Abstract

We report the implementation of a method which can yield the 3D distribution of the phase (refractive index) of a weakly absorbing object from a single tomographic data set. In order to reduce the residual absorption artifact (due to the fact that only one data set is used) the original algorithm presented by A. V. Bronnikov is amended by adding in the filter a factor found by using a semi empirical approach. The quality of the reconstruction is largely sufficient for optimal segmentation and further postprocessing even though the filter correction is based on assumption of constant absorption. This one step approach allows keeping radiation dose to the minimum. Spatial resolution is comparable to the conventional absorption based technique. The performance of the method is validated by using an established phase contrast technique.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  22. The computation of the two-dimensional Radon transform and backprojection over the angle ω are combined in a single step of the algorithm [20].
  23. J. M. Cowley, Diffraction Physics (North-Holland, 1995), Chap. 3.4.
  24. A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instr. 68,2774-2782 (1997).
    [CrossRef]

2006

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

2005

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

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

2004

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, "Exit wave reconstruction at atomic resolution," Ultramicroscopy 100, 91-104 (2004).
[CrossRef] [PubMed]

W. K. Hsieh, F. R. Chen, J. J. Kai, A. I. Kirkland, "Resolution extension and exit wave reconstruction in complex HREM," Ultramicroscopy 98, 99-114 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Optics Commun. 231,53-70 (2004).
[CrossRef]

2002

A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. of the Opt. Soc. of Am. A 19, 472-480 (2002).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

2001

U. Bonse, and F. Beckmann, "Multiple-beam X-ray interferometry for phase-contrast microtomography," Journal of Synchrotron Radiat. 8, 1-5 (2001).
[CrossRef]

1999

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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, "Reconstruction formulas in phase-contrast tomography," Optics Communications 171, 239-244 (1999).
[CrossRef]

1997

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instr. 68,2774-2782 (1997).
[CrossRef]

1996

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

U. Bonse and F. Bush, "X-ray computed microtomography using synchrotron radiation," Prog. Biophys. Mol. Biol., 65, 133-169 (1996).
[CrossRef] [PubMed]

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

1995

A. Momose, "Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer," Nuclear Instruments and Methods A 352, 622-628 (1995).
[CrossRef]

1983

M. R. Teague, "Deterministic phase retrieval - a Green-function solution," J. of the Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

1965

U. Bonse and M. Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Abela, R.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Allen, L. J.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, "Exit wave reconstruction at atomic resolution," Ultramicroscopy 100, 91-104 (2004).
[CrossRef] [PubMed]

Barnea, Z.

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

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Beckmann, F.

U. Bonse, and F. Beckmann, "Multiple-beam X-ray interferometry for phase-contrast microtomography," Journal of Synchrotron Radiat. 8, 1-5 (2001).
[CrossRef]

Bonse, U.

U. Bonse, and F. Beckmann, "Multiple-beam X-ray interferometry for phase-contrast microtomography," Journal of Synchrotron Radiat. 8, 1-5 (2001).
[CrossRef]

U. Bonse and F. Bush, "X-ray computed microtomography using synchrotron radiation," Prog. Biophys. Mol. Biol., 65, 133-169 (1996).
[CrossRef] [PubMed]

U. Bonse and M. Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Borchert, G.

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Bronnikov, A. V.

A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. of the Opt. Soc. of Am. A 19, 472-480 (2002).
[CrossRef]

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Optics Communications 171, 239-244 (1999).
[CrossRef]

Bush, F.

U. Bonse and F. Bush, "X-ray computed microtomography using synchrotron radiation," Prog. Biophys. Mol. Biol., 65, 133-169 (1996).
[CrossRef] [PubMed]

Chen, F. R.

W. K. Hsieh, F. R. Chen, J. J. Kai, A. I. Kirkland, "Resolution extension and exit wave reconstruction in complex HREM," Ultramicroscopy 98, 99-114 (2004).
[CrossRef] [PubMed]

Cloetens, P.

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

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Cookson, D. F.

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

David, C.

Davis, T. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

DeCarlo, F.

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

Dhal, B. B.

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

Diaz, A.

Gao, D.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instr. 68,2774-2782 (1997).
[CrossRef]

Groso, A.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Optics Commun. 231,53-70 (2004).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

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

Hart, M.

U. Bonse and M. Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Hsieh, W. K.

W. K. Hsieh, F. R. Chen, J. J. Kai, A. I. Kirkland, "Resolution extension and exit wave reconstruction in complex HREM," Ultramicroscopy 98, 99-114 (2004).
[CrossRef] [PubMed]

Hunt, S.

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Kai, J. J.

W. K. Hsieh, F. R. Chen, J. J. Kai, A. I. Kirkland, "Resolution extension and exit wave reconstruction in complex HREM," Ultramicroscopy 98, 99-114 (2004).
[CrossRef] [PubMed]

Kirkland, A. I.

W. K. Hsieh, F. R. Chen, J. J. Kai, A. I. Kirkland, "Resolution extension and exit wave reconstruction in complex HREM," Ultramicroscopy 98, 99-114 (2004).
[CrossRef] [PubMed]

Linga, S.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Mahon, P. J.

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

Mayo, S. C.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

McBride, W.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, "Exit wave reconstruction at atomic resolution," Ultramicroscopy 100, 91-104 (2004).
[CrossRef] [PubMed]

Miller, P. R.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

Momose, A.

A. Momose, "Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer," Nuclear Instruments and Methods A 352, 622-628 (1995).
[CrossRef]

Müller, R.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

Nugent, K. A.

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

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

O’Leary, N. L.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, "Exit wave reconstruction at atomic resolution," Ultramicroscopy 100, 91-104 (2004).
[CrossRef] [PubMed]

Oxley, M. P.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, "Exit wave reconstruction at atomic resolution," Ultramicroscopy 100, 91-104 (2004).
[CrossRef] [PubMed]

Paganin, D.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. 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, A. Pogany, D. M. Paganin and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Optics Commun. 231,53-70 (2004).
[CrossRef]

Parry, D. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

Patterson, B.

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Peele, A. G.

A. G. Peele, F. DeCarlo, P. J. Mahon, B. B. Dhal and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 083707 (2005).
[CrossRef]

Pfeiffer, F.

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Optics Commun. 231,53-70 (2004).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instr. 68,2774-2782 (1997).
[CrossRef]

Rüegsegger, P.

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Schneider, P.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

Stampanoni, M.

A. Groso, M. Stampanoni, R. Abela, P. Schneider, S. Linga, and R. Müller, "Phase contrast tomography: an alternative approach," Appl. Phys. Lett. 88, 214104 (2006).
[CrossRef]

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

M. Stampanoni, G. Borchert, P. Wyss, R. Abela, B. Patterson, S. Hunt, D. Vermeulen and P. Rüegsegger, "High resolution X-ray detector for synchrotron-based microtomography," Nucl Instrum Methods A 491,291-301 (2002).
[CrossRef]

Stevenson, A. W.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207,79-96 (2002).
[CrossRef] [PubMed]

Teague, M. R.

M. R. Teague, "Deterministic phase retrieval - a Green-function solution," J. of the Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van 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]

Van Landuyt, J.

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Figures (4)

Fig. 1.
Fig. 1.

Tomographic reconstruction of a 350 microns thin wood sample: (a) absorption contrast obtained with 10 keV photon energy, (b) phase contrast, obtained with 25 keV. Effective pixel size is 0.7 microns. The length of the scale bar is 50 μm. Sample by courtesy of P.Trtik, ETHZ

Fig. 2.
Fig. 2.

Values for the correction factor αsimul as function of transmission found by simulations.

Fig. 3.
Fig. 3.

Phase tomography reconstruction (a) and the 3D rendering (b) of a 350 microns thin wood sample using modified filter given in the Eq. (8). The length of the scale bar is 50 μm.

Fig. 4.
Fig. 4.

Validation of the MBA method: (a) Phase tomographic reconstruction of sample consisting of polyacrylate, starch and cross-linked rubber matrix obtained using DPC and (b) using MBA. The length of the scale bar is 100 μm.

Equations (10)

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I θ , z = d ( x , y ) = I θ , z = 0 ( x , y ) λd 2 π [ I θ , z = 0 ( x , y ) ϕ θ ( x , y ) ]
I θ , z = d ( x , y ) = I θ , z = 0 ( x , y ) [ 1 λd 2 π 2 ϕ θ ( x , y ) ]
2 s 2 δ ̂ s θ ω = 1 d g ̂ θ ( s , ω )
δ ( x 1 , x 2 , x 3 ) = 1 4 π 2 d 0 π ( q * * g θ )
q ( x , y ) y x 2 + y 2
q ( ξ , η ) = ξ ξ 2 + η 2
g θ ( x , y ) = I θ , z = d ( x , y ) 1 , θ [ 0 , π ]
q ( ξ , η ) = ξ ξ 2 + η 2 + α exp
SSE = ( δ reconstructed δ simulated ) 2 ( δ simulated ) 2
α exp = α simul pixel simul array simul pixel exp array exp

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