Abstract

We describe an x-ray differential phase-contrast imaging method based on two-dimensional transmission gratings that are directly resolved by an x-ray camera. X-ray refraction and diffraction in the sample lead to variations of the positions and amplitudes of the grating fringes on the camera. These effects can be quantified through spatial harmonic analysis. The use of 2D gratings allows differential phase contrast in several directions to be obtained from a single image. When compared to previous grating-based interferometry methods, this approach obviates the need for multiple exposures and separate measurements for different directions and thereby accelerates imaging speed.

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H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, Radiology (Oak Brook, Ill.) 251, 910 (2009).

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

2008

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

2007

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

C. J. Martin, Radiat. Protect. Dosimetry 128, 389 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef] [PubMed]

2005

2004

2003

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, Opt. Express 11, 2289 (2003).
[CrossRef] [PubMed]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

2001

B. C. Platt and R. Shack, J. Refract. Surgery 17, S573 (2001).

1998

1997

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

1971

Anastasio, M. A.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Arvanitis, C. D.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Bennett, E.

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

Bennett, E. E.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, Radiology (Oak Brook, Ill.) 251, 910 (2009).

Bohndiek, S. E.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Brankov, J. G.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Bronnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

Carroll, S. C.

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

Chapman, D.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Clark, A. T.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Clauser, J. F.

J. F. Clauser, “Ultrahigh resolution interferometric x-ray imaging,” U. S. patent 5,812,629 (September 22, 1998).

Cloetens, P.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

David, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef] [PubMed]

Davis, T. J.

Diaz, A.

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Galatsanos, N. P.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996).
[CrossRef]

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Grunzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Gureyev, T. E.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, Opt. Express 11, 2289 (2003).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Hegedus, M. M.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, Radiology (Oak Brook, Ill.) 251, 910 (2009).

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

Huang, Z. F.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Kang, K. J.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Koch, A.

Kottler, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef] [PubMed]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Martin, C. J.

C. J. Martin, Radiat. Protect. Dosimetry 128, 389 (2007).
[CrossRef]

Mayo, S. C.

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Miller, P. R.

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Muehleman, C.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef] [PubMed]

Olivo, A.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Oltulu, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Paganin, D.

Pfeiffer, F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef] [PubMed]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Platt, B. C.

B. C. Platt and R. Shack, J. Refract. Surgery 17, S573 (2001).

Pogany, A.

Prydderch, M.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Rapacchi, S.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, Radiology (Oak Brook, Ill.) 251, 910 (2009).

Raven, C.

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Sexton, B.

Shack, R.

B. C. Platt and R. Shack, J. Refract. Surgery 17, S573 (2001).

Snigirev, A.

Spanne, P.

Speller, R. D.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Stampanoni, M.

Stevenson, A. W.

Suzuki, T.

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Turchetta, R.

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Wang, Z. T.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

Wen, H.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, Radiology (Oak Brook, Ill.) 251, 910 (2009).

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

Wernick, M. N.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Wilkins, S. W.

Wirjadi, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Yang, Y. Y.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

Yokozeki, S.

Zhang, L.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

Zhong, Z.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, Phys. Med. Biol. 48, 3875(2003).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef] [PubMed]

Zhu, P. P.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

Ziegler, E.

Appl. Opt.

IEEE Trans. Med. Imaging

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, IEEE Trans. Med. Imaging 27, 997 (2008).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

Z. T. Wang, Z. F. Huang, L. Zhang, K. J. Kang, and P. P. Zhu, IEEE Trans. Nucl. Sci. 56, 1383 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Refract. Surgery

B. C. Platt and R. Shack, J. Refract. Surgery 17, S573 (2001).

Jpn. J. Appl. Phys.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

Nat. Mater.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, Nat. Mater. 7, 134 (2008).
[CrossRef] [PubMed]

Nature

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996).
[CrossRef]

Nucl. Instrum. Meth. Phys. Res. A

A. Olivo, C. D. Arvanitis, S. E. Bohndiek, A. T. Clark, M. Prydderch, R. Turchetta, and R. D. Speller, Nucl. Instrum. Meth. Phys. Res. A 581, 776 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

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Figures (4)

Fig. 1
Fig. 1

Imaging device includes the x-ray tube, transmission grating, and x-ray camera. The camera resolution is sufficient to resolve the projected grating lines.

Fig. 2
Fig. 2

Spatial harmonic analysis begins with Fourier transformation of (a) the raw image into (b) the spatial frequency domain. The discrete peaks in (b) are spaced by the primary frequencies of the projection of the 2D grating and labeled by their positions. Inverse Fourier transformation of the subregion surrounding the ( 1 , 0 ) peak results in the ( 1 , 0 ) harmonic image. It is a complex image of (c) magnitude and (d) phase.

Fig. 3
Fig. 3

DPC images of glass beads from the ( 1 , 0 ) and ( 0 , 1 ) harmonic images of Fig. 2. They are magnified eight times by the cone beam. Different segments of their boundaries are highlighted in different DPC images.

Fig. 4
Fig. 4

Complete set of results from a single raw image of a rat’s paw, including the absorption image derived from the ( 0 , 0 ) harmonic peak, the DPC, and the diffraction images from the ( 0 , 1 ) and ( 1 , 0 ) harmonic peaks.

Equations (2)

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I M , N ( x , y ) = Fourier transform [ f ( k x , k y ) ] , k x [ 2 π M π P , 2 π M + π P ] , k y [ 2 π N π P , 2 π N + π P ] ,
D M , N ( x , y ) = ln [ | I M , N ( x , y ) | / | I 0 , 0 ( x , y ) | ] ,

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