detector position uncertainty of (a) error level 1, (b) error level 2, and (c) error level 3. The corresponding estimates of ϕ(x, y) are contained in subfigures (d)–(f).

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Tables (1)

Table 1. Error levels in the detector positions in Geometry ‘A’.

View Table

Equations (97)

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

n (\vec{r}) \equiv 1 - δ (\vec{r}) + j β (\vec{r}),

μ (\vec{r}) = 2 k β (\vec{r}),

U_{o} (x, y) = T (x, y) U_{i}

T (x, y) = M (x, y) \exp [j ϕ (x, y)] .

A (x, y) = k \int d z β (\vec{r})

ϕ (x, y) = - k \int d z δ (\vec{r})

U_{m} (x, y) = G_{m} (x, y) * U_{o} (x, y),

K_{m} (x, y) \equiv 1 - \frac{I_{m} (x, y)}{I_{i}},

{\tilde{K}}_{m} (u, v) = \iint_{- \infty}^{\infty} d x d y K_{m} (x, y) \exp [- j 2 π (ux + vy)] .

{\tilde{K}}_{m} (u, v) = 2 {\tilde{G}}_{m}^{a} (u, v) \tilde{A} (u, v) + 2 {\tilde{G}}_{m}^{p} (u, v) \tilde{ϕ} (u, v)

{\tilde{G}}_{m}^{a} (u, v) = \frac{1}{2} [{\tilde{G}}_{m} (u, v) + {\tilde{G}}_{m}^{*} (- u, - v)],

{\tilde{G}}_{m}^{p} (u, v) = \frac{1}{2 j} [{\tilde{G}}_{m} (u, v) - {\tilde{G}}_{m}^{*} (- u, - v)] .

\tilde{A} (u, v) = \frac{{\tilde{G}}_{n}^{p} (u, v) {\tilde{K}}_{m} (u, v) - {\tilde{G}}_{m}^{p} (u, v) {\tilde{K}}_{n} (u, v)}{2 [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]}

\tilde{ϕ} (u, v) = \frac{- {\tilde{G}}_{n}^{a} (u, v) {\tilde{K}}_{m} (u, v) + {\tilde{G}}_{m}^{a} (u, v) {\tilde{K}}_{n} (u, v)}{2 [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]} .

{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v) = 0 .

{\tilde{ϕ}}_{m, n} (u, v) = α_{l, m}^{ϕ} (u, v) {\tilde{ϕ}}_{l, m} (u, v) + α_{l, n}^{ϕ} (u, v) {\tilde{ϕ}}_{l, n} (u, v)

{\tilde{A}}_{m, n} (u, v) = α_{l, m}^{a} (u, v) {\tilde{A}}_{l, m} (u, v) + α_{l, n}^{a} (u, v) {\tilde{A}}_{l, n} (u, v)

α_{l, m}^{ϕ} (u, v) = \frac{- {\tilde{G}}_{n}^{a} (u, v) [{\tilde{G}}_{l}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v) - {\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{l}^{p} (u, v)]}{{\tilde{G}}_{l}^{a} (u, v) [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]}

α_{l, n}^{ϕ} (u, v) = \frac{{\tilde{G}}_{m}^{a} (u, v) [{\tilde{G}}_{l}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{l}^{p} (u, v)]}{{\tilde{G}}_{l}^{a} (u, v) [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]},

α_{l, m}^{a} (u, v) = \frac{- {\tilde{G}}_{n}^{p} (u, v) [{\tilde{G}}_{l}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v) - {\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{l}^{p} (u, v)]}{{\tilde{G}}_{l}^{p} (u, v) [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]},

α_{l, n}^{a} (u, v) = \frac{{\tilde{G}}_{m}^{p} (u, v) [{\tilde{G}}_{l}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{l}^{p} (u, v)]}{{\tilde{G}}_{l}^{p} (u, v) [{\tilde{G}}_{m}^{a} (u, v) {\tilde{G}}_{n}^{p} (u, v) - {\tilde{G}}_{n}^{a} (u, v) {\tilde{G}}_{m}^{p} (u, v)]},

α_{l, m}^{ϕ} (u, v) + α_{l, n}^{ϕ} (u, v) \equiv 1

α_{l, m}^{a} (u, v) + α_{l, n}^{a} (u, v) \equiv 1 .

\tilde{ϕ} (u, v) = ω_{1, 2}^{ϕ} (u, v) {\tilde{ϕ}}_{1, 2} (u, v) + ω_{1, 3}^{ϕ} (u, v) {\tilde{ϕ}}_{1, 3} (u, v)

\tilde{A} (u, v) = ω_{1, 2}^{a} (u, v) {\tilde{A}}_{1, 2} (u, v) + ω_{1, 3}^{a} (u, v) {\tilde{A}}_{1, 3} (u, v),

ω_{1, 2}^{ϕ} + ω_{1, 3}^{ϕ} = 1

ω_{1, 2}^{a} + ω_{1, 3}^{a} = 1 .

σ_{m, n}^{2} (u, v) \equiv Var {{\tilde{ϕ}}_{m, n} (u, v)}

ρ_{k, l; m, n}^{(r)} (u, v) + j ρ_{k, l; m, n}^{(i)} (u, v) \equiv Cov {{\tilde{ϕ}}_{k, l} (u, v), {\tilde{ϕ}}_{m, n} (u, v)}

R_{m, n} (u, v) + j I_{m, n} (u, v) \equiv ω_{m, n}^{ϕ} (u, v),

Var {\tilde{ϕ} (u, v)} = {∣ ω_{1, 2}^{ϕ} (u, v) ∣}^{2} σ_{1, 2}^{2} + {∣ ω_{1, 3}^{ϕ} (u, v) ∣}^{2} σ_{1, 3}^{2}

+ 2 Re [ω_{1, 2}^{ϕ} (u, v) {[ω_{1, 3}^{ϕ} (u, v)]}^{*} Cov {{\tilde{ϕ}}_{1, 2} (u, v), {\tilde{ϕ}}_{1, 3} (u, v)}]

{\frac{\partial Var {\tilde{ϕ}}}{\partial R_{1, 2}} ∣}_{R_{1, 2}^{(op)}} = 0

{\frac{\partial Var {\tilde{ϕ}}}{\partial I_{1, 2}} ∣}_{I_{1, 2}^{(op)}} = 0,

R_{1, 2}^{(op)} (u, v) = \frac{σ_{1, 3}^{2} - ρ_{1, 2; 1, 3}^{(r)}}{σ_{1, 2}^{2} + σ_{1, 3}^{2} - 2 ρ_{1, 2; 1, 3}^{(r)}}

I_{1, 2}^{(op)} (u, v) = \frac{ρ_{1, 2; 1, 3}^{(i)}}{σ_{1, 2}^{2} + σ_{1, 3}^{2} - 2 ρ_{1, 2; 1, 3}^{(r)}} .

G_{m} (x, y) = \frac{\exp [j {kz}_{m}]}{j λ z_{m}} \exp [j π \frac{x^{2} + y^{2}}{λ z_{m}}],

{\tilde{G}}_{m} (u, v) = \exp [j {kz}_{m} - j π λ z_{m} (u^{2} + v^{2})] .

{\tilde{ϕ}}_{m, n} (u, v) = \frac{- \cos (π λ z_{n} f^{2}) {\tilde{K}}_{m} (u, v) + \cos (π λ z_{m} f^{2}) {\tilde{K}}_{n} (u, v)}{D_{m, n}}

{\tilde{A}}_{m, n} (u, v) = \frac{- \sin (π λ z_{m} f^{2}) {\tilde{K}}_{n} (u, v) + \sin (π λ z_{n} f^{2}) {\tilde{K}}_{m} (u, v)}{D_{m, n}},

D_{m, n} (u, v) \equiv 2 \sin (π λ f^{2} (z_{m} - z_{n})) .

u^{2} + v^{2} = \frac{l}{λ (z_{m} - z_{n})},

u_{M}^{2} + v_{M}^{2} \geq \frac{1}{λ (z_{m} - z_{n})} .

Var {{\tilde{ϕ}}_{m, n} (u, v)} = \frac{\cos^{2} (π λ z_{n} f^{2}) Var {{\tilde{I}}_{m} (u, v)} + \cos^{2} (π λ z_{m} f^{2}) Var {{\tilde{I}}_{n} (u, v)}}{D_{m, n}^{2}}

Var {{\tilde{A}}_{m, n} (u, v)} = \frac{\sin^{2} (π λ z_{n} f^{2}) Var {{\tilde{I}}_{m} (u, v)} + \sin^{2} (π λ z_{m} f^{2}) Var {{\tilde{I}}_{n} (u, v)}}{D_{m, n}^{2}}

Cov {{\tilde{ϕ}}_{1, 2} (u, v), {\tilde{ϕ}}_{1, 3} (u, v)} = \frac{\cos (π λ z_{2} f^{2}) \cos (π λ z_{3} f^{2}) Var {{\tilde{I}}_{1} (u, v)}}{D_{1, 2} D_{1, 3}}

Cov {{\tilde{A}}_{1, 2} (u, v), {\tilde{A}}_{1, 3} (u, v)} = \frac{\sin (π λ z_{2} f^{2}) \sin (π λ z_{3} f^{2}) Var {{\tilde{I}}_{1} (u, v)}}{D_{1, 2} D_{1, 3}}

Var {{\tilde{ϕ}}_{m, n} (u, v)} \propto \frac{1}{D_{m, n}^{2} (u, v)}

Var {{\tilde{A}}_{m, n} (u, v)} \propto \frac{1}{D_{m, n}^{2} (u, v)} .

ω_{1, 2}^{heur} (u, v) = \frac{D_{1, 2}^{2} + α_{1, 2}^{ϕ} D_{2, 3}^{2}}{D_{1, 2}^{2} + D_{1, 3}^{2} + D_{2, 3}^{2}}

ω_{1, 3}^{heur} (u, v) = \frac{D_{1, 3}^{2} + α_{1, 3}^{ϕ} D_{2, 3}^{2}}{D_{1, 2}^{2} + D_{1, 3}^{2} + D_{2, 3}^{2}},

I_{m} [r, s] = {I_{m} (x, y) ∣}_{x = r Δ x, y = s Δ y},

I_{m} [r, s] = I_{m}^{0} [r, s] + n_{m} [r, s],

Var {n_{m} [r, s]} = {(I_{m}^{0} [r, s])}^{2} σ^{2} (z_{m}),

Cov {n_{m} [r, s], n_{m'} [r', s']} = Var {n_{m} [r, s]} δ_{rr'} δ_{ss'} δ_{mm'},

{\tilde{I}}_{m} [p, q] = \tilde{I_{m}^{0}} [p, q] + {\tilde{n}}_{m} [p, q]

{\tilde{I}}_{m} [p, q] = \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} I_{m} [r, s] \exp [- j \frac{2 π}{N} (pr + qs)]

\tilde{I_{m}^{0}} [p, q] = \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} I_{m}^{0} [r, s] \exp [- j \frac{2 π}{N} (pr + qs)]

{\tilde{n}}_{m} [p, q] = \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} n_{m} [r, s] \exp [- j \frac{2 π}{N} (pr + qs)],

Var {{\tilde{n}}_{m} [p, q]} = \sum_{r, r' = 0}^{N - 1} \sum_{s, s' = 0}^{N - 1} \exp [- j \frac{2 π}{N} (p (r - r') + q (s - s'))] Cov {n_{m} [r, s], n_{m} [r', s']}

= \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} E {{(n_{m} [r, s])}^{2}} = \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I_{m}^{0} [r, s])}^{2} σ^{2} (z_{m}),

{{\tilde{I}}_{m} (u, v) ∣}_{u = p Δ u, v = q Δ v} \approx \frac{L^{2}}{N^{2}} {\tilde{I}}_{m} [p, q],

{Var {{\tilde{I}}_{m} (u, v)} ∣}_{u = p Δ u, v = q Δ v} \approx \frac{L^{4}}{N^{4}} Var {{\tilde{I}}_{m} [p, q]} = \frac{L^{4}}{N^{4}} σ^{2} (z_{m}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I_{m}^{0} [r, s])}^{2} .

{Var {{\tilde{ϕ}}_{m, n} (u, v)} ∣}_{u = p Δ u, v = q Δ v} \approx

\frac{L^{4}}{N^{4}} \frac{\cos^{2} (π λ z_{n} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{m}])}^{2} σ^{2} (z_{m}) + \cos^{2} (π λ z_{m} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{n}])}^{2} σ^{2} (z_{n})}{D_{m, n}^{2}}

{Var {{\tilde{A}}_{m, n} (u, v)} ∣}_{u = p Δ u, v = q Δ v} \approx

\frac{L^{4}}{N^{4}} \frac{\sin^{2} (π λ z_{n} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{m}])}^{2} σ^{2} (z_{m}) + \sin^{2} (π λ z_{m} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{n}])}^{2} σ^{2} (z_{n})}{D_{m, n}^{2}}

Cov {{\tilde{ϕ}}_{1, 2}, {\tilde{ϕ}}_{1, 3}} \approx \frac{L^{4}}{N^{4}} {\frac{\cos (π λ z_{2} f^{2}) \cos (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ^{2} (z_{1})}{D_{1, 2} D_{1, 3}} ∣}_{u = p Δ u, v = q Δ v}

Cov {{\tilde{A}}_{1, 2}, {\tilde{A}}_{1, 3}} \approx \frac{L^{4}}{N^{4}} {\frac{\sin (π λ z_{2} f^{2}) \sin (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ^{2} (z_{1})}{D_{1, 2} D_{1, 3}} ∣}_{u = p Δ u, v = q Δ v} .

ω_{1, 2}^{ϕ} (u, v) = \frac{K_{1}^{ϕ} - K_{2}^{ϕ}}{K_{1}^{ϕ} + K_{3}^{ϕ} - 2 K_{2}^{ϕ}}

ω_{1, 2}^{ϕ} (u, v) = \frac{K_{1}^{a} - K_{2}^{a}}{K_{1}^{a} + K_{3}^{a} - 2 K_{2}^{a}}

K_{1}^{ϕ} = D_{1, 2}^{2} [\cos^{2} (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2} + \cos^{2} (π λ z_{1} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{3}])}^{2} σ_{3}^{2}]

K_{2}^{ϕ} = D_{1, 2} D_{1, 3} [\cos (π λ z_{2} f^{2}) \cos (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2}]

K_{3}^{ϕ} = D_{1, 3}^{2} [\cos^{2} (π λ z_{2} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2} + \cos^{2} (π λ z_{1} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{2}])}^{2} σ_{2}^{2}],

K_{1}^{a} = D_{1, 2}^{2} [\sin^{2} (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2} + \sin^{2} (π λ z_{1} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{3}])}^{2} σ_{3}^{2}]

K_{2}^{a} = D_{1, 2} D_{1, 3} [\sin (π λ z_{2} f^{2}) \sin (π λ z_{3} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2}]

K_{3}^{a} = D_{1, 3}^{2} [\sin^{2} (π λ z_{2} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{1}])}^{2} σ_{1}^{2} + \sin^{2} (π λ z_{1} f^{2}) \sum_{r = 0}^{N - 1} \sum_{s = 0}^{N - 1} {(I^{0} [r, s; z_{2}])}^{2} σ_{2}^{2}],

ω_{1, 2}^{ϕ} (u, v) \approx

\frac{D_{1, 2}^{2} [\cos^{2} (π λ z_{3} f^{2}) σ_{1}^{2} + \cos^{2} (π λ z_{1} f^{2}) σ_{3}^{2}] - D_{1, 2} D_{1, 3} [\cos (π λ z_{2} f^{2}) \cos (π λ z_{3} f^{2}) σ_{1}^{2}]}{\sum_{\begin{matrix} m, n = 2 \\ m \neq n \end{matrix}}^{3} D_{1, m}^{2} [\cos^{2} (π λ z_{n} f^{2}) σ_{1}^{2} + \cos^{2} (π λ z_{1} f^{2}) σ_{n}^{2}] - 2 D_{1} D_{1, 3} [\cos (π λ z_{2} f^{2}) \cos (π λ z_{3} f^{2}) σ_{1}^{2}]} .

\tilde{ϕ} (u, v) = \sum_{m = 1}^{M - 1} \sum_{n = m + 1}^{M} {\hat{ω}}_{m, n}^{ϕ} (u, v) {\hat{ϕ}}_{m, n} (u, v),

\sum_{m = 1}^{M - 1} \sum_{n = m + 1}^{M} {\hat{ω}}_{m, n}^{ϕ} (u, v) = 1 .

\tilde{ϕ} (u, v) = \sum_{m = 2}^{M} ω_{1, m}^{ϕ} (u, v) {\tilde{ϕ}}_{1, m} (u, v) .

Var {\tilde{ϕ} (u, v)} = \sum_{l = 2}^{M} {∣ ω_{1, l}^{ϕ} (u, v) ∣}^{2} Var {{\tilde{ϕ}}_{1, l} (u, v)}

+ 2 Re [\sum_{m = 2}^{M - 1} \sum_{n = m + 1}^{M} ω_{1, m}^{ϕ} (u, v) ω_{1, n}^{ϕ *} (u, v) Cov {{\tilde{ϕ}}_{1, m} (u, v) {\tilde{ϕ}}_{1, n} (u, v)}],

σ_{1, m}^{2} (u, v) \equiv Var {{\tilde{ϕ}}_{1, m} [u, v]},

ρ_{1, m; 1, n}^{(r)} (u, v) + j ρ_{1, m; 1, n}^{(i)} (u, v) \equiv Cov {{\tilde{ϕ}}_{1, m} [u, v], {\tilde{ϕ}}_{1, n} [u, v]},

R_{1, m} (u, v) + j I_{1, m} (u, v) \equiv ω_{1, m}^{ϕ} (u, v) .

Var {\tilde{ϕ}} = \sum_{l = 2}^{M} (R_{1, l}^{2} + I_{1, l}^{2}) σ_{1, l}^{2}

+ 2 {\sum_{m = 2}^{M - 1} \sum_{n = m + 1}^{M} [ρ_{1, m; 1, n}^{(r)} (R_{1, m} R_{1, n} + I_{1, m} I_{1, n}) - ρ_{1, m; 1, n}^{(i)} (R_{1, n} I_{1, m} - R_{1, m} I_{1, n})]} .

{\frac{\partial Var {\tilde{ϕ}}}{\partial R_{1, m}} ∣}_{R_{1, m}^{(op)}} = 0

{\frac{\partial Var {\tilde{ϕ}}}{\partial I_{1, m}} ∣}_{I_{1, m}^{(op)}} = 0,

Hx = b

H = (\begin{matrix} h_{11} & h_{12} & \dots & h_{1, 2 M - 4} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ h_{2 M - 4, 1} & h_{2 M - 4, 2} & \dots & h_{2 M - 4, 2 M - 4} \end{matrix}),

x = {(\begin{matrix} R_{1, 2}^{(op)} & I_{1, 2}^{(op)} & R_{1, 3}^{(op)} & I_{1, 3}^{(op)} & \dots & R_{1, M - 1}^{(op)} & I_{1, M - 1}^{(op)} \end{matrix})}^{T},

b = {(\begin{matrix} b_{1} & b_{2} & \dots & b_{2 M - 4} \end{matrix})}^{T},

x_{m} (u, v) = \frac{\det (H_{m})}{\det (H)} m = 1, 2, 3, \dots, 2 M - 4

ω_{1, m}^{heur} (u, v) = \frac{D_{1, m}^{2} + α_{1, m} \sum_{\begin{matrix} n = 2 \\ n \neq m \end{matrix}}^{M} D_{m, n}^{2}}{\sum_{l = 1}^{M - 1} \sum_{m = l + 1}^{M} D_{l, m}^{2}}

Geometry ‘A’	ε(z ₁)	ε(z ₂)	ε(z ₃)
error level 1	-3 mm	+2 mm	-5 mm
error level 2	-5 mm	+4 mm	+10 mm
error level 3	-19 mm	+12 mm	+17 mm