Abstract

We present a digital signal processing technique that reduces the speckle content in reconstructed digital holograms. The method is based on sequential sampling of the discrete Fourier transform of the reconstructed image field. Speckle reduction is achieved at the expense of a reduced intensity and resolution, but this trade-off is shown to be greatly superior to that imposed by the traditional mean and median filtering techniques. In particular, we show that the speckle can be reduced by half with no loss of resolution (according to standard definitions of both metrics).

© 2007 Optical Society of America

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References

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  1. J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, 1984).
  2. P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).
  3. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).
  4. J. W. Goodman, Speckle Phenomena: Theory and Applications (Roberts & Company, 2006).
  5. R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: Mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
    [CrossRef] [PubMed]
  6. J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
    [CrossRef]
  7. D. Kim, "Reduction of coherent artifacts in dynamic holographic three-dimensional displays by diffraction-specific pseudorandom diffusion," Opt. Lett. 29, 611-613 (2004).
    [CrossRef] [PubMed]
  8. N. Bertaux, Y. Frauel, P. Réfrégier, and B. Javidi, "Speckle removal using a maximum-likelihood technique with isoline gray-level regularization," J. Opt. Soc. Am. A 21, 2283-2291 (2004).
    [CrossRef]
  9. J. C. Dainty and W. T. Welford, "Reduction of speckle in image plane hologram reconstruction by use of a moving pupil," Opt. Commun. 3, 289-294 (1971).
    [CrossRef]
  10. P. Hariharan and Z. S. Hegedus, "Reduction of speckle in coherent imaging by spatial frequency sampling," Opt. Acta 21, 345-356 (1974).
    [CrossRef]
  11. T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
    [CrossRef] [PubMed]
  12. J. Maycock, C. M. Elhinney, B. M. Hennelly, T. J. Naughton, J. McDonald, and B. Javidi, "Three-dimensional scene reconstruction of partially occluded objects using digital holograms," Appl. Opt. 45, 2975-2985 (2006).
    [CrossRef] [PubMed]
  13. A. Shortt, T. J. Naughton, and B. Javidi, "Compression of digital holograms of three-dimensional objects using wavelets," Opt. Express 14, 2625-2630 (2006).
    [CrossRef] [PubMed]
  14. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  15. S. Lowenthal and H. Arsenault, "Image formation for coherent diffuse objects: Statistical properties," J. Opt. Soc. Am. 60, 1478-1483 (1970).
    [CrossRef]
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw Hill, 1965).
  17. I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theory 8, 194-195 (1962).
    [CrossRef]
  18. B. M. Hennelly and J. T. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 22, 917-927 (2005).
    [CrossRef]
  19. L. S. Lim, "Techniques for speckle noise removal," Opt. Eng. (Bellingham) 20, 670-678 (1981).
  20. T. J. Crimmins, "Geometric filter for speckle reduction," Appl. Opt. 24, 1438-1443 (1985).
    [CrossRef] [PubMed]
  21. O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
    [CrossRef] [PubMed]
  22. T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. (Bellingham) 43, 2233-2238 (2004).
    [CrossRef]
  23. B. Javidi, P. Ferraro, S.-H. Hong, S. De Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
    [CrossRef] [PubMed]
  24. I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase in phase-shifting digital holography," Appl. Opt. 45, 975-983 (2006).
    [CrossRef] [PubMed]
  25. A. E. Shortt, T. J. Naughton, and B. Javidi, "A companding approach for nonuniform quantization of digital holograms of three-dimensional objects," Opt. Express 14, 5129-5134 (2006).
    [CrossRef] [PubMed]
  26. E. Darakis and J. J. Soraghan, "Compression of interference patterns with application to phase-shifting digital holography," Appl. Opt. 45, 2437-2443 (2006).
    [CrossRef] [PubMed]
  27. E. Darakis and J. J. Soraghan, "Use of Fresnelets for phase-shifting digital hologram compression," IEEE Trans. Image Process. 15, 3804-3811 (2006).
    [CrossRef] [PubMed]
  28. F. Charrière, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Hegera, E., A. D. Mitchell, P. Marquet, and B. Rappaz, "Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba," Opt. Express 14, 7005-7013 (2006).
    [CrossRef] [PubMed]
  29. Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001).
    [CrossRef]
  30. I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
    [CrossRef]
  31. T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
    [CrossRef] [PubMed]

2006 (9)

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: Mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
[CrossRef]

J. Maycock, C. M. Elhinney, B. M. Hennelly, T. J. Naughton, J. McDonald, and B. Javidi, "Three-dimensional scene reconstruction of partially occluded objects using digital holograms," Appl. Opt. 45, 2975-2985 (2006).
[CrossRef] [PubMed]

A. Shortt, T. J. Naughton, and B. Javidi, "Compression of digital holograms of three-dimensional objects using wavelets," Opt. Express 14, 2625-2630 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase in phase-shifting digital holography," Appl. Opt. 45, 975-983 (2006).
[CrossRef] [PubMed]

A. E. Shortt, T. J. Naughton, and B. Javidi, "A companding approach for nonuniform quantization of digital holograms of three-dimensional objects," Opt. Express 14, 5129-5134 (2006).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, "Compression of interference patterns with application to phase-shifting digital holography," Appl. Opt. 45, 2437-2443 (2006).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, "Use of Fresnelets for phase-shifting digital hologram compression," IEEE Trans. Image Process. 15, 3804-3811 (2006).
[CrossRef] [PubMed]

F. Charrière, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Hegera, E., A. D. Mitchell, P. Marquet, and B. Rappaz, "Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba," Opt. Express 14, 7005-7013 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (3)

2003 (1)

2002 (2)

2001 (1)

1997 (1)

1985 (1)

1981 (1)

L. S. Lim, "Techniques for speckle noise removal," Opt. Eng. (Bellingham) 20, 670-678 (1981).

1974 (1)

P. Hariharan and Z. S. Hegedus, "Reduction of speckle in coherent imaging by spatial frequency sampling," Opt. Acta 21, 345-356 (1974).
[CrossRef]

1971 (1)

J. C. Dainty and W. T. Welford, "Reduction of speckle in image plane hologram reconstruction by use of a moving pupil," Opt. Commun. 3, 289-294 (1971).
[CrossRef]

1970 (1)

1962 (1)

I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theory 8, 194-195 (1962).
[CrossRef]

Alfieri, D.

Arsenault, H.

Bertaux, N.

Castaneda, R.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
[CrossRef]

Castro, M.-A.

Charrière, F.

Colomb, T.

Crimmins, T. J.

Dainty, J. C.

J. C. Dainty and W. T. Welford, "Reduction of speckle in image plane hologram reconstruction by use of a moving pupil," Opt. Commun. 3, 289-294 (1971).
[CrossRef]

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, 1984).

Darakis, E.

E. Darakis and J. J. Soraghan, "Compression of interference patterns with application to phase-shifting digital holography," Appl. Opt. 45, 2437-2443 (2006).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, "Use of Fresnelets for phase-shifting digital hologram compression," IEEE Trans. Image Process. 15, 3804-3811 (2006).
[CrossRef] [PubMed]

De Nicola, S.

Depeursinge, C.

Elhinney, C. M.

Ferraro, P.

Finizio, A.

Frauel, Y.

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena: Theory and Applications (Roberts & Company, 2006).

Hariharan, P.

P. Hariharan and Z. S. Hegedus, "Reduction of speckle in coherent imaging by spatial frequency sampling," Opt. Acta 21, 345-356 (1974).
[CrossRef]

Hegedus, Z. S.

P. Hariharan and Z. S. Hegedus, "Reduction of speckle in coherent imaging by spatial frequency sampling," Opt. Acta 21, 345-356 (1974).
[CrossRef]

Hegera, T. J.

Hennelly, B. M.

Hong, S.-H.

Javidi, B.

J. Maycock, C. M. Elhinney, B. M. Hennelly, T. J. Naughton, J. McDonald, and B. Javidi, "Three-dimensional scene reconstruction of partially occluded objects using digital holograms," Appl. Opt. 45, 2975-2985 (2006).
[CrossRef] [PubMed]

A. Shortt, T. J. Naughton, and B. Javidi, "Compression of digital holograms of three-dimensional objects using wavelets," Opt. Express 14, 2625-2630 (2006).
[CrossRef] [PubMed]

A. E. Shortt, T. J. Naughton, and B. Javidi, "A companding approach for nonuniform quantization of digital holograms of three-dimensional objects," Opt. Express 14, 5129-5134 (2006).
[CrossRef] [PubMed]

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

B. Javidi, P. Ferraro, S.-H. Hong, S. De Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. (Bellingham) 43, 2233-2238 (2004).
[CrossRef]

N. Bertaux, Y. Frauel, P. Réfrégier, and B. Javidi, "Speckle removal using a maximum-likelihood technique with isoline gray-level regularization," J. Opt. Soc. Am. A 21, 2283-2291 (2004).
[CrossRef]

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
[CrossRef] [PubMed]

O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
[CrossRef] [PubMed]

Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001).
[CrossRef]

Kameda, M.

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

Kelly, D. P.

Kim, D.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

Lim, L. S.

L. S. Lim, "Techniques for speckle noise removal," Opt. Eng. (Bellingham) 20, 670-678 (1981).

Liu, Y.

Lowenthal, S.

Marquet, P.

Matoba, O.

Maycock, J.

McDonald, J.

McDonald, J. B.

Mills, G. A.

Mitchell, E. A. D.

Morimoto, Y.

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

Naughton, T. J.

Nomura, I. T.

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

Okazaki, A.

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

O'Neill, F. T.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw Hill, 1965).

Patten, R. F.

Pavillon, N.

Pierattini, G.

Ramírez, J. H.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
[CrossRef]

Rappaz, B.

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

Reed, I. S.

I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theory 8, 194-195 (1962).
[CrossRef]

Réfrégier, P.

Sheridan, J. T.

Shortt, A.

Shortt, A. E.

Soraghan, J. J.

E. Darakis and J. J. Soraghan, "Use of Fresnelets for phase-shifting digital hologram compression," IEEE Trans. Image Process. 15, 3804-3811 (2006).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, "Compression of interference patterns with application to phase-shifting digital holography," Appl. Opt. 45, 2437-2443 (2006).
[CrossRef] [PubMed]

Tajahuerce, E.

Welford, W. T.

J. C. Dainty and W. T. Welford, "Reduction of speckle in image plane hologram reconstruction by use of a moving pupil," Opt. Commun. 3, 289-294 (1971).
[CrossRef]

Yamaguchi, I.

Yamamoto, K.

Yokota, M.

Zhang, T.

Appl. Opt. (8)

T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
[CrossRef] [PubMed]

J. Maycock, C. M. Elhinney, B. M. Hennelly, T. J. Naughton, J. McDonald, and B. Javidi, "Three-dimensional scene reconstruction of partially occluded objects using digital holograms," Appl. Opt. 45, 2975-2985 (2006).
[CrossRef] [PubMed]

T. J. Crimmins, "Geometric filter for speckle reduction," Appl. Opt. 24, 1438-1443 (1985).
[CrossRef] [PubMed]

O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
[CrossRef] [PubMed]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase in phase-shifting digital holography," Appl. Opt. 45, 975-983 (2006).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, "Compression of interference patterns with application to phase-shifting digital holography," Appl. Opt. 45, 2437-2443 (2006).
[CrossRef] [PubMed]

Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001).
[CrossRef]

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (1)

E. Darakis and J. J. Soraghan, "Use of Fresnelets for phase-shifting digital hologram compression," IEEE Trans. Image Process. 15, 3804-3811 (2006).
[CrossRef] [PubMed]

IRE Trans. Inf. Theory (1)

I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theory 8, 194-195 (1962).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

P. Hariharan and Z. S. Hegedus, "Reduction of speckle in coherent imaging by spatial frequency sampling," Opt. Acta 21, 345-356 (1974).
[CrossRef]

Opt. Commun. (2)

J. C. Dainty and W. T. Welford, "Reduction of speckle in image plane hologram reconstruction by use of a moving pupil," Opt. Commun. 3, 289-294 (1971).
[CrossRef]

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, "Incoherent recovering of the spatial resolution in digital holography," Opt. Commun. 260, 62-67 (2006).
[CrossRef]

Opt. Eng. (Bellingham) (3)

L. S. Lim, "Techniques for speckle noise removal," Opt. Eng. (Bellingham) 20, 670-678 (1981).

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. (Bellingham) 43, 2233-2238 (2004).
[CrossRef]

I. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. (Bellingham) 44, 075801 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Other (5)

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, 1984).

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

J. W. Goodman, Speckle Phenomena: Theory and Applications (Roberts & Company, 2006).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw Hill, 1965).

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Figures (4)

Fig. 1
Fig. 1

(a) Coherent optical imaging system with a moving filter in the Fourier plane. The hologram plane denotes the location that our DH would occupy in such a system. (b) Schematic of the DFF technique, which is the discrete analog of the optical technique in (a), starting from the hologram plane. DLCT, discrete linear canonical transform; IDFT, inverse discrete Fourier transform.

Fig. 2
Fig. 2

Nonfiltered version of the USAF 1951, three-bar resolving power test chart ( 2048 × 2048 pixels in size), (b) zoomed in 115 × 115 pixel region of the chart showing the smallest details on the chart, (c) zoomed in results of applying on (a) the DFF technique with aperture width of 512, (d) zoomed in result of applying on (a) a median filter with a neighborhood size of 3 × 3 , and (e) zoomed in result of applying on (a) a mean filter with a neighborhood size of 3 × 3 .

Fig. 3
Fig. 3

(a) Shows the original reconstruction and (b) shows the result of applying the DFF technique.

Fig. 4
Fig. 4

Graph showing the results of DFF technique (points are labeled with the aperture size used), and the median and the mean filters (points are labeled with the neighborhood sizes used).

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

f ( r ̇ ) = t ( r ) d ( r ) ,
R f f ( r 1 , r 2 ) = f ( r 1 ) f * ( r 2 ) = t ( r 1 ) d ( r 1 ) t * ( r 2 ) d * ( r 2 ) ,
R f f ( r 1 , r 2 ) = t ( r 1 ) t * ( r 2 ) d ( r 1 ) d * ( r 2 ) = t ( r 1 ) t * ( r 2 ) R d d ( r 1 , r 2 ) .
R d d ( r 1 , r 2 ) = d ( r 1 ) d * ( r 2 ) = δ ( r 1 r 2 ) .
I ( r ) = j I j ( r ) ,
R I I ( r 1 , r 2 ) = m , n R I m I n ( r 1 , r 2 ) , = m , n g m ( r 1 ) 2 g n ( r 2 ) 2 ,
X 1 2 X 2 2 = X 1 2 X 2 2 + X 1 X 2 * 2 .
R I I ( r 1 , r 2 ) = m , n g m ( r 1 ) 2 g n ( r 2 ) 2 + m , n g m * ( r 1 ) g n ( r 2 ) 2 , = m , n I m ( r 1 ) I n ( r 2 ) + m , n R g m g n ( r 1 , r 2 ) 2 .
Ω ( u 1 , u 2 ) = F { R I I ( r 1 , r 2 ) } , = F { m , n I m ( r 1 ) I n ( r 2 ) } + F { m , n R g m g n ( r 1 , r 2 ) 2 } ,
Ω 1 ( u 1 , u 2 ) = m , n [ T ( u 1 ) T * ( u 1 ) ] [ H m ( u 1 ) H m * ( u 1 ) ] [ T ( u 2 ) T * ( u 2 ) ] [ H n ( u 2 ) H n * ( u 2 ) ] ,
Ω 2 ( u 1 , u 2 ) = m , n F { R f f ( r 1 , r 2 ) r 1 h m ( r 1 ) r 2 h n * ( r 2 ) 2 } ,
Ω 2 ( u 1 , u 2 ) = m , n [ Γ ( u 1 , u 2 ) H m ( u 1 ) H n * ( u 2 ) ] [ Γ * ( u 1 , u 2 ) H m * ( u 1 ) H n ( u 2 ) ] ,
Ω 2 ( u , u ) = R t t ( 0 ) 2 m , n [ H m ( u ) H n * ( u ) ] [ H m * ( u ) H n ( u ) ] .
Ω 2 ( u , u ) = R t t ( 0 ) 2 m x , m y , n x , n y [ rect ( u x m x Δ u x w x , u y m y Δ u y w y ) rect ( u x n x Δ u x w x , u y n y Δ u y w y ) ] * [ rect ( u x m x Δ u x w x , u y m y Δ u y w y ) rect ( u x n x Δ u x w x , u y n y Δ u y w y ) ] ,
Ω 2 ( u , u ) = R u ( 0 ) 2 m x , m y , n x , n y [ 1 u x Δ u x ( m + n ) w x Δ u x m n ] [ 1 u y Δ u y ( m + n ) w y Δ u y m n ] .
R = 1 X ,

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