Abstract

A modified fringe-adjusted joint transform correlator is proposed that is able to accommodate noise in the input scene. The effect of noise in the input scene on the performance of the joint transform correlator is analyzed and quantified. When the target is embedded in a severely noise-corrupted input scene, it is shown that the proposed modified fringe-adjusted filter joint transform correlator delivers a better correlation performance and the capacity to accommodate this noise than does the fringe-adjusted filter-based correlator. When the power spectra of the input image and the reference image are subtracted from the power spectrum of the joint-input image, it is found that the noise effect on the output plane is independent of the objects in the input scene and originates from the convolution of the reference image and noise in the input scene.

© 1996 Optical Society of America

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References

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  1. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  2. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  3. M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
    [CrossRef]
  4. F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
    [CrossRef]
  5. B. Javidi, C. J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
    [CrossRef] [PubMed]
  6. W. B. Hahn, D. L. Flannery, “Design elements of a binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  7. F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effect of fringe binarization on multi-object joint transform correlation,” Appl. Opt. 28, 2988–2990 (1988).
    [CrossRef]
  8. J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
    [CrossRef]
  9. M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint Fourier transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
    [CrossRef]
  10. M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe adjusted joint transform correlator,” Opt. Eng. 33, 522–527 (1994).
    [CrossRef]
  11. C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
    [CrossRef]
  12. M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
    [CrossRef] [PubMed]
  13. M. S. Alam, M. A. Karim, “Joint transform correlation under varying illumination,” Appl. Opt. 32, 4351–4356 (1993).
    [CrossRef] [PubMed]
  14. D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
    [CrossRef]
  15. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  16. H. K. Liu, T. H. Chao, “Optical image subtraction techniques, 1975–1985,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 55–65 (1986).
  17. A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
    [CrossRef] [PubMed]
  18. M. A. Flavin, J. L. Horner, “Amplitude-encoded phase-only filters,”Appl. Opt. 28, 1692–1696 (1989).
    [CrossRef] [PubMed]
  19. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  20. B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
    [CrossRef]
  21. R. K. Wang, C. R. Chatwin, M. Y. Huang, “Modified filter synthetic discriminant functions for improved optical correlator performance,” Appl. Opt. 33, 7646–7654 (1994).
    [CrossRef] [PubMed]
  22. R. K. Wang, C. R. Chatwin, “Multilevel phase- and amplitude-encoded modified-filter synthetic-discriminant-function filters,” Appl. Opt. 34, 4094–4104 (1995).
    [CrossRef] [PubMed]

1995 (1)

1994 (3)

R. K. Wang, C. R. Chatwin, M. Y. Huang, “Modified filter synthetic discriminant functions for improved optical correlator performance,” Appl. Opt. 33, 7646–7654 (1994).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe adjusted joint transform correlator,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

1993 (2)

1992 (4)

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
[CrossRef]

W. B. Hahn, D. L. Flannery, “Design elements of a binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint Fourier transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
[CrossRef]

1991 (3)

M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
[CrossRef]

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
[CrossRef]

1990 (2)

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

1989 (1)

1988 (2)

1984 (1)

1966 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Alam, M. S.

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe adjusted joint transform correlator,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Joint transform correlation under varying illumination,” Appl. Opt. 32, 4351–4356 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint Fourier transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
[CrossRef]

M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
[CrossRef]

Awwal, A. A. S.

M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
[CrossRef]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

Barnes, T. H.

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

Bunch, R. M.

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Chao, T. H.

H. K. Liu, T. H. Chao, “Optical image subtraction techniques, 1975–1985,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 55–65 (1986).

Chatwin, C. R.

Cheng, F.

Cotrell, D. M.

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Davis, J. A.

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Eiju, T.

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

Feng, D.

D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
[CrossRef]

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of a binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Flavin, M. A.

Gianino, P. D.

Goodman, J. W.

Gregory, D. A.

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of a binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Haskell, T. G.

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

Horner, J. L.

Huang, M. Y.

Jahan, S. R.

Javidi, B.

B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
[CrossRef]

B. Javidi, C. J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
[CrossRef] [PubMed]

Johnson, F. T. J.

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

Karim, M. A.

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe adjusted joint transform correlator,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Joint transform correlation under varying illumination,” Appl. Opt. 32, 4351–4356 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint Fourier transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
[CrossRef]

M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
[CrossRef]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

Kuo, C. J.

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

B. Javidi, C. J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
[CrossRef] [PubMed]

Liu, H. K.

H. K. Liu, T. H. Chao, “Optical image subtraction techniques, 1975–1985,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 55–65 (1986).

Matsuda, K.

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

Merrill, E. A.

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Nagata, T.

Parchekani, F.

B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
[CrossRef]

Tang, Q.

B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Wang, R. K.

Weaver, C. S.

Xia, S.

D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
[CrossRef]

Yu, F. T. S.

Zhao, H.

D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
[CrossRef]

Appl. Opt. (11)

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

B. Javidi, C. J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
[CrossRef] [PubMed]

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effect of fringe binarization on multi-object joint transform correlation,” Appl. Opt. 28, 2988–2990 (1988).
[CrossRef]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Joint transform correlation under varying illumination,” Appl. Opt. 32, 4351–4356 (1993).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

M. A. Flavin, J. L. Horner, “Amplitude-encoded phase-only filters,”Appl. Opt. 28, 1692–1696 (1989).
[CrossRef] [PubMed]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

R. K. Wang, C. R. Chatwin, M. Y. Huang, “Modified filter synthetic discriminant functions for improved optical correlator performance,” Appl. Opt. 33, 7646–7654 (1994).
[CrossRef] [PubMed]

R. K. Wang, C. R. Chatwin, “Multilevel phase- and amplitude-encoded modified-filter synthetic-discriminant-function filters,” Appl. Opt. 34, 4094–4104 (1995).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

M. S. Alam, A. A. S. Awwal, M. A. Karim, “Improved correlation discrimination using a joint transform optical correlator,” Microwave Opt. Technol. Lett. 4, 103–106 (1991).
[CrossRef]

Opt. Commun. (1)

D. Feng, H. Zhao, S. Xia, “Amplitude-modulated JTC for improving correlation discrimination,” Opt. Commun. 86, 260–264 (1991).
[CrossRef]

Opt. Eng. (6)

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe adjusted joint transform correlator,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

B. Javidi, F. Parchekani, Q. Tang, “Gray-scale nonlinear joint transform correlator,” Opt. Eng. 31, 888–895 (1992).
[CrossRef]

F. T. J. Johnson, T. H. Barnes, T. Eiju, T. G. Haskell, K. Matsuda, “Analysis of a joint transform correlator using a phase-only spatial light modulator,” Opt. Eng. 30, 1947–1957 (1991).
[CrossRef]

J. A. Davis, E. A. Merrill, D. M. Cotrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint transform correlation,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

W. B. Hahn, D. L. Flannery, “Design elements of a binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Opt. Laser Technol. (1)

M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint Fourier transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
[CrossRef]

Other (1)

H. K. Liu, T. H. Chao, “Optical image subtraction techniques, 1975–1985,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 55–65 (1986).

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Figures (9)

Fig. 1
Fig. 1

Schematic drawing of a fringe-adjusted JTC (see Ref. 12).

Fig. 2
Fig. 2

Schematic diagram of an alternative real-time fringe-adjusted JTC.

Fig. 3
Fig. 3

Bradley APC images used in the simulation: (a) a noise-free image, and (b) a noise-corrupted image with a signal–energy-to-noise–energy ratio of 0.21.

Fig. 4
Fig. 4

Three-dimensional plots of the correlation output functions when the input scene is free of noise: (a) from the MFAF-based JTC, and (b) from the FAF-based JTC.

Fig. 5
Fig. 5

Three-dimensional plots of the correlation output functions with no power-spectra subtraction when the input scene is noise corrupted: (a) from the MFAF-based JTC, and (b) from the FAF-based JTC.

Fig. 6
Fig. 6

Three-dimensional plots of the correlation output functions with power-spectra subtraction from a noise-corrupted input scene: (a) from the MFAF-based JTC, and (b) from the FAF-based JTC.

Fig. 7
Fig. 7

Noise-free multiple-object input scene used for the simulation.

Fig. 8
Fig. 8

Noise-corrupted multiple-object input scene used in the simulation.

Fig. 9
Fig. 9

Three-dimensional plots of the correlation output functions with power-spectra subtraction from a noise-corrupted multi-object input scene: (a) from the MFAF-based JTC, and (b) from the FAF-based JTC.

Tables (3)

Tables Icon

Table 1 Quantified Results from an Input Scene with a Noise-Free Single Object

Tables Icon

Table 2 Quantified Results from an Input Scene with a Noise-Corrupted Single Object

Tables Icon

Table 3 Quantified Results from an Input Scene with Multiple Noise-Corrupted Objects

Equations (24)

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f ( x , y ) = r ( x , y + y 0 ) + t ( x , y y 0 ) .
F ( u , υ ) = R ( u , υ ) exp ( j υ y 0 ) + T ( u , υ ) exp ( j υ y 0 ) ,
| F ( u , υ ) | 2 = | R ( u , υ ) | 2 + | T ( u , υ ) | 2 + R ( u , υ ) T * ( u , υ ) exp ( j 2 υ y 0 ) + R * ( u , υ ) T ( u , υ ) exp ( j 2 υ y 0 ) ,
H amf ( u , υ ) = 1 | R ( u , υ ) | 2 .
H faf ( u , υ ) = B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | 2 ,
t ( x , y ) = t ( x , y ) + n ( x , y ) .
f ( x , y ) = r ( x , y + y 0 ) + t ( x , y y 0 ) + n ( x , y y 0 ) .
F ( u , υ ) = R ( u , υ ) exp ( j υ y 0 ) + T ( u , υ ) exp ( j υ y 0 ) + N ( u , υ ) exp ( j υ y 0 ) ,
| F ( u , υ ) | 2 = | R ( u , υ ) | 2 + | T ( u , υ ) | 2 + 2 Re { R * ( u , υ ) T ( u , υ ) } cos ( 2 υ y 0 ) + 2 Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) + 2 Re { N ( u , υ ) T * ( u , υ ) } + | N ( u , υ ) | 2 .
H mfaf ( u , υ ) = B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | .
G ( u , υ ) = H mfaf ( u , υ ) | F ( u , υ ) | 2 = B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | [ | R ( u , υ ) | 2 + | T ( u , υ ) | 2 + 2 Re { R * ( u , υ ) T ( u , υ ) } cos ( 2 υ y 0 ) ] ,
G ( u , υ ) = B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | [ | R ( u , υ ) | 2 + | T ( u , υ ) | 2 + 2 Re { R * ( u , υ ) T ( u , υ ) } cos ( 2 υ y 0 ) + 2 Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) + 2 Re { N ( u , υ ) T * ( u , υ ) } + | N ( u , υ ) | 2 ] ,
G ( u , υ ) 2 | R ( u , υ ) | + 2 | R ( u , υ ) | [ cos ( 2 υ y 0 ) ] + | N ( u , υ ) | 2 | R ( u , υ ) | + 2 Re { N ( u , υ ) exp ( i ϕ r ) } × [ 1 + cos ( 2 υ y 0 ) ] ,
2 | R ( u , υ ) | [ cos ( 2 υ y 0 ) ] = exp ( i ϕ r ) R ( u , υ ) exp ( i 2 υ y 0 ) + [ exp ( i ϕ r ) R ( u , υ ) exp ( i 2 υ y 0 ) ] * .
G ( u , υ ) 2 + 2 cos ( 2 υ y 0 ) + | N ( u , υ ) | 2 | R ( u , υ ) | 2 + 2 | R ( u , υ ) | Re { N ( u , υ ) exp ( i ϕ r ) } × [ 1 + cos ( 2 υ y 0 ) ] .
| I ( u , υ ) | 2 = | T ( u , υ ) exp ( j υ y 0 ) + N ( u , υ ) exp ( j υ y 0 ) | 2 = | T ( u , υ ) | 2 + | N ( u , υ ) | 2 + 2 Re { N ( u , υ ) T * ( u , υ ) } ,
P ( u , υ ) = | F ( u , υ ) | 2 | R ( u , υ ) | 2 | I ( u , υ ) | 2 = 2 Re { T ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) + 2 Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) ,
P ( u , υ ) = 2 B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | × [ Re { T ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) + Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) ] .
C ( u , υ ) = { + 1 , P ( u , υ ) 0 1 otherwise ,
P ( u , υ ) = { P ( u , υ ) P ( u , υ ) 0 P ( u , υ ) , otherwise ,
f ( x , y ) = r ( x , y + y 0 ) + i = 1 n t i ( x x i , y y i ) + n ( x , y y 0 ) ,
| F ( u , υ ) | 2 = | R ( u , υ ) | 2 + i = 1 n | T i ( u , υ ) | 2 + 2 i = 1 n Re { T i ( u , υ ) R * ( u , υ ) } cos [ u x i + υ ( y 0 + y i ) ] + i = 1 n k = 1 n Re { T i ( u , υ ) } × cos [ u x i + υ ( y i y k ) ] + 2 Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) + | N ( u , υ ) | 2 + 2 i = 1 n Re { N ( u , υ ) T i * ( u , υ ) } cos [ u x 0 + υ ( y 0 y i ) ] ,
P ( u , υ ) = 2 i = 1 n Re { T i ( u , υ ) R * ( u , υ ) } cos [ u x i + υ ( y 0 + y i ) ] + 2 Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) .
G ( u , υ ) = 2 B ( u , υ ) A ( u , υ ) + | R ( u , υ ) | × [ i = 1 n Re { T i ( u , υ ) R * ( u , υ ) } cos [ u x i + υ ( y 0 + y i ) ] + Re { N ( u , υ ) R * ( u , υ ) } cos ( 2 υ y 0 ) ] .

Metrics