Abstract

In this paper, we describe various numerical space-shifting manipulations of the reconstructed images to remove the dc noise in the reconstruction, in terms of the periodicity characteristics of images in digital holography. The theoretical interpretation on different reconstruction periods of the image and the dc noise is described in detail for the first time, to the best of our knowledge. It is related to CCD sampling periods or frequencies for the fringes and the dc term of a hologram. With the calculations of Hadamard product of two different spatially shifted images and subsequent extraction of the root of it, the dc noise can be suppressed effectively and a clear image with the original intensity contrast can be obtained at the center in the hologram reconstruction, particularly when the image and the dc noise are completely or partially superposed with each other. The experiments for both in-line and off-axis imaging cases show that all results are completely consistent with theoretical predictions.

© 2011 Optical Society of America

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    [CrossRef]
  37. C.-S. Guo, L. Zhang, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
    [CrossRef]

2011

2010

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12, 015301 (2010).
[CrossRef]

Y. Dong and J. Wu, “Space-shifting digital holography with dc term removal,” Opt. Lett. 35, 1287–1289 (2010).
[CrossRef] [PubMed]

2009

2008

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” Appl. Phys. B 90, 527–532 (2008).
[CrossRef]

J. Weng, J. Zhong, and C. Hu, “Digital reconstruction based on angular spectrum diffraction with the ridge of wavelet transform in holographic phase-contrast microscopy,” Opt. Express 16, 21971–21981 (2008).
[CrossRef] [PubMed]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

J. Hahn, H. Kim, S.-W. Cho, and B. Lee, “Phase-shifting interferometry with genetic algorithm-based twin image noise elimination,” Appl. Opt. 47, 4068–4076 (2008).
[CrossRef] [PubMed]

2007

2006

2005

W. Wang, W. Yue, and L. Ai, “A new method for eliminating zero-order and conjugate image in digital holography,” Proc. SPIE 5636, 849–855 (2005).
[CrossRef]

Q. Lu, B. Ge, J. Jiang, and Y. Zhang, “Design of the experimental system of digital off-axis holography and analysis of zero-order image,” Proc. SPIE 5636, 254–259 (2005).
[CrossRef]

2004

Y. Zhang, Q. Lu, and B. Ge, “Elimination of zero-order diffraction in digital off-axis holography,” Opt. Commun. 240, 261–267 (2004).
[CrossRef]

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353–1358 (2004).
[CrossRef]

P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
[CrossRef] [PubMed]

2003

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

C.-S. Guo, L. Zhang, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

2002

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
[CrossRef]

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

2000

1997

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

T. M. Kreis and W. P. O. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360(1997).
[CrossRef]

Ahrenberg, L.

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Ai, L.

W. Wang, W. Yue, and L. Ai, “A new method for eliminating zero-order and conjugate image in digital holography,” Proc. SPIE 5636, 849–855 (2005).
[CrossRef]

Asundi, A.

A. Asundi and V. R. Singh, “Sectioning of amplitude images in digital holography,” Meas. Sci. Technol. 17, 75–78(2006).
[CrossRef]

Atlan, M.

Awatsuji, Y.

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Bo, F.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Cai, L. Z.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12, 015301 (2010).
[CrossRef]

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34, 1210–1212 (2009).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[CrossRef] [PubMed]

Chang, C.-C.

W.-T. Hsieh, M.-K. Kuo, H. F. Yau, and C.-C. Chang, “A simple arbitrary phase-step digital holographic reconstruction approach without blurring using two holograms,” Opt. Rev. 16, 466–471 (2009).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” Appl. Phys. B 90, 527–532 (2008).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express 15, 8851–8856 (2007).
[CrossRef] [PubMed]

Chen, G.-L.

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” Appl. Phys. B 90, 527–532 (2008).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express 15, 8851–8856 (2007).
[CrossRef] [PubMed]

Chen, P.-J.

Cheng, X.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Cheng, X. C.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

Cho, H.

H. Cho, J.-K. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Cho, S.-W.

Cuche, E.

Depeursinge, C.

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2009, ISBI’09 (2009), pp. 201–204.
[CrossRef]

Devaney, A. J.

Dong, G. Y.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

Dong, Y.

Fujii, A.

Garcia, J.

Garcia-Sucerquia, J.

J. A. H. Ramirez and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259–263 (2007).
[CrossRef]

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353–1358 (2004).
[CrossRef]

Ge, B.

Q. Lu, B. Ge, J. Jiang, and Y. Zhang, “Design of the experimental system of digital off-axis holography and analysis of zero-order image,” Proc. SPIE 5636, 254–259 (2005).
[CrossRef]

Y. Zhang, Q. Lu, and B. Ge, “Elimination of zero-order diffraction in digital off-axis holography,” Opt. Commun. 240, 261–267 (2004).
[CrossRef]

Gross, M.

Guo, C.-S.

C.-S. Guo, L. Zhang, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

Guo, J. P.

Guo, P.

Hahn, J.

Hennelly, B. M.

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Hsieh, W.-T.

W.-T. Hsieh, M.-K. Kuo, H. F. Yau, and C.-C. Chang, “A simple arbitrary phase-step digital holographic reconstruction approach without blurring using two holograms,” Opt. Rev. 16, 466–471 (2009).
[CrossRef]

Hu, C.

Javidi, B.

Jhou, G.-S.

Jiang, J.

Q. Lu, B. Ge, J. Jiang, and Y. Zhang, “Design of the experimental system of digital off-axis holography and analysis of zero-order image,” Proc. SPIE 5636, 254–259 (2005).
[CrossRef]

Juptner, W. P. O.

T. M. Kreis and W. P. O. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360(1997).
[CrossRef]

Jüptner, W.

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars and W. Jüptner, Digital Holography (Springer2005).

Kim, D.

H. Cho, J.-K. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Kim, H.

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
[CrossRef]

T. M. Kreis and W. P. O. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360(1997).
[CrossRef]

Kubota, T.

Kuo, M.-K.

W.-T. Hsieh, M.-K. Kuo, H. F. Yau, and C.-C. Chang, “A simple arbitrary phase-step digital holographic reconstruction approach without blurring using two holograms,” Opt. Rev. 16, 466–471 (2009).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” Appl. Phys. B 90, 527–532 (2008).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express 15, 8851–8856 (2007).
[CrossRef] [PubMed]

Lee, B.

Li, A. M.

Li, Y.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Lin, C.-Y.

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” Appl. Phys. B 90, 527–532 (2008).
[CrossRef]

G.-L. Chen, C.-Y. Lin, M.-K. Kuo, and C.-C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express 15, 8851–8856 (2007).
[CrossRef] [PubMed]

Liu, C.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Liu, J.-P.

Liu, Q.

Liu, Z.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Lu, Q.

Q. Lu, B. Ge, J. Jiang, and Y. Zhang, “Design of the experimental system of digital off-axis holography and analysis of zero-order image,” Proc. SPIE 5636, 254–259 (2005).
[CrossRef]

Y. Zhang, Q. Lu, and B. Ge, “Elimination of zero-order diffraction in digital off-axis holography,” Opt. Commun. 240, 261–267 (2004).
[CrossRef]

Lu, W.

Marquet, P.

Matoba, O.

McElhinney, C.

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Meng, X. F.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34, 1210–1212 (2009).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

Mico, V.

Naughton, T. J.

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Pan, W.

Pavillon, N.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2009, ISBI’09 (2009), pp. 201–204.
[CrossRef]

Peng, X.

Poon, T.-C.

Prieto, D. V.

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353–1358 (2004).
[CrossRef]

Ramirez, J. A. H.

J. A. H. Ramirez and J. Garcia-Sucerquia, “Digital off-axis holography without zero-order diffraction via phase manipulation,” Opt. Commun. 277, 259–263 (2007).
[CrossRef]

J. Garcia-Sucerquia, J. A. H. Ramirez, and D. V. Prieto, “DC term filtering techniques in digital holography,” Proc. SPIE 5622, 1353–1358 (2004).
[CrossRef]

Sasada, M.

Schnars, U.

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars and W. Jüptner, Digital Holography (Springer2005).

Seelamantula, C. S.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2009, ISBI’09 (2009), pp. 201–204.
[CrossRef]

Shen, X. X.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[CrossRef]

Shin, S.

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[CrossRef]

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Opt. Rev.

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of CCD sampling periods for dc background term and fringes of a hologram.

Fig. 2
Fig. 2

Hologram reconstruction with Fresnel transformation approach for in-line digital holography (a) without any dc noise suppression and (b) with presuppression of the dc noise using the intensity-averaging technique.

Fig. 3
Fig. 3

Space-shifting manipulations of the reconstructed images (a) in the horizontal direction and (b) in the perpendicular direction.

Fig. 4
Fig. 4

Zero-order noise removal with Hadamard product processing of Figs. 3a, 3b.

Fig. 5
Fig. 5

Zero-order noise removal for off-axis digital holography: (a) presuppression of the dc noise using intensity-averaging technique, (b) and (c) space-shifting manipulations of the images in the horizontal and perpendicular directions, (d) dc noise removal from the Hadamard product of (b) and (c), (e) two-dimensional space-shifting processing of the image, and (f) the dc noise removal from the Hadamard product of (a) and (e).

Equations (18)

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Γ ( x , y ) = C ( x , y ) + + u ( ξ , η ) exp [ j 2 π λ d ( ξ x + η y ) ] d ξ d η ,
C ( x , y ) = j λ d exp ( j 2 π λ d ) exp [ j π λ d ( x 2 + y 2 ) ] ,
u ( ξ , η ) = R ( ξ , η ) h ( ξ , η ) exp [ j π / ( λ d ) ( ξ 2 + η 2 ) ] ,
Γ ( x , y ) = I 1 { u ( ξ , η ) } = I 1 { R ( ξ , η ) h ( ξ , η ) exp [ j π / ( λ d ) ( ξ 2 + η 2 ) ] } .
Γ ( x , y ) = Γ 0 ( x , y ) + Γ 1 ( x , y ) ,
Γ 0 ( x , y ) = I 1 { R ( ξ , η ) h 0 exp [ j π / ( λ d ) ( ξ 2 + η 2 ) ] } ,
Γ 1 ( x , y ) = I 1 { R ( ξ , η ) h 1 ( ξ , η ) exp [ j π / ( λ d ) ( ξ 2 + η 2 ) ] } .
h CCD ( ξ , η ) = h ( ξ , η ) × g CCD ( ξ , η ) = h ( ξ , η ) [ rect ( ξ α · Δ ξ , η β · Δ η ) * comb ( ξ L ξ , η L η ) ] × rect ( ξ M Δ ξ , η N Δ η ) .
Γ ( x , y ) = I 1 { u ( ξ , η ) } = I 1 { g CCD ( ξ , η ) · U ( ξ , η ) } = I 1 { g CCD ( ξ , η ) } * I 1 { U ( ξ , η ) } ,
I 1 { U ( ξ , η ) } = I 1 { R ( ξ , η ) h ( ξ , η ) exp [ j π / ( λ d ) ( ξ 2 + η 2 ) ] } ,
I 1 { g CCD ( ξ , η ) } = I 1 { [ rect ( ξ α · Δ ξ , η β · Δ η ) * comb ( ξ L ξ , η L η ) ] × rect ( ξ M Δ ξ , η N Δ η ) } = K · [ sinc ( Δ ξ λ d x , Δ η λ d y ) × comb ( L ξ λ d x , L η λ d y ) ] * sinc ( M · Δ ξ λ d x , N · Δ η λ d y ) ,
I 1 { g CCD ( ξ , η ) } = K · m = n = { sinc ( m 2 , n 2 ) × sinc ( M · Δ ξ λ d x M · Δ ξ L ξ m , N · Δ η λ d y N · Δ η L η n ) } ,
IDFT { g CCD ( m , n ) } = K · m = n = { sinc ( m 2 , n 2 ) × sinc ( m M p m , n N q n ) } ,
Γ ( m , n ) = IDFT { g CCD ( k , l ) · U ( k , l ) } = IDFT { g CCD ( k , l ) } * IDFT { U ( k , l ) } = IDFT { g CCD , h 0 ( k , l ) } * IDFT { U h 0 ( k , l ) } + IDFT { g CCD , h 1 ( k , l ) } * IDFT { U h 1 ( k , l ) } ,
g ( x + x 0 , y ) = I 1 { G ( ω x , ω y ) exp [ j ω ( x 0 ) ] }
g ( x , y + y 0 ) = I 1 { G ( ω x , ω y ) exp [ j ω ( y 0 ) ] } ,
Γ ( m + M / 2 , n ) = IDFT { ( 1 ) k u ( k , l ) }
Γ ( m , n + N / 2 ) = IDFT { ( 1 ) l u ( k , l ) } .

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