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.

© 2007 Optical Society of America

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2007 (1)

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/analyserbased 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 edgeenhancement 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)

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, "Laserbased 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)

Appl. Phys. lett. (1)

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]

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)

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]

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)

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]

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. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D (4)

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).

Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, "Polychromaticity in the combined propagation-based/analyserbased 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).

Med. Phys. (4)

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

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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).
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Nucl. Instrum. Methods Phys. Res. A (1)

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

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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).
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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)

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).
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Figures (12)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

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

Fig. 4.
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.

Fig. 5.
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).

Fig. 6.
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.

Fig. 7.
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).

Fig. 8.
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).

Fig. 9.
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.

Fig. 10.
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).

Fig. 11.
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.

Fig. 12.
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).

Tables (1)

Tables Icon

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

Equations (97)

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n ( r ) 1 δ ( r ) + j β ( r ) ,
μ ( r ) = 2 k β ( 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 d z β ( r )
ϕ ( x , y ) = k d z δ ( r )
U m ( x , y ) = G m ( x , y ) * U o ( x , y ) ,
K m ( x , y ) 1 I m ( x , y ) I i ,
K ˜ m ( u , v ) = d x d y K m ( x , y ) exp [ j 2 π ( ux + vy ) ] .
K ˜ m ( u , v ) = 2 G ˜ m a ( u , v ) A ˜ ( u , v ) + 2 G ˜ m p ( u , v ) ϕ ˜ ( u , v )
G ˜ m a ( u , v ) = 1 2 [ G ˜ m ( u , v ) + G ˜ m * ( u , v ) ] ,
G ˜ m p ( u , v ) = 1 2 j [ G ˜ m ( u , v ) G ˜ m * ( u , v ) ] .
A ˜ ( u , v ) = G ˜ n p ( u , v ) K ˜ m ( u , v ) G ˜ m p ( u , v ) K ˜ n ( u , v ) 2 [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ]
ϕ ˜ ( u , v ) = G ˜ n a ( u , v ) K ˜ m ( u , v ) + G ˜ m a ( u , v ) K ˜ n ( u , v ) 2 [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ] .
G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) = 0 .
ϕ ˜ m , n ( u , v ) = α l , m ϕ ( u , v ) ϕ ˜ l , m ( u , v ) + α l , n ϕ ( u , v ) ϕ ˜ l , n ( u , v )
A ˜ m , n ( u , v ) = α l , m a ( u , v ) A ˜ l , m ( u , v ) + α l , n a ( u , v ) A ˜ l , n ( u , v )
α l , m ϕ ( u , v ) = G ˜ n a ( u , v ) [ G ˜ l a ( u , v ) G ˜ m p ( u , v ) G ˜ m a ( u , v ) G ˜ l p ( u , v ) ] G ˜ l a ( u , v ) [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ]
α l , n ϕ ( u , v ) = G ˜ m a ( u , v ) [ G ˜ l a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ l p ( u , v ) ] G ˜ l a ( u , v ) [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ] ,
α l , m a ( u , v ) = G ˜ n p ( u , v ) [ G ˜ l a ( u , v ) G ˜ m p ( u , v ) G ˜ m a ( u , v ) G ˜ l p ( u , v ) ] G ˜ l p ( u , v ) [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ] ,
α l , n a ( u , v ) = G ˜ m p ( u , v ) [ G ˜ l a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ l p ( u , v ) ] G ˜ l p ( u , v ) [ G ˜ m a ( u , v ) G ˜ n p ( u , v ) G ˜ n a ( u , v ) G ˜ m p ( u , v ) ] ,
α l , m ϕ ( u , v ) + α l , n ϕ ( u , v ) 1
α l , m a ( u , v ) + α l , n a ( u , v ) 1 .
ϕ ˜ ( u , v ) = ω 1 , 2 ϕ ( u , v ) ϕ ˜ 1 , 2 ( u , v ) + ω 1 , 3 ϕ ( u , v ) ϕ ˜ 1 , 3 ( u , v )
A ˜ ( u , v ) = ω 1 , 2 a ( u , v ) A ˜ 1 , 2 ( u , v ) + ω 1 , 3 a ( u , v ) 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 ) Var { ϕ ˜ m , n ( u , v ) }
ρ k , l ; m , n ( r ) ( u , v ) + j ρ k , l ; m , n ( i ) ( u , v ) Cov { ϕ ˜ k , l ( u , v ) , ϕ ˜ m , n ( u , v ) }
R m , n ( u , v ) + j I m , n ( u , v ) ω m , n ϕ ( u , v ) ,
Var { 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 { ϕ ˜ 1 , 2 ( u , v ) , ϕ ˜ 1 , 3 ( u , v ) } ]
Var { ϕ ˜ } R 1 , 2 R 1 , 2 ( op ) = 0
Var { ϕ ˜ } I 1 , 2 I 1 , 2 ( op ) = 0 ,
R 1 , 2 ( op ) ( u , v ) = 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 ) = 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 ) = exp [ j kz m ] j λ z m exp [ j π x 2 + y 2 λ z m ] ,
G ˜ m ( u , v ) = exp [ j kz m j π λ z m ( u 2 + v 2 ) ] .
ϕ ˜ m , n ( u , v ) = cos ( π λ z n f 2 ) K ˜ m ( u , v ) + cos ( π λ z m f 2 ) K ˜ n ( u , v ) D m , n
A ˜ m , n ( u , v ) = sin ( π λ z m f 2 ) K ˜ n ( u , v ) + sin ( π λ z n f 2 ) K ˜ m ( u , v ) D m , n ,
D m , n ( u , v ) 2 sin ( π λ f 2 ( z m z n ) ) .
u 2 + v 2 = l λ ( z m z n ) ,
u M 2 + v M 2 1 λ ( z m z n ) .
Var { ϕ ˜ m , n ( u , v ) } = cos 2 ( π λ z n f 2 ) Var { I ˜ m ( u , v ) } + cos 2 ( π λ z m f 2 ) Var { I ˜ n ( u , v ) } D m , n 2
Var { A ˜ m , n ( u , v ) } = sin 2 ( π λ z n f 2 ) Var { I ˜ m ( u , v ) } + sin 2 ( π λ z m f 2 ) Var { I ˜ n ( u , v ) } D m , n 2
Cov { ϕ ˜ 1 , 2 ( u , v ) , ϕ ˜ 1 , 3 ( u , v ) } = cos ( π λ z 2 f 2 ) cos ( π λ z 3 f 2 ) Var { I ˜ 1 ( u , v ) } D 1 , 2 D 1 , 3
Cov { A ˜ 1 , 2 ( u , v ) , A ˜ 1 , 3 ( u , v ) } = sin ( π λ z 2 f 2 ) sin ( π λ z 3 f 2 ) Var { I ˜ 1 ( u , v ) } D 1 , 2 D 1 , 3
Var { ϕ ˜ m , n ( u , v ) } 1 D m , n 2 ( u , v )
Var { A ˜ m , n ( u , v ) } 1 D m , n 2 ( u , v ) .
ω 1 , 2 heur ( u , v ) = 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 ) = 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 ,
I ˜ m [ p , q ] = I m 0 ˜ [ p , q ] + n ˜ m [ p , q ]
I ˜ m [ p , q ] = r = 0 N 1 s = 0 N 1 I m [ r , s ] exp [ j 2 π N ( pr + qs ) ]
I m 0 ˜ [ p , q ] = r = 0 N 1 s = 0 N 1 I m 0 [ r , s ] exp [ j 2 π N ( pr + qs ) ]
n ˜ m [ p , q ] = r = 0 N 1 s = 0 N 1 n m [ r , s ] exp [ j 2 π N ( pr + qs ) ] ,
Var { n ˜ m [ p , q ] } = r , r = 0 N 1 s , s = 0 N 1 exp [ j 2 π N ( p ( r r ) + q ( s s ) ) ] Cov { n m [ r , s ] , n m [ r , s ] }
= r = 0 N 1 s = 0 N 1 E { ( n m [ r , s ] ) 2 } = r = 0 N 1 s = 0 N 1 ( I m 0 [ r , s ] ) 2 σ 2 ( z m ) ,
I ˜ m ( u , v ) u = p Δ u , v = q Δ v L 2 N 2 I ˜ m [ p , q ] ,
Var { I ˜ m ( u , v ) } u = p Δ u , v = q Δ v L 4 N 4 Var { I ˜ m [ p , q ] } = L 4 N 4 σ 2 ( z m ) r = 0 N 1 s = 0 N 1 ( I m 0 [ r , s ] ) 2 .
Var { ϕ ˜ m , n ( u , v ) } u = p Δ u , v = q Δ v
L 4 N 4 cos 2 ( π λ z n f 2 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z m ] ) 2 σ 2 ( z m ) + cos 2 ( π λ z m f 2 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z n ] ) 2 σ 2 ( z n ) D m , n 2
Var { A ˜ m , n ( u , v ) } u = p Δ u , v = q Δ v
L 4 N 4 sin 2 ( π λ z n f 2 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z m ] ) 2 σ 2 ( z m ) + sin 2 ( π λ z m f 2 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z n ] ) 2 σ 2 ( z n ) D m , n 2
Cov { ϕ ˜ 1 , 2 , ϕ ˜ 1 , 3 } L 4 N 4 cos ( π λ z 2 f 2 ) cos ( π λ z 3 f 2 ) r = 0 N 1 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 { A ˜ 1 , 2 , A ˜ 1 , 3 } L 4 N 4 sin ( π λ z 2 f 2 ) sin ( π λ z 3 f 2 ) r = 0 N 1 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 ) = K 1 ϕ K 2 ϕ K 1 ϕ + K 3 ϕ 2 K 2 ϕ
ω 1 , 2 ϕ ( u , v ) = 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 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z 1 ] ) 2 σ 1 2 + cos 2 ( π λ z 1 f 2 ) r = 0 N 1 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 ) r = 0 N 1 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 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z 1 ] ) 2 σ 1 2 + cos 2 ( π λ z 1 f 2 ) r = 0 N 1 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 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z 1 ] ) 2 σ 1 2 + sin 2 ( π λ z 1 f 2 ) r = 0 N 1 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 ) r = 0 N 1 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 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z 1 ] ) 2 σ 1 2 + sin 2 ( π λ z 1 f 2 ) r = 0 N 1 s = 0 N 1 ( I 0 [ r , s ; z 2 ] ) 2 σ 2 2 ] ,
ω 1 , 2 ϕ ( u , v )
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 ] m , n = 2 m n 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 ] .
ϕ ˜ ( u , v ) = m = 1 M 1 n = m + 1 M ω ̂ m , n ϕ ( u , v ) ϕ ̂ m , n ( u , v ) ,
m = 1 M 1 n = m + 1 M ω ̂ m , n ϕ ( u , v ) = 1 .
ϕ ˜ ( u , v ) = m = 2 M ω 1 , m ϕ ( u , v ) ϕ ˜ 1 , m ( u , v ) .
Var { ϕ ˜ ( u , v ) } = l = 2 M ω 1 , l ϕ ( u , v ) 2 Var { ϕ ˜ 1 , l ( u , v ) }
+ 2 Re [ m = 2 M 1 n = m + 1 M ω 1 , m ϕ ( u , v ) ω 1 , n ϕ * ( u , v ) Cov { ϕ ˜ 1 , m ( u , v ) ϕ ˜ 1 , n ( u , v ) } ] ,
σ 1 , m 2 ( u , v ) Var { ϕ ˜ 1 , m [ u , v ] } ,
ρ 1 , m ; 1 , n ( r ) ( u , v ) + j ρ 1 , m ; 1 , n ( i ) ( u , v ) Cov { ϕ ˜ 1 , m [ u , v ] , ϕ ˜ 1 , n [ u , v ] } ,
R 1 , m ( u , v ) + j I 1 , m ( u , v ) ω 1 , m ϕ ( u , v ) .
Var { ϕ ˜ } = l = 2 M ( R 1 , l 2 + I 1 , l 2 ) σ 1 , l 2
+ 2 { m = 2 M 1 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 ) ] } .
Var { ϕ ˜ } R 1 , m R 1 , m ( op ) = 0
Var { ϕ ˜ } I 1 , m I 1 , m ( op ) = 0 ,
Hx = b
H = ( h 11 h 12 h 1 , 2 M 4 h 2 M 4 , 1 h 2 M 4 , 2 h 2 M 4 , 2 M 4 ) ,
x = ( R 1 , 2 ( op ) I 1 , 2 ( op ) R 1 , 3 ( op ) I 1 , 3 ( op ) R 1 , M 1 ( op ) I 1 , M 1 ( op ) ) T ,
b = ( b 1 b 2 b 2 M 4 ) T ,
x m ( u , v ) = det ( H m ) det ( H ) m = 1 , 2 , 3 , , 2 M 4
ω 1 , m heur ( u , v ) = D 1 , m 2 + α 1 , m n = 2 n m M D m , n 2 l = 1 M 1 m = l + 1 M D l , m 2

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