Abstract

Binarization of Fresnel holograms by direct thresholding based on the polarity of the fringe pattern is studied. It is found that if the hologram is binarized (i.e., for black and white hologram pixels) in this manner, only the edges of the object are preserved in the reconstructed image. To alleviate the errors caused by binarization, the use of error diffusion has been routinely employed. However, the reconstructed image using such standard technique is heavily contaminated with random noise. In this paper, we propose a novel noniterative method for generating Fresnel holograms that are suitable for binarization. Our method is capable of preserving good visual quality on the reconstructed images.

© 2011 Optical Society of America

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References

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  1. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748(1967).
    [CrossRef]
  2. B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
    [CrossRef]
  3. V. Guarnieri and F. Francini, “Computer generated holograms (CGH) realization: the integration of dedicated software tool with digital slides printer,” Proc. SPIE 3190, 391–401(1997).
    [CrossRef]
  4. Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
    [CrossRef]
  5. H. Yoshikawa and M. Tachinami, “Development of direct fringe printer for computer-generated holograms,” Proc. SPIE 5742, 259–266 (2005).
    [CrossRef]
  6. V. V. Krishna and P. S. Naidu, “An overview of quantization in detour phase binary holograms,” Proc. SPIE 813, 397–398(1987).
  7. E. Zhang, S. Noehte, C. H. Dietrich, and R. Manner, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
    [CrossRef]
  8. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
    [CrossRef]
  9. G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
    [CrossRef]
  10. A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.
  11. S. Nozaki and Y.-W. Chen, “Evolutionary perturbation of simulated annealing in optimization of kinoforms,” in Fifth International Conference on Natural Computation (2009), Vol.  6, pp. 13–16.
  12. I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
    [CrossRef]
  13. M. P. Chang and O. K. Ersoy, “Iterative interlacing error diffusion for synthesis of computer-generated holograms,” Appl. Opt. 32, 3122–3129 (1993).
    [CrossRef]
  14. E. Zhang, S. Noehte, C. H. Dietrich, and R. Männer, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
    [CrossRef]
  15. B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
    [CrossRef]
  16. L. C. Ferri, “Visualization of 3D information with digital holography using laser printers,” Comp. Graph. 25, 309–321(2001).
  17. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. Soc. Inf. Display 17, 75–77 (1976).
  18. R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710(1991).
    [CrossRef]
  19. R. Eschbach and Z. G. Fan, “Complex-valued error diffusion for off-axis computer-generated holograms,” Appl. Opt. 32, 3130–3136 (1993).
    [CrossRef]
  20. R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
    [CrossRef]
  21. F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
    [CrossRef]
  22. T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
  23. T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
    [CrossRef]

2010 (1)

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

2008 (1)

T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
[CrossRef]

2005 (2)

H. Yoshikawa and M. Tachinami, “Development of direct fringe printer for computer-generated holograms,” Proc. SPIE 5742, 259–266 (2005).
[CrossRef]

G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
[CrossRef]

2004 (1)

Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
[CrossRef]

2001 (2)

B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
[CrossRef]

L. C. Ferri, “Visualization of 3D information with digital holography using laser printers,” Comp. Graph. 25, 309–321(2001).

1997 (2)

V. Guarnieri and F. Francini, “Computer generated holograms (CGH) realization: the integration of dedicated software tool with digital slides printer,” Proc. SPIE 3190, 391–401(1997).
[CrossRef]

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef]

1996 (1)

R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
[CrossRef]

1995 (3)

1993 (2)

1991 (1)

1987 (1)

V. V. Krishna and P. S. Naidu, “An overview of quantization in detour phase binary holograms,” Proc. SPIE 813, 397–398(1987).

1976 (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

1969 (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

1967 (1)

Albero, J.

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

Brown, B. R.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Bryngdahl, O.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

Buckley, E.

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

Cable, A. J.

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

Chang, M. P.

Chen, Y.-W.

S. Nozaki and Y.-W. Chen, “Evolutionary perturbation of simulated annealing in optimization of kinoforms,” in Fifth International Conference on Natural Computation (2009), Vol.  6, pp. 13–16.

Chhetri, B. B.

B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
[CrossRef]

Dietrich, C. H.

Easton, R. L.

R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
[CrossRef]

Ersoy, O. K.

Eschbach, R.

R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
[CrossRef]

R. Eschbach and Z. G. Fan, “Complex-valued error diffusion for off-axis computer-generated holograms,” Appl. Opt. 32, 3130–3136 (1993).
[CrossRef]

R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710(1991).
[CrossRef]

Fan, Z. G.

Ferri, L. C.

L. C. Ferri, “Visualization of 3D information with digital holography using laser printers,” Comp. Graph. 25, 309–321(2001).

Fetthauer, F.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

Floyd, R. W.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

Francini, F.

V. Guarnieri and F. Francini, “Computer generated holograms (CGH) realization: the integration of dedicated software tool with digital slides printer,” Proc. SPIE 3190, 391–401(1997).
[CrossRef]

Gorecki, C.

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

Guarnieri, V.

V. Guarnieri and F. Francini, “Computer generated holograms (CGH) realization: the integration of dedicated software tool with digital slides printer,” Proc. SPIE 3190, 391–401(1997).
[CrossRef]

Krishna, V. V.

V. V. Krishna and P. S. Naidu, “An overview of quantization in detour phase binary holograms,” Proc. SPIE 813, 397–398(1987).

Lawrence, N. A.

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

Lohmann, A. W.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748(1967).
[CrossRef]

Manner, R.

Männer, R.

Marsh, P.

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

Martínez-García, A.

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

Mills, G. A.

Moreno, I.

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

Morishima, M.

Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
[CrossRef]

Nagarajan, R.

R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
[CrossRef]

Naidu, P. S.

V. V. Krishna and P. S. Naidu, “An overview of quantization in detour phase binary holograms,” Proc. SPIE 813, 397–398(1987).

Nieradko, L.

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

Noehte, S.

Nozaki, S.

S. Nozaki and Y.-W. Chen, “Evolutionary perturbation of simulated annealing in optimization of kinoforms,” in Fifth International Conference on Natural Computation (2009), Vol.  6, pp. 13–16.

Paris, D. P.

Poon, T.-C.

T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
[CrossRef]

T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).

Sakamoto, Y.

Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
[CrossRef]

Shimomura, T.

B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
[CrossRef]

Steinberg, L.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

Surrel, Y.

Tachinami, M.

H. Yoshikawa and M. Tachinami, “Development of direct fringe printer for computer-generated holograms,” Proc. SPIE 5742, 259–266 (2005).
[CrossRef]

Usui, M. A.

Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
[CrossRef]

Weissbach, S.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

Wilkinson, T. D

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

Yamaguchi, I.

Yang, S.

B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
[CrossRef]

Yoshikawa, H.

H. Yoshikawa and M. Tachinami, “Development of direct fringe printer for computer-generated holograms,” Proc. SPIE 5742, 259–266 (2005).
[CrossRef]

Zhang, E.

Zhao, B.

Am. J. Phys. (1)

T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
[CrossRef]

Appl. Opt. (8)

Comp. Graph. (1)

L. C. Ferri, “Visualization of 3D information with digital holography using laser printers,” Comp. Graph. 25, 309–321(2001).

IBM J. Res. Dev. (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid Pub. 5, 10011 (2010).
[CrossRef]

J. Mod. Opt. (1)

R. L. Easton, R. Eschbach, and R. Nagarajan, “Error diffusion in cell-oriented Fourier-transform computer-generated holograms to compensate for printing constraints,” J. Mod. Opt. 43, 1219–1236 (1996).
[CrossRef]

Opt. Commun. (1)

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

Opt. Eng. (1)

B. B. Chhetri, S. Yang, and T. Shimomura, “Iterative stepwise binarization of digital amplitude holograms with added energy to the signal window,” Opt. Eng. 40, 2718–2725(2001).
[CrossRef]

Proc. Soc. Inf. Display (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

Proc. SPIE (4)

V. Guarnieri and F. Francini, “Computer generated holograms (CGH) realization: the integration of dedicated software tool with digital slides printer,” Proc. SPIE 3190, 391–401(1997).
[CrossRef]

Y. Sakamoto, M. Morishima, and M. A. Usui, “Computer-generated holograms on a CD-R disk,” Proc. SPIE 5290, 42–49 (2004).
[CrossRef]

H. Yoshikawa and M. Tachinami, “Development of direct fringe printer for computer-generated holograms,” Proc. SPIE 5742, 259–266 (2005).
[CrossRef]

V. V. Krishna and P. S. Naidu, “An overview of quantization in detour phase binary holograms,” Proc. SPIE 813, 397–398(1987).

Other (3)

A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, and T. D Wilkinson, “Real-time binary hologram generation for high-quality video projection applications,” in SID International Symposium Digest of Technical Papers (2004), pp. 1431–1433.

S. Nozaki and Y.-W. Chen, “Evolutionary perturbation of simulated annealing in optimization of kinoforms,” in Fifth International Conference on Natural Computation (2009), Vol.  6, pp. 13–16.

T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).

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Figures (11)

Fig. 1
Fig. 1

(a) Intensity image of a solid square positioned at 0.4 m from the hologram. (b) Original hologram (real part of the complex Fresnel hologram based on Eq. (1b) of the solid square. (c) Sign binarization of the real part of the complex Fresnel hologram of the solid square.

Fig. 2
Fig. 2

(a) Reconstructed intensity image of the original hologram in Fig. 1b. (b) Reconstructed intensity image of the binarized hologram in Fig. 1c.

Fig. 3
Fig. 3

(a) Hologram profile [according to Eqs. (1)] along the horizontal dotted line of the hologram in Fig. 1b. (b) Hologram profile [according to Eqs. (1a, 1b)] along the horizontal dotted line of the hologram in Fig. 1c.

Fig. 4
Fig. 4

(a) Reconstructed intensity image of the hologram of a single point. (b) Reconstructed intensity image of the binarized hologram of a single point.

Fig. 5
Fig. 5

(a) Line formed by a sequence of 128 points. (b) Reconstructed intensity image of the hologram of the line shown in (a). (c) Reconstructed intensity image of the binarized hologram of the line shown in (a).

Fig. 6
Fig. 6

(a) Dotted line formed by a sequence of eight points with a regular spacing of 16 pixels. (b) Reconstructed intensity image of the hologram of the dotted line shown in (a). (c) Reconstructed intensity image of the binarized hologram of the dotted line shown in (a).

Fig. 7
Fig. 7

(a) Original oversampled signal I ( t ) . (b) Spectrum of I ( t ) with f S as a sampling frequency. (c) Subsampling of the original signal I 1 ( t ) . (d) Spectrum of I ( t ) after subsampling by M times.

Fig. 8
Fig. 8

(a) Horizontal line traces across the center of the original hologram of the solid square in Fig. 1a after down-sampling by 16 times. (b) Horizontal line traces across the center of the binarized hologram of the solid square in Fig. 1a after down-sampling by 16 times.

Fig. 9
Fig. 9

(a) Reconstructed intensity image of the hologram representing the square image in Fig. 1a, binarized with thresholding. (b) Reconstructed intensity image of the hologram representing the square image in Fig. 1a, binarized with error diffusion. (c) Reconstructed intensity image of the hologram representing the square image in Fig. 1a, binarized with our proposed method with a down-sampling factor of 8 on the source image. (d) Optical reconstruction of the intensity image of the binarized (by direct thresholding) hologram of a white square. (e) Optical reconstruction of the intensity image of the binarized (by error diffusion) hologram of a white square. (f) Optical reconstruction of the intensity image of the binarized hologram of a white square being down-sampled by 8 times.

Fig. 10
Fig. 10

(a) Source image comprising an upper row of text characters “CTU” at z 0 = 0.25 m , and a lower row of characters “HK” at z 1 = 0.3 m . (b) Reconstructed image of the hologram representing the image in (a), and at z 0 = 0.25 m . (c) Reconstructed image of the hologram representing the image in (a), and at z 1 = 0.3 m .

Fig. 11
Fig. 11

(a) Source image after applying the antialiasing filter and down-sampling. (b) Reconstructed image at z 0 = 0.25 m of the hologram representing the image in Fig. 10a, and binarized with thresholding. (c) Reconstructed image at z 0 = 0.25 m of the hologram representing the image in Fig. 10a, and binarized with error diffusion. (d) Reconstructed image at z 0 = 0.25 m of the hologram representing the image in Fig. 10a, and binarized with thresholding after subsampling to the source image (our proposed method). (e) Reconstructed image at z 1 = 0.3 m of the hologram representing the image in Fig. 10a, and binarized with thresholding. (f) Reconstructed image at z 1 = 0.3 m of the hologram representing the image in Fig. 10a, and binarized with error diffusion. (g) Reconstructed image at z 1 = 0.3 m of the hologram representing the image in Fig. 10a, and binarized with thresholding after subsampling to the source image (our proposed method).

Tables (4)

Tables Icon

Table 1 Optical Setting Adopted in the Hologram Generation Process

Tables Icon

Table 2 Optical Setting Adopted in the Experimental Evaluation

Tables Icon

Table 3 Comparison of the RMSE Between the Square Image Reconstructed Based on Eq. (1b), and Each of the Binarization Methods

Tables Icon

Table 4 Comparison of the RMSE Between the “CTU” and the “HK” Images Reconstructed Based on Eq. (1b), and Each of the Binarization Methods

Equations (8)

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H F ( m , n ) = O ( m , n ; z i ) R * ( m , n ) ,
H ( m , n ) = Re { O ( m , n ; z i ) R * ( m , n ) } ,
O ( m , n ; z i ) = u = 0 X 1 v = 0 Y 1 I ( u , v ; z i ) exp ( i k r ( m u , n v ; z i ) ) r ( m u , n v ; z i ) ,
O ( m , n ; z i ) = I ( m , n ; z i ) * F ( m , n ; z i ) ,
I D ( m , n ; z i ) = I 1 ( m , n ; z i ) I 2 ( m , n ; z i ) ,
I 1 ( m , n ; ; z i ) = { I ( m , n ; z i ) m = τ M 0 otherwise ,
I 2 ( m , n ; z i ) = { I ( m , n ; z i ) n = τ M 0 otherwise ,
I 1 ( m ) = I ( m ) r = δ ( m r M ) .

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