Abstract

Intensity diffraction tomography (I-DT) reconstruction theory provides a mathematical mapping between two in-line intensity measurements acquired at a given tomographic view angle and Fourier components of the object function. Poles in this mapping will cause certain Fourier components to contain greatly amplified noise levels when applied to noisy measurement data, which can result in noisy and distorted images in practice. We investigate the statistically principled use of multiple in-line intensity measurements in I-DT. Reconstruction methods are developed that exploit the statistical structure of the in-line measurements to minimize the variance of the estimated Fourier components of the object function.

© 2007 Optical Society of America

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  10. T. Beetz, C. Jacobsen, and A. Stein, "Soft x-ray diffraction tomography: simulations and first experimerimental results," J. Physiol. Paris IV104, 31-34 (2003).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. C. E. Metz and X. Pan, "A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT," IEEE Trans. Med. Imaging 14, 643-658 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  37. B. Chen and J. J. Stamnes, "Validity of diffraction tomography based on the first-Born and first-Rytov approximations," Appl. Opt. 37, 2996-3006 (1998).
    [CrossRef]

2006 (2)

M. A. Anastasio, D. Shi, and G. Gbur, "Multi-spectral intensity diffraction tomography reconstruction theory: I. Quasi-nondispersive objects," J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

2005 (3)

2004 (3)

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

2003 (2)

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

T. Beetz, C. Jacobsen, and A. Stein, "Soft x-ray diffraction tomography: simulations and first experimerimental results," J. Physiol. Paris IV104, 31-34 (2003).

2002 (2)

2001 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
[CrossRef]

2000 (4)

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

M. A. Anastasio and X. Pan, "Computationally efficient and statistically robust image reconstruction in 3D diffraction tomography," J. Opt. Soc. Am. A 17, 391-400 (2000).
[CrossRef]

1998 (3)

X. Pan, "A unified reconstruction theory for diffraction tomography with considerations of noise control," J. Opt. Soc. Am. A 15, 2312-2326 (1998).
[CrossRef]

B. Chen and J. J. Stamnes, "Validity of diffraction tomography based on the first-Born and first-Rytov approximations," Appl. Opt. 37, 2996-3006 (1998).
[CrossRef]

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

1996 (2)

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

1995 (3)

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
[CrossRef]

C. E. Metz and X. Pan, "A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT," IEEE Trans. Med. Imaging 14, 643-658 (1995).
[CrossRef] [PubMed]

1993 (1)

1992 (2)

A. J. Devaney and A. Schatzberg, "Coherent optical tomographic microscope," Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

1986 (1)

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak, and L. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-874 (1984).
[CrossRef]

1982 (1)

1970 (1)

1969 (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Anastasio, M. A.

M. A. Anastasio, D. Shi, and G. Gbur, "Multi-spectral intensity diffraction tomography reconstruction theory: I. Quasi-nondispersive objects," J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

M. A. Anastasio and X. Pan, "Computationally efficient and statistically robust image reconstruction in 3D diffraction tomography," J. Opt. Soc. Am. A 17, 391-400 (2000).
[CrossRef]

M. A. Anastasio, Y. Huang, D. Shi, and G. Gbur, "Noise properties of intensity diffraction tomography," in Medical Imaging 2004: Physics of Medical Imaging, M.I.Yaffe and M.J.Flynn, eds., Proc. SPIE 5368, 272-280 (2004).

Arsenault, H.

Baranov, V.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Barty, A.

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

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Beetz, T.

T. Beetz, C. Jacobsen, and A. Stein, "Soft x-ray diffraction tomography: simulations and first experimerimental results," J. Physiol. Paris IV104, 31-34 (2003).

Borisov, V.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Born, M.

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

Bulyshev, A.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Calef, B.

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

Carney, P. S.

M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

Chen, B.

Chommeloux, L.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

Devaney, A. J.

P. Guo and A. J. Devaney, "Comparison of reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 22, 2338-2347 (2005).
[CrossRef]

M. Maleki and A. J. Devaney, "Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

A. J. Devaney and A. Schatzberg, "Coherent optical tomographic microscope," Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

Dezern, K.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Duchene, B.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

Fienup, J. R.

Gbur, G.

M. A. Anastasio, D. Shi, and G. Gbur, "Multi-spectral intensity diffraction tomography reconstruction theory: I. Quasi-nondispersive objects," J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
[CrossRef]

G. Gbur and E. Wolf, "Hybrid diffraction tomography without phase information," J. Opt. Soc. Am. A 19, 2194-2202 (2002).
[CrossRef]

G. Gbur and E. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
[CrossRef]

M. A. Anastasio, Y. Huang, D. Shi, and G. Gbur, "Noise properties of intensity diffraction tomography," in Medical Imaging 2004: Physics of Medical Imaging, M.I.Yaffe and M.J.Flynn, eds., Proc. SPIE 5368, 272-280 (2004).

Guo, P.

Howells, M. R.

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

Huang, Y.

M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio, Y. Huang, D. Shi, and G. Gbur, "Noise properties of intensity diffraction tomography," in Medical Imaging 2004: Physics of Medical Imaging, M.I.Yaffe and M.J.Flynn, eds., Proc. SPIE 5368, 272-280 (2004).

Jacobsen, C.

T. Beetz, C. Jacobsen, and A. Stein, "Soft x-ray diffraction tomography: simulations and first experimerimental results," J. Physiol. Paris IV104, 31-34 (2003).

Jacobsen, C. J.

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

Joachimowicz, N.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak, and L. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-874 (1984).
[CrossRef]

Larsen, L.

M. Slaney, A. C. Kak, and L. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-874 (1984).
[CrossRef]

Lathi, B. P.

B. P. Lathi, Signal Processing and Linear Systems (Oxford U. Press, 1998).

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
[CrossRef]

Lesselier, D.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

Lowenthal, S.

Maleki, M.

Mcmahon, P. I.

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

Metz, C. E.

C. E. Metz and X. Pan, "A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT," IEEE Trans. Med. Imaging 14, 643-658 (1995).
[CrossRef] [PubMed]

Nugent, K. A.

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

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Paganin, D.

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

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Pan, X.

Pichot, Ch.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
[CrossRef]

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Schatzberg, A.

A. J. Devaney and A. Schatzberg, "Coherent optical tomographic microscope," Proc. SPIE 1767, 62-71 (1992).
[CrossRef]

Schotland, J. C.

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

Semenov, S.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Shi, D.

M. A. Anastasio, D. Shi, and G. Gbur, "Multi-spectral intensity diffraction tomography reconstruction theory: I. Quasi-nondispersive objects," J. Opt. Soc. Am. A 23, 1359-1368 (2006).
[CrossRef]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," Proc. SPIE 5535, 361-368 (2004).
[CrossRef]

M. A. Anastasio, Y. Huang, D. Shi, and G. Gbur, "Noise properties of intensity diffraction tomography," in Medical Imaging 2004: Physics of Medical Imaging, M.I.Yaffe and M.J.Flynn, eds., Proc. SPIE 5368, 272-280 (2004).

Sizov, Y.

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
[CrossRef] [PubMed]

Slaney, M.

M. Slaney, A. C. Kak, and L. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. 32, 860-874 (1984).
[CrossRef]

Souvorov, A.

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S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
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T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
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T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
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T. C. Wedberg and J. J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
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M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yun, W.

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

AIP Conf. Proc. (1)

M. R. Howells, B. Calef, C. J. Jacobsen, J. H. Spence, and W. Yun, "A modern approach to x-ray holography," AIP Conf. Proc. 507, 587-592 (2000).
[CrossRef]

Appl. Opt. (2)

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P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, V. Borisov, Y. Sizov, A. Starostin, K. Dezern, G. Tatsis, and V. Baranov, "Microwave tomography: two-dimensional system for biological imaging," IEEE Trans. Biomed. Eng. 43, 869-877 (1996).
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IEEE Trans. Image Process. (1)

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Image Process. 1, 221-228 (1992).
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IEEE Trans. Med. Imaging (1)

C. E. Metz and X. Pan, "A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT," IEEE Trans. Med. Imaging 14, 643-658 (1995).
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Inf. Sci. (N.Y.) (1)

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
[CrossRef]

Inverse Probl. (2)

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1998).
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A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
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V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
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[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

J. Physiol. Paris (1)

T. Beetz, C. Jacobsen, and A. Stein, "Soft x-ray diffraction tomography: simulations and first experimerimental results," J. Physiol. Paris IV104, 31-34 (2003).

Meas. Sci. Technol. (1)

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

Opt. Commun. (2)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

T. C. Wedberg and J. J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
[CrossRef]

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D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
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M. A. Anastasio, Y. Huang, G. Gbur, and P. S. Carney, "Investigation of 3D microscopy using intensity diffraction tomography," Proc. SPIE 6090, 77-84 (2006).

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T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods for optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

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E. Wolf, "Principles and development of diffraction tomography," in Trends in Optics, A.Consortini, ed. (Academic, 1996).
[CrossRef]

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

M. A. Anastasio, Y. Huang, D. Shi, and G. Gbur, "Noise properties of intensity diffraction tomography," in Medical Imaging 2004: Physics of Medical Imaging, M.I.Yaffe and M.J.Flynn, eds., Proc. SPIE 5368, 272-280 (2004).

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Figures (12)

Fig. 1
Fig. 1

Measurement geometry of I-DT. In principle, intensity measurements are required on only two detector planes. In practice, additional measurements are warranted to mitigate amplification of data noise.

Fig. 2
Fig. 2

(a) Plots of the optimal combination coefficients (for u = 0 ) as described at the end of Subsection 4E. The solid, dashed, and dashed–dotted curves depict ω 1 , 2 = R 1 , 2 , ω 1 , 3 = R 1 , 3 , and ω 2 , 3 = R 2 , 3 , respectively. (b) Plots of Var { F m , n [ u , v ] } that were computed analytically according to Eq. (65), for u = 0 and a given tomographic view angle. The variances Var { F 1 , 2 [ 0 , v ] } , Var { F 1 , 3 [ 0 , v ] } , and Var { F 2 , 3 [ 0 , v ] } are described by the solid, dashed, and dashed–dotted curves, respectively. For a given detector pair, note the complementary behavior between the plots in (a) and (b).

Fig. 3
Fig. 3

x = 0 slice of the 3D phantom object representing f ( r ) in the computer simulations. Its real and imaginary components are shown in (a) and (b), respectively.

Fig. 4
Fig. 4

Empirical variance estimates of the different Fourier data computed at a given view angle as described at the beginning of Section 6. Profiles corresponding to u = 0 are displayed for Var { F 1 , 2 [ u , v ] } (dashed curve), Var { F 1 , 3 [ u , v ] } (dashed–dotted curve), Var { F 2 , 3 [ u , v ] } (dotted curve), Var { F 1 3 [ u , v ] } (point–plus curve), and Var { F opt [ u , v ] } (solid curve). As expected, the value of Var { F opt [ u , v ] } is lower than the values associated with the other estimates for all frequencies.

Fig. 5
Fig. 5

Comparison of empirically determined and analytically calculated variance maps for a given view angle. Images of log [ Var { F emp opt [ u , v ] } ] and log [ Var { F opt [ u , v ] } analytic ] are shown in (a) and (b), respectively. Profiles through the two images are contained in (c), and the variances Var { F emp opt [ 0 , v ] } , Var { F opt [ 0 , v ] } analytic are depicted by the dashed and solid curves, respectively. For all frequency components, the agreement between the empirical and analytic variance maps is excellent.

Fig. 6
Fig. 6

Images of the x = 0 slice of f ( r ) that were reconstructed from noiseless Fourier data. The real and imaginary components of f ( r ) are shown in (a) and (b), respectively. From left to right in the figures, the images were reconstructed from the data F 1 , 2 [ u , v ] , F 1 , 3 [ u , v ] , F 2 , 3 [ u , v ] , F 1 3 [ u , v ] , and F opt [ u , v ] , respectively, obtained at each tomographic view angle.

Fig. 7
Fig. 7

Images of the x = 0 slice of f ( r ) that were reconstructed from noisy Fourier data corresponding to noise level 1. The real and imaginary components of f ( r ) are shown in Figs. 6a, 6b, respectively. From left to right in the figures, the images were reconstructed from noisy realizations of F 1 , 2 [ u , v ] , F 1 , 3 [ u , v ] , F 2 , 3 [ u , v ] , F 1 3 [ u , v ] , and F opt [ u , v ] , respectively, obtained at each tomographic view angle.

Fig. 8
Fig. 8

Dashed profiles shown in (a) and (b) correspond to central rows of the images in Fig. 7, right-most panel, which were reconstructed from the Fourier data F opt [ u , v ] corresponding to noise level 1.

Fig. 9
Fig. 9

Images of the x = 0 slice of f ( r ) that were reconstructed from noisy Fourier data corresponding to noise level 2. The ordering of the images is the same as in Fig. 7.

Fig. 10
Fig. 10

Empirical estimates of (a) Var { f 1 , 2 ( r ) } , (b) Var { f 1 , 3 ( r ) } , (c) Var { f 2 , 3 ( r ) } , (d) Var { f 1 3 ( r ) } , and (e) Var { f opt ( r ) } corresponding to noise level 1.

Fig. 11
Fig. 11

Series of images of the x = 0 slice of f ( r ) reconstructed from noiseless versions of F opt [ u , v ] corresponding to measurement geometries with different detector-plane spacings. The value of Δ 1 , 3 = 0.095 is fixed and Δ 1 , 2 and Δ 2 , 3 are varied. From left to right, the detector plane spacings correspond to Δ 1 , 2 = 0.026 m , Δ 1 , 2 = 0.038 m , Δ 1 , 2 = 0.0475 m , Δ 1 , 2 = 0.057 m , and Δ 1 , 2 = 0.069 m , respectively. Subfigures (a) and (b) depict the real and imaginary components, respectively, of the reconstructed object function.

Fig. 12
Fig. 12

Noisy versions of the images shown in Fig. 11, corresponding to noise level 3.

Equations (90)

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f ( r ) = k 2 4 π [ n 2 ( r ) 1 ] .
U ( x , y ; z ) = U i ( x , y , z ) exp [ ψ ( x , y ; z ) ] ,
Ψ ( u , v ; z ) = 1 ( 2 π ) 2 R 2 d x d y ψ ( x , y ; z ) exp [ i ( u x + v y ) ] ,
F ( K ) = 1 ( 2 π ) 3 V d r f ( r ) exp [ i K r ] ,
Ψ ( u , v ; z = d ) = 4 π 2 w i F ( K = u s ̂ 1 + v s ̂ 2 + ( w k ) s ̂ 0 ) exp [ i ( w k ) d ] ,
w k 2 u 2 v 2 .
d I ( x , y ; z ) log [ I ( x , y ; z ) ] = ψ ( x , y ; z ) + ψ * ( x , y ; z ) ,
D I ( u , v ; z ) = ( 2 π ) 2 w 2 i { w F [ u , v ] exp [ i ( w k ) z ] w F * [ u , v ] exp [ i ( w k ) z ] } ,
F [ u , v ] F ( u s ̂ 1 + v s ̂ 2 + ( w k ) s ̂ 0 )
D Δ ( u , v ; d ) D I ( u , v ; d ) D I ( u , v ; d + Δ ) exp [ i ( w k ) Δ ] .
F [ u , v ] = D Δ ( u , v ; d ) w exp [ i ( w k ) d ] ( 2 π ) 2 i ( 1 exp [ 2 i ( w k ) Δ ] ) ,
( w k ) Δ = n π ,
Var { F [ u , v ] } = w 2 ( 2 π ) 4 Var { D Δ ( u , v ; d ) } 2 [ 1 cos ( 2 ( w k ) Δ ) ] ,
F [ u , v ] = ω 1 , 2 ( u , v ) F 1 , 2 [ u , v ] + ω 1 , 3 ( u , v ) F 1 , 3 [ u , v ] + ω 2 , 3 ( u , v ) F 2 , 3 [ u , v ] ,
ω 1 , 2 ( u , v ) + ω 1 , 3 ( u , v ) + ω 2 , 3 ( u , v ) = 1 .
Var { Ψ ( u , v ; z ) } = 16 π 4 w 2 Var { F [ u , v ] } .
1 ( 2 π ) 4 R 2 d x d y Var { ψ ( x , y ; z ) } = R 2 d u d v Var { Ψ ( u , v ; z ) } ,
Var { F [ u , v ] } = ω 1 , 2 ( u , v ) 2 Var { F 1 , 2 [ u , v ] } + ω 1 , 3 ( u , v ) 2 Var { F 1 , 3 [ u , v ] } + ω 2 , 3 ( u , v ) 2 Var { F 2 , 3 [ u , v ] } + 2 Re [ ω 1 , 2 ( u , v ) ω 1 , 3 * ( u , v ) Cov { F 1 , 2 [ u , v ] , F 1 , 3 [ u , v ] } + ω 1 , 2 ( u , v ) ω 2 , 3 * ( u , v ) Cov { F 1 , 2 [ u , v ] , F 2 , 3 [ u , v ] } + ω 1 , 3 ( u , v ) ω 2 , 3 * ( u , v ) Cov { F 1 , 3 [ u , v ] , F 2 , 3 [ u , v ] } ] .
Var { F m , n [ u , v ] } σ m , n 2 ( u , v ) ,
Cov { F l , j [ u , v ] , F m , n [ u , v ] } ρ l , j ; m , n ( r ) ( u , v ) + i ρ l , j ; m , n ( i ) ( u , v ) ,
ω m , n ( u , v ) R m , n ( u , v ) + i I m , n ( u , v ) ,
R 2 , 3 ( u , v ) = 1 R 1 , 2 ( u , v ) R 1 , 3 ( u , v ) ,
I 2 , 3 ( u , v ) = I 1 , 2 ( u , v ) I 1 , 3 ( u , v ) .
Var { F [ u , v ] } = σ 1 , 2 2 ( R 1 , 2 2 + I 1 , 2 2 ) + σ 1 , 3 2 ( R 1 , 3 2 + I 1 , 3 2 ) + σ 2 , 3 2 [ ( 1 R 1 , 2 R 1 , 3 ) 2 + ( I 1 , 2 + I 1 , 3 ) 2 ] + 2 [ ρ 1 , 2 ; 1 , 3 ( r ) ( R 1 , 2 R 1 , 3 + I 1 , 2 I 1 , 3 ) + ρ 1 , 2 ; 1 , 3 ( i ) ( R 1 , 2 I 1 , 3 R 1 , 3 I 1 , 2 ) + ρ 1 , 2 ; 2 , 3 ( r ) ( R 1 , 2 R 1 , 2 2 R 1 , 2 R 1 , 3 I 1 , 2 2 I 1 , 2 I 1 , 3 ) ρ 1 , 2 ; 2 , 3 ( i ) ( I 1 , 2 R 1 , 3 I 1 , 2 + R 1 , 2 I 1 , 3 ) + ρ 1 , 3 ; 2 , 3 ( r ) ( R 1 , 3 R 1 , 2 R 1 , 3 R 1 , 3 2 I 1 , 2 I 1 , 3 I 1 , 3 2 ) ρ 1 , 3 ; 2 , 3 ( i ) ( I 1 , 3 R 1 , 2 I 1 , 3 + R 1 , 3 I 1 , 2 ) ] .
Var { F } R 1 , 2 R 1 , 2 = R 1 , 2 ( op ) = 0 ;
Var { F } I 1 , 2 I 1 , 2 = I 1 , 2 ( op ) = 0 ;
Var { F } R 1 , 3 R 1 , 3 = R 1 , 3 ( op ) = 0 ;
Var { F } I 1 , 3 I 1 , 3 = I 1 , 3 ( op ) = 0 .
A x = b ,
x = ( R 1 , 2 I 1 , 2 R 1 , 3 I 1 , 3 ) ,
b = ( σ 2 , 3 2 ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( i ) σ 2 , 3 2 ρ 1 , 3 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) ) ,
A = [ σ 1 , 2 2 + σ 2 , 3 2 2 ρ 1 , 2 ; 2 , 3 ( r ) 0 σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) ρ 1 , 2 ; 1 , 3 ( i ) ρ 1 , 2 ; 2 , 3 ( i ) + ρ 1 , 3 ; 2 , 3 ( i ) 0 σ 1 , 2 2 + σ 2 , 3 2 2 ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( i ) ρ 1 , 2 ; 1 , 3 ( i ) ρ 1 , 3 ; 2 , 3 ( i ) σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( i ) ρ 1 , 2 ; 1 , 3 ( i ) ρ 1 , 3 ; 2 , 3 ( i ) σ 1 , 3 2 + σ 2 , 3 2 2 ρ 1 , 3 ; 2 , 3 ( r ) 0 ρ 1 , 2 ; 1 , 3 ( i ) ρ 1 , 2 ; 2 , 3 ( i ) + ρ 1 , 3 ; 2 , 3 ( i ) σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) 0 σ 1 , 3 2 + σ 2 , 3 2 2 ρ 1 , 3 ; 2 , 3 ( r ) ] .
R 1 , 2 = A 1 A ,
I 1 , 2 = A 2 A ,
R 1 , 3 = A 3 A ,
I 1 , 3 = A 4 A ,
A r x r = b r ,
x r = ( R 1 , 2 R 1 , 3 ) ,
b r = ( σ 2 , 3 2 ρ 1 , 2 ; 2 , 3 ( r ) σ 2 , 3 2 ρ 1 , 3 ; 2 , 3 ( r ) ) ,
A r = [ σ 1 , 2 2 + σ 2 , 3 2 2 ρ 1 , 2 ; 2 , 3 ( r ) σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) σ 2 , 3 2 + ρ 1 , 2 ; 1 , 3 ( r ) ρ 1 , 2 ; 2 , 3 ( r ) ρ 1 , 3 ; 2 , 3 ( r ) σ 1 , 3 2 + σ 2 , 3 2 2 ρ 1 , 3 ; 2 , 3 ( r ) ] .
R 1 , 2 ( u , v ) = A 1 r A r ,
R 1 , 3 ( u , v ) = A 2 r A r ,
Var { F m , n [ u , v ] } w 2 32 π 4 [ 1 cos ( 2 ( w k ) Δ m , n ) ] .
ω m , n heur ( u , v ) = 1 cos ( 2 ( w k ) Δ m , n ) 3 cos ( 2 ( w k ) Δ 1 , 2 ) cos ( 2 ( w k ) Δ 1 , 3 ) cos ( 2 ( w k ) Δ 2 , 3 ) ,
I [ r , s ; z ] = I ( x , y ; z ) x = r Δ x , y = s Δ y ,
I [ r , s ; z ] = I [ r , s ; z ] + n [ r , s ; z ] ,
E { n [ r , s ; z ] } = 0 ,
Var { n [ r , s ; z ] } = I 2 [ r , s ; z ] σ 2 ( z ) ,
Cov { n [ r , s ; z ] , n [ r , s ; z + Δ ] } = Var { n [ r , s ; z ] } δ r r δ s s δ Δ 0 ,
δ a b { 1 , if a = b 0 , if a b } ,
Cov { I [ r , s ; z ] , I [ r , s , z + Δ ] } = Var { I [ r , s , z ] } δ r r δ s s δ Δ 0 .
n [ r , s ; z ] n [ r , s ; z ] I [ r , s ; z ] .
I [ r , s ; z ] = I [ r , s ; z ] ( 1 + n [ r , s ; z ] ) .
n [ r , s ; z ] 1 ,
log ( 1 + n [ r , s ; z ] ) n [ r , s ; z ] ,
log I [ r , s ; z ] = log I [ r , s ; z ] + n [ r , s ; z ] ,
d I [ r , s ; z ] = d I [ r , s ; z ] + n [ r , s ; z ] ,
d I [ r , s ; z ] log I [ r , s ; z ]
D I [ p , q ; z ] = D I [ p , q ; z ] + n ̂ [ p , q ; z ] ,
D I [ p , q ; z ] = r = 0 N 1 s = 0 N 1 d I [ r , s ; z ] exp [ i 2 π N ( p r + q s ) ] ,
D I [ p , q ; z ] = r = 0 N 1 s = 0 N 1 d I [ r , s ; z ] exp [ i 2 π N ( p r + q s ) ] ,
n ̂ [ p , q ; z ] = r = 0 N 1 s = 0 N 1 n [ r , s ; z ] exp [ i 2 π N ( p r + q s ) ] ,
Var { n ̂ [ p , q ; z ] } = r , r = 0 N 1 s , s = 0 N 1 exp [ i 2 π N ( p ( r r ) + q ( s s ) ) ] Cov { n [ r , s ; z ] , n [ r , s ; z ] } ,
= r = 0 N 1 s = 0 N 1 E { ( n [ r , s ; z ] ) 2 } = N 2 σ 2 ( z ) ,
Var { D I [ p , q ; z ] } = N 2 σ 2 ( z ) .
D I ( u , v ; z ) u = p Δ u , v = q Δ v L 2 N 2 D I [ p , q ; z ] ,
Var { D I ( u , v ; z ) } u = p Δ u , v = q Δ u L 4 N 4 Var { D I [ p , q ; z ] } = L 4 N 2 σ 2 ( z ) .
Var { D Δ ( u , v ; z ) } u = p Δ u , v = q Δ v = [ Var { D I ( u , v ; d ) } + Var { D I ( u , v ; d + Δ ) } ] u = p Δ u , v = q Δ v ,
Var { D Δ ( u , v ; z ) } u = p Δ u , v = q Δ v L 4 N 2 [ σ 2 ( d ) + σ 2 ( d + Δ ) ] .
Var { F m , n [ u , v ] } u = p Δ u , v = q Δ v w 2 L 4 [ σ 2 ( z = z m ) + σ 2 ( z = z n ) ] 32 N 2 π 4 [ 1 cos ( 2 ( w k ) Δ m , n ) ] ,
Cov { F 1 , 2 [ u , v ] , F 1 , 3 [ u , v ] } u = p Δ u , v = q Δ v w 2 L 4 σ 2 ( z = z 1 ) N 2 ( 2 π ) 4 ( 1 exp [ 2 i ( w k ) Δ 1 , 2 ] ) ( 1 exp [ 2 i ( w k ) Δ 1 , 3 ] ) u = p Δ u , v = q Δ v ,
Cov { F 1 , 2 [ u , v ] , F 2 , 3 [ u , v ] } u = p Δ u , v = q Δ v w 2 L 4 σ 2 ( z = z 2 ) exp [ 2 i ( w k ) Δ 1 , 2 ] N 2 ( 2 π ) 4 ( 1 exp [ 2 i ( w k ) Δ 1 , 2 ] ) ( 1 exp [ 2 i ( w k ) Δ 2 , 3 ] ) u = p Δ u , v = q Δ v ,
Cov { F 1 , 3 [ u , v ] , F 2 , 3 [ u , v ] } u = p Δ u , v = q Δ v w 2 L 4 σ 2 ( z = z 3 ) exp [ 2 i ( w k ) Δ 1 , 2 ] N 2 ( 2 π ) 4 ( 1 exp [ 2 i ( w k ) Δ 1 , 3 ] ) ( 1 exp [ 2 i ( w k ) Δ 2 , 3 ] ) u = p Δ u , v = q Δ v ,
R 1 , 2 = σ 3 2 [ 1 cos ( 2 ( w k ) Δ 1 , 2 ) ] σ 1 2 [ 1 cos ( 2 ( w k ) Δ 2 , 3 ) ] + σ 2 2 [ 1 cos ( 2 ( w k ) Δ 1 , 3 ) ] + σ 3 2 [ 1 cos ( 2 ( w k ) Δ 1 , 2 ) ] ,
R 1 , 3 = σ 2 2 [ 1 cos ( 2 ( w k ) Δ 1 , 3 ) ] σ 1 2 [ 1 cos ( 2 ( w k ) Δ 2 , 3 ) ] + σ 2 2 [ 1 cos ( 2 ( w k ) Δ 1 , 3 ) ] + σ 3 2 [ 1 cos ( 2 ( w k ) Δ 1 , 2 ) ] ,
R 2 , 3 = σ 1 2 [ 1 cos ( 2 ( w k ) Δ 2 , 3 ) ] σ 1 2 [ 1 cos ( 2 ( w k ) Δ 2 , 3 ) ] + σ 2 2 [ 1 cos ( 2 ( w k ) Δ 1 , 3 ) ] + σ 3 2 [ 1 cos ( 2 ( w k ) Δ 1 , 2 ) ] ,
F [ u , v ] = m = 1 M 1 n = m + 1 M ω m , n ( u , v ) F m , n [ u , v ] ,
m = 1 M 1 n = m + 1 M ω m , n ( u , v ) = 1 .
Var { F [ u , v ] } = m = 1 M 1 n = m + 1 M ω m , n ( u , v ) 2 Var { F m , n [ u , v ] } + 2 Re [ l = 1 M 2 j = l + 1 M 1 n = j + 1 M ω l , j ( u , v ) ω l , n * ( u , v ) Cov { F l , j [ u , v ] F l , n [ u , v ] } + l = 1 M 2 j = l + 1 M m = l + 1 M 1 n = m + 1 M ω l , j ( u , v ) ω m , n * ( u , v ) Cov { F l , j [ u , v ] F m , n [ u , v ] } ] ,
Var { F m , n [ u , v ] } σ m , n 2 ( u , v ) ,
Cov { F l , j [ u , v ] , F m , n [ u , v ] } ρ l , j ; m , n ( r ) ( u , v ) + i ρ l , j ; m , n ( i ) ( u , v ) ,
ω m , n ( u , v ) R m , n ( u , v ) + i I m , n ( u , v ) .
Var { F } = m = 1 M 1 n = m + 1 M ( R m , n 2 + I m , n 2 ) σ m , n + 2 { l = 1 M 2 j = l + 1 M 1 n = j + 1 M [ ρ l , j ; l , n ( r ) ( R l , j R l , n + I l , j I l , n ) ρ l , j ; l , n ( i ) ( R l , n I l , j R l , j I l , n ) ] + l = 1 M 2 j = l + 1 M m = l + 1 M 1 n = m + 1 M [ ρ l , j ; m , n ( r ) ( R l , j R m , n + I l , j I m , n ) ρ l , j ; m , n ( i ) ( R m , n I l , j R l , j I m , n ) ] } .
Var { F } R m , n R m , n = R m , n ( op ) = 0 ,
Var { F } I m , n I m , n = I m , n ( op ) = 0 ,
A x = b ,
A = [ a 11 a 12 a 1 , M 2 M 2 a M 2 M 2 , 1 a M 2 M 2 , 2 a M 2 M 2 , M 2 M 2 ] ,
x = ( R 1 , 2 ( op ) I 1 , 2 ( op ) R 1 , 3 ( op ) I 1 , 3 ( op ) R M 2 , M ( op ) I M 2 , M ( op ) ) ,
b = ( b 1 b 2 b M 2 M 2 ) .
ω m , n heur ( u , v ) = 2 [ 1 cos ( 2 ( w k ) Δ m , n ) ] M 2 M 2 [ m = 1 M 1 n = m + 1 M cos ( 2 ( w k ) Δ m , n ) ] ,

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