Abstract

An adaptive joint transform correlator for real-time pattern recognition is presented. A reference image for the correlator is generated with a new iterative algorithm. The training algorithm is based on synthetic discriminant functions. The obtained reference image contains the information needed to reliably discriminate a target against known false objects and a cluttered background. Calibration lookup tables of all optoelectronics elements used are included in the design of the adaptive joint transform correlator. Two methods for the implementation of the proposed joint transform correlator in an optodigital setup are considered. Experimental results are provided and compared with those of computer simulations.

© 2006 Optical Society of America

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  1. A. Vanderlugt, "Signal detection by complex filtering," IEEE Trans. Inf. Theory IT-10, 139-145 (1964).
  2. C. S. Weaver and J. L. Goodman, "Technique for optically convolving two functions," Appl. Opt. 5, 1248-1249 (1966).
    [CrossRef] [PubMed]
  3. J. Nicolás, J. Campos, C. Iemmi, I. Moreno, and M. J. Yzuel, "Convergent optical correlator alignment based on frequency filtering," Appl. Opt. 41, 1505-1514 (2002).
    [CrossRef] [PubMed]
  4. B. V. K. Vijaya Kumar and L. Hassebrook, "Performance measures for correlation filters," Appl. Opt. 29, 2997-3005 (1990).
    [CrossRef] [PubMed]
  5. L. P. Yaroslavsky, "The theory of optimal methods for localization of objects in pictures," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1993), Vol. XXXII, pp. 145-201.
    [CrossRef]
  6. V. Kober, L. P. Yaroslavsky, J. Campos, and M. J. Yzuel, "Optimal filter approximation by means of a phase-only filter with quantization," Opt. Lett. 19, 978-980 (1994).
    [CrossRef] [PubMed]
  7. C. F. Hester and D. Casasent, "Multivariant technique for multiclass pattern recognition," Appl. Opt. 19, 1758-1761 (1980).
    [CrossRef] [PubMed]
  8. D. Casasent, "Unified synthetic discriminant function computational formulation," Appl. Opt. 23, 1620-1627 (1984).
    [CrossRef] [PubMed]
  9. A. Mahalanobis, B. V. K. Vijaya Kumar, and D. Casasent, "Minimum average correlation filters," Appl. Opt. 26, 3633-3640 (1987).
    [CrossRef] [PubMed]
  10. B. V. K. Vijaya Kumar, "Tutorial survey of composite filter designs for optical correlators," Appl. Opt. 31, 4773-4801 (1992).
    [CrossRef] [PubMed]
  11. B. Javidi, "Nonlinear joint power spectrum based optical correlation," Appl. Opt. 28, 2358-2366 (1989).
    [CrossRef] [PubMed]
  12. B. Javidi and J. Wang, "Binary nonlinear joint transform correlation with median and subset median thresholding," Appl. Opt. 30, 967-976 (1991).
    [CrossRef] [PubMed]
  13. L. P. Yaroslavky and E. Marom, "Nonlinearity optimization in nonlinear joint transform correlators," Appl. Opt. 36, 4816-4822 (1997).
    [CrossRef]
  14. Q. Tang and B. Javidi, "Sensitivity of the nonlinear joint transform correlator: experimental investigations," Appl. Opt. 31, 4016-4024 (1992).
    [CrossRef] [PubMed]
  15. M. S. Alam and M. A. Karim, "Fringe-adjusted joint transform correlator," Appl. Opt. 32, 4344-4350 (1993).
    [CrossRef] [PubMed]
  16. J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
    [CrossRef]
  17. V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Real-time pattern recognition with adaptive correlation filters," in Applications of Digital Image Processing XXVIII, A. G. Tescher. ed., Proc. SPIE 5909,5909211-5909217 (2005).
  18. B. Javidi and J. Wang, "Limitation of the classic definition of the correlation signal-to-noise-ratio in optical pattern recognition with disjoint signal and scene noise," Appl. Opt. 31, 6826 (1992).
    [CrossRef] [PubMed]
  19. B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A. 11, 2604-2612 (1994).
    [CrossRef]
  20. V. Kober and J. Campos, "Accuracy of location measurement of a noisy target in a nonoverlapping background," J. Opt. Soc. Am. A 13, 1653-1666 (1996).
    [CrossRef]
  21. H. K. Liu and T. H. Chao, "Liquid crystal television spatial light modulators," Appl. Opt. 28, 4772-4780 (1989).
    [CrossRef] [PubMed]
  22. K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-247 (1990).
    [CrossRef]
  23. C. Soutar and K. Lu, "Determination of the physical properties of an arbitrary twisted-nematic liquid crystal display cell," Opt. Eng. 33, 2704-2712 (1994).
    [CrossRef]
  24. R. Dou and M. K. Giles, "Simple technique for measuring the phase property of a twisted nematic liquid crystal television," Opt. Eng. 35, 808-813 (1996).
    [CrossRef]
  25. H. Kim and Y. H. Lee, "Unique measurement of the parameters of a twisted-nematic liquid-crystal display," Appl. Opt. 44, 1642-1649 (2005).
    [CrossRef] [PubMed]
  26. I. Moreno, J. Campos, M. J. Yzuel, and V. Kober, "Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition," Opt. Eng. 37, 144-150 (1998).
    [CrossRef]
  27. J. Campos, A. Márquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, and I. Moreno, "Fully complex synthetic discriminant functions written onto phase-only modulators," Appl. Opt. 39, 5965-5970 (2000).
    [CrossRef]
  28. B. Javidi and J. L. Horner, "Single SLM joint transform correlator," Appl. Opt. 28, 1027-1032 (1989).
    [CrossRef] [PubMed]

2005 (2)

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

H. Kim and Y. H. Lee, "Unique measurement of the parameters of a twisted-nematic liquid-crystal display," Appl. Opt. 44, 1642-1649 (2005).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

1998 (1)

I. Moreno, J. Campos, M. J. Yzuel, and V. Kober, "Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition," Opt. Eng. 37, 144-150 (1998).
[CrossRef]

1997 (1)

1996 (2)

R. Dou and M. K. Giles, "Simple technique for measuring the phase property of a twisted nematic liquid crystal television," Opt. Eng. 35, 808-813 (1996).
[CrossRef]

V. Kober and J. Campos, "Accuracy of location measurement of a noisy target in a nonoverlapping background," J. Opt. Soc. Am. A 13, 1653-1666 (1996).
[CrossRef]

1994 (3)

B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A. 11, 2604-2612 (1994).
[CrossRef]

C. Soutar and K. Lu, "Determination of the physical properties of an arbitrary twisted-nematic liquid crystal display cell," Opt. Eng. 33, 2704-2712 (1994).
[CrossRef]

V. Kober, L. P. Yaroslavsky, J. Campos, and M. J. Yzuel, "Optimal filter approximation by means of a phase-only filter with quantization," Opt. Lett. 19, 978-980 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (3)

1991 (1)

1990 (2)

B. V. K. Vijaya Kumar and L. Hassebrook, "Performance measures for correlation filters," Appl. Opt. 29, 2997-3005 (1990).
[CrossRef] [PubMed]

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-247 (1990).
[CrossRef]

1989 (3)

1987 (1)

1984 (1)

1980 (1)

1966 (1)

1964 (1)

A. Vanderlugt, "Signal detection by complex filtering," IEEE Trans. Inf. Theory IT-10, 139-145 (1964).

Alam, M. S.

Álvarez-Borrego, J.

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Real-time pattern recognition with adaptive correlation filters," in Applications of Digital Image Processing XXVIII, A. G. Tescher. ed., Proc. SPIE 5909,5909211-5909217 (2005).

Campos, J.

Casasent, D.

Chao, T. H.

Cottrell, D. M.

Davis, J. A.

Díaz-Ramírez, V. H.

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Real-time pattern recognition with adaptive correlation filters," in Applications of Digital Image Processing XXVIII, A. G. Tescher. ed., Proc. SPIE 5909,5909211-5909217 (2005).

Dou, R.

R. Dou and M. K. Giles, "Simple technique for measuring the phase property of a twisted nematic liquid crystal television," Opt. Eng. 35, 808-813 (1996).
[CrossRef]

Giles, M. K.

R. Dou and M. K. Giles, "Simple technique for measuring the phase property of a twisted nematic liquid crystal television," Opt. Eng. 35, 808-813 (1996).
[CrossRef]

Gonzalez-Fraga, J. A.

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

Goodman, J. L.

Hassebrook, L.

Hester, C. F.

Horner, J. L.

Iemmi, C.

Javidi, B.

Karim, M. A.

Kim, H.

Kober, V.

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, and V. Kober, "Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition," Opt. Eng. 37, 144-150 (1998).
[CrossRef]

V. Kober and J. Campos, "Accuracy of location measurement of a noisy target in a nonoverlapping background," J. Opt. Soc. Am. A 13, 1653-1666 (1996).
[CrossRef]

V. Kober, L. P. Yaroslavsky, J. Campos, and M. J. Yzuel, "Optimal filter approximation by means of a phase-only filter with quantization," Opt. Lett. 19, 978-980 (1994).
[CrossRef] [PubMed]

V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Real-time pattern recognition with adaptive correlation filters," in Applications of Digital Image Processing XXVIII, A. G. Tescher. ed., Proc. SPIE 5909,5909211-5909217 (2005).

Kumar, B. V. K. Vijaya

Lee, Y. H.

Liu, H. K.

Lu, K.

C. Soutar and K. Lu, "Determination of the physical properties of an arbitrary twisted-nematic liquid crystal display cell," Opt. Eng. 33, 2704-2712 (1994).
[CrossRef]

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-247 (1990).
[CrossRef]

Mahalanobis, A.

Marom, E.

Márquez, A.

Moreno, I.

Nicolás, J.

Saleh, B. E. A.

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-247 (1990).
[CrossRef]

Soutar, C.

C. Soutar and K. Lu, "Determination of the physical properties of an arbitrary twisted-nematic liquid crystal display cell," Opt. Eng. 33, 2704-2712 (1994).
[CrossRef]

Tang, Q.

Vanderlugt, A.

A. Vanderlugt, "Signal detection by complex filtering," IEEE Trans. Inf. Theory IT-10, 139-145 (1964).

Wang, J.

B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A. 11, 2604-2612 (1994).
[CrossRef]

B. Javidi and J. Wang, "Limitation of the classic definition of the correlation signal-to-noise-ratio in optical pattern recognition with disjoint signal and scene noise," Appl. Opt. 31, 6826 (1992).
[CrossRef] [PubMed]

B. Javidi and J. Wang, "Binary nonlinear joint transform correlation with median and subset median thresholding," Appl. Opt. 30, 967-976 (1991).
[CrossRef] [PubMed]

Weaver, C. S.

Yaroslavky, L. P.

Yaroslavsky, L. P.

V. Kober, L. P. Yaroslavsky, J. Campos, and M. J. Yzuel, "Optimal filter approximation by means of a phase-only filter with quantization," Opt. Lett. 19, 978-980 (1994).
[CrossRef] [PubMed]

L. P. Yaroslavsky, "The theory of optimal methods for localization of objects in pictures," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1993), Vol. XXXII, pp. 145-201.
[CrossRef]

Yzuel, M. J.

Appl. Opt. (17)

C. F. Hester and D. Casasent, "Multivariant technique for multiclass pattern recognition," Appl. Opt. 19, 1758-1761 (1980).
[CrossRef] [PubMed]

D. Casasent, "Unified synthetic discriminant function computational formulation," Appl. Opt. 23, 1620-1627 (1984).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. Vijaya Kumar, and D. Casasent, "Minimum average correlation filters," Appl. Opt. 26, 3633-3640 (1987).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, "Tutorial survey of composite filter designs for optical correlators," Appl. Opt. 31, 4773-4801 (1992).
[CrossRef] [PubMed]

B. Javidi, "Nonlinear joint power spectrum based optical correlation," Appl. Opt. 28, 2358-2366 (1989).
[CrossRef] [PubMed]

B. Javidi and J. Wang, "Binary nonlinear joint transform correlation with median and subset median thresholding," Appl. Opt. 30, 967-976 (1991).
[CrossRef] [PubMed]

L. P. Yaroslavky and E. Marom, "Nonlinearity optimization in nonlinear joint transform correlators," Appl. Opt. 36, 4816-4822 (1997).
[CrossRef]

Q. Tang and B. Javidi, "Sensitivity of the nonlinear joint transform correlator: experimental investigations," Appl. Opt. 31, 4016-4024 (1992).
[CrossRef] [PubMed]

M. S. Alam and M. A. Karim, "Fringe-adjusted joint transform correlator," Appl. Opt. 32, 4344-4350 (1993).
[CrossRef] [PubMed]

C. S. Weaver and J. L. Goodman, "Technique for optically convolving two functions," Appl. Opt. 5, 1248-1249 (1966).
[CrossRef] [PubMed]

J. Nicolás, J. Campos, C. Iemmi, I. Moreno, and M. J. Yzuel, "Convergent optical correlator alignment based on frequency filtering," Appl. Opt. 41, 1505-1514 (2002).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar and L. Hassebrook, "Performance measures for correlation filters," Appl. Opt. 29, 2997-3005 (1990).
[CrossRef] [PubMed]

B. Javidi and J. Wang, "Limitation of the classic definition of the correlation signal-to-noise-ratio in optical pattern recognition with disjoint signal and scene noise," Appl. Opt. 31, 6826 (1992).
[CrossRef] [PubMed]

H. K. Liu and T. H. Chao, "Liquid crystal television spatial light modulators," Appl. Opt. 28, 4772-4780 (1989).
[CrossRef] [PubMed]

H. Kim and Y. H. Lee, "Unique measurement of the parameters of a twisted-nematic liquid-crystal display," Appl. Opt. 44, 1642-1649 (2005).
[CrossRef] [PubMed]

J. Campos, A. Márquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, and I. Moreno, "Fully complex synthetic discriminant functions written onto phase-only modulators," Appl. Opt. 39, 5965-5970 (2000).
[CrossRef]

B. Javidi and J. L. Horner, "Single SLM joint transform correlator," Appl. Opt. 28, 1027-1032 (1989).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

A. Vanderlugt, "Signal detection by complex filtering," IEEE Trans. Inf. Theory IT-10, 139-145 (1964).

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A. 11, 2604-2612 (1994).
[CrossRef]

Lect. Notes Comput. Sci. (1)

J. A. Gonzalez-Fraga, V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Improving the discrimination capability with an adaptive synthetic discriminant function filter," Lect. Notes Comput. Sci. 3523, 83-90 (2005).
[CrossRef]

Opt. Eng. (4)

I. Moreno, J. Campos, M. J. Yzuel, and V. Kober, "Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition," Opt. Eng. 37, 144-150 (1998).
[CrossRef]

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-247 (1990).
[CrossRef]

C. Soutar and K. Lu, "Determination of the physical properties of an arbitrary twisted-nematic liquid crystal display cell," Opt. Eng. 33, 2704-2712 (1994).
[CrossRef]

R. Dou and M. K. Giles, "Simple technique for measuring the phase property of a twisted nematic liquid crystal television," Opt. Eng. 35, 808-813 (1996).
[CrossRef]

Opt. Lett. (1)

Other (2)

L. P. Yaroslavsky, "The theory of optimal methods for localization of objects in pictures," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1993), Vol. XXXII, pp. 145-201.
[CrossRef]

V. H. Díaz-Ramírez, V. Kober, and J. Álvarez-Borrego, "Real-time pattern recognition with adaptive correlation filters," in Applications of Digital Image Processing XXVIII, A. G. Tescher. ed., Proc. SPIE 5909,5909211-5909217 (2005).

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Figures (21)

Fig. 1
Fig. 1

Block diagram of the JTC.

Fig. 2
Fig. 2

Block diagram of the nonlinear JTC.

Fig. 3
Fig. 3

Block diagram of the fringe-adjusted JTC.

Fig. 4
Fig. 4

Block diagram of the proposed iterative algorithm.

Fig. 5
Fig. 5

(Color online) Compact JTC used in experiments.

Fig. 6
Fig. 6

Intensity response of a twisted nematic LCD captured with a CCD camera.

Fig. 7
Fig. 7

Input scene containing two objects with the same shape but with different information contents.

Fig. 8
Fig. 8

Computer simulation results obtained for the scene in Fig. 7 with (a) binary JTC, (b) fringe-adjusted JTC.

Fig. 9
Fig. 9

Bipolar reference image obtained for the scene in Fig. 7 with the proposed method.

Fig. 10
Fig. 10

Computer simulation result obtained for the scene in Fig. 7 with the proposed JTC.

Fig. 11
Fig. 11

Input scene containing three objects with similar shapes but with different information contents.

Fig. 12
Fig. 12

Rotation-invariant reference image obtained for the scene in Fig. 11 with the proposed method.

Fig. 13
Fig. 13

Correlation intensity distributions obtained with the reference image in Fig. 12 and with the scene in Fig. 11 for the target rotated by degrees of (a) 0, (b) 2, (c) 4, (d) 6, (e) 8, and (f) 10.

Fig. 14
Fig. 14

Scale-invariant reference image obtained for the scene in Fig. 11 with the proposed method.

Fig. 15
Fig. 15

Correlation intensity distributions obtained with the reference image in Fig. 14 and with the scene in Fig. 11 for the target scaled by factors of (a) 1, (b) 0.98, (c) 0.99, (d) 1.05, (e) 1.1, and (f) 1.2.

Fig. 16
Fig. 16

Joint images composed with the scene in Fig. 7 and with (a) the positive part of the reference image in Fig. 9 and (b) the negative part of the reference image in Fig. 9.

Fig. 17
Fig. 17

Cross-correlation intensity planes obtained in optodigital JTC for the joint images shown in (a) Figs. 16(a) and (b) 16(b).

Fig. 18
Fig. 18

Cross-correlation intensity plane after elementwise postprocessing: (a) intensity plane, (b) intensity distribution.

Fig. 19
Fig. 19

Joint image formed for the constant addition method.

Fig. 20
Fig. 20

Cross-correlation intensity plane obtained in optodigital JTC for the joint image shown in Fig. 19.

Fig. 21
Fig. 21

Cross-correlation plane obtained from the plane in Fig. 20 after postprocessing; (a) intensity plane, (b) intensity distribution.

Equations (39)

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

f ( x , y ) = s ( x , y d ) + t ( x , y + d ) ,
F ( μ , ν ) = S ( μ , ν ) exp ( i d ν ) + T ( μ , ν ) exp ( i d ν ) .
E ( μ , ν ) = | F ( μ , ν ) | 2 = S ( μ , ν ) S * ( μ , ν ) + T ( μ , ν ) T * ( μ , ν ) + S ( μ , ν ) T * ( μ , ν ) exp ( i 2 d ν ) + S * ( μ , ν ) T ( μ , ν ) exp ( i 2 d ν ) .
e ( x , y ) = s ( x , y ) s ( x , y ) + t ( x , y ) t ( x , y ) + s ( x , y 2 d ) t ( x , y 2 d ) + s ( x , y + 2 d ) t ( x , y + 2 d ) ,
DC = 1 | C B ( 0 , 0 ) | 2 | C T ( 0 , 0 ) | 2 ,
f ( x , y ) = s ( x , y d ) + b ˜ ( x , y d ) + t ( x , y + d ) ,
b ˜ ( x , y ) = w ( x x 0 , y y 0 ) b ( x , y ) ,
w ( x x 0 , y y 0 ) = { 0 , within   the   object   area 1 , otherwise .
| F ( μ , ν ) | 2 = S ( μ , ν ) 2 + T ( μ , ν ) 2 + B ˜ ( μ , ν ) 2 + [ T ( μ , ν ) S * ( μ , ν ) + T ( μ , ν ) B ˜ * ( μ , ν ) + S ( μ , ν ) B ˜ * ( μ , ν ) ] exp ( i 2 d ν ) + [ T * ( μ , ν ) S ( μ , ν ) + T * ( μ , ν ) B ˜ ( μ , ν ) + S * ( μ , ν ) B ˜ ( μ , ν ) ] exp ( i 2 d ν ) .
G ( ω ) = g ( E ) exp ( i ω E ) d E .
g ( E ) = 1 2 π G ( ω ) exp ( i ω E ) d ω .
g ( E ) = 1 2 π G ( ω ) exp { i ω [ S 2 ( μ , ν ) + T 2 ( μ , ν ) ] } × exp { i 2 ω S ( μ , ν ) T ( μ , ν ) × cos [ 2 d ν + ϕ S ( μ , ν ) ϕ T ( μ , ν ) ] } d ω ,
G ( ω ) = 2 ( i ω ) k + 1 Γ ( k + 1 ) ,
g k ( E ) = v = 0 ϵ v Γ ( k + 1 ) [ S ( μ , ν ) T ( μ , ν ) ] k 2 k Γ ( 1 v k 2 ) Γ ( 1 + v + k 2 ) × cos [ 2 v d ν + v ϕ S ( μ , ν ) v ϕ T ( μ , ν ) ] ,
ϵ v = { 1 , for     v = 0 2 , for v > 0 .
H f a f ( μ , ν ) = B ( μ , ν ) A ( μ , ν ) + | T ( μ , ν ) | 2 ,
H f a f ( μ , ν ) 1 | T ( μ , ν ) | 2 .
G ( μ , ν ) 2 { 1 + cos [ ϕ S ( μ , ν ) ϕ T ( μ , ν ) + 2 ν d ] } .
h ( x , y ) = i = 1 N a i s i ( x , y ) ,
s i h = c i .
h = R a ,
c = R + h ,
a = ( R + R ) 1 c ,
h = R ( R + R ) 1 c .
c = [ 1 , 1 , , 1 ] T .
c = [ 1 , 1 , . . . , 1 , 0 , 0 , , 0 ] T .
h ( x , y ) = h + ( x , y ) h ( x , y ) ,
h + ( x , y ) = { h ( x , y ) , h ( x , y ) 0 0 , otherwise ,
h ( x , y ) = { h ( x , y ) , h ( x , y ) < 0 0 , otherwise .
c ( x , y ) = | s ( x , y ) [ h + ( x , y ) h ( x , y ) ] | 2 = | s ( x , y ) h + ( x , y ) | 2 + | s ( x , y ) h ( x , y ) | 2 2 [ | s ( x , y ) h + ( x , y ) | 2 ] 1 / 2 × [ | s ( x , y ) h ( x , y ) | 2 ] 1 / 2 .
f ( x , y ) = s ˜ ( x , y d ) + h ˜ ( x , y + d ) ,
c ( x , y ) = | s ˜ ( x , y ) s ˜ ( x , y ) + h ˜ ( x , y ) h ˜ ( x , y ) | 2 + | s ˜ ( x , y 2 d ) h ˜ ( x , y 2 d ) | 2 + | h ˜ ( x , y + 2 d ) s ˜ ( x , y + 2 d ) | 2 .
| s ( x , y ) h ( x , y ) | 2 = | [ s ( x , y ) + c c ] [ h ( x , y ) + c c ] | 2 = | [ s ˜ ( x , y ) c ] [ h ˜ ( x , y ) c ] | 2 = | s ˜ ( x , y ) h ˜ ( x , y ) | 2 + | h ˜ ( x , y ) c | 2 + | s ˜ ( x , y ) c | 2 + | c c | 2 2 { s ˜ ( x , y ) h ˜ ( x , y ) h ˜ ( x , y ) c } 2 { s ˜ ( x , y ) h ˜ ( x , y ) s ˜ ( x , y ) c } + 2 { s ˜ ( x , y ) h ˜ ( x , y ) c c } + 2 { h ˜ ( x , y ) c s ˜ ( x , y ) c } 2 { h ˜ ( x , y ) c c c } 2 { s ˜ ( x , y ) c c c } .
| s ( x , y ) h ( x , y ) | 2 = | s ˜ ( x , y ) h ˜ ( x , y ) | 2 + C 1 2 + C 2 2 + C 3 2 2 { s ˜ ( x , y ) h ˜ ( x , y ) C 1 } 2 { s ˜ ( x , y ) h ˜ ( x , y ) C 2 } + 2 { s ˜ ( x , y ) h ˜ ( x , y ) C 3 } + 2 { C 1 C 2 } 2 { C 1 C 3 } 2 { C 2 C 3 } .
C 1 = α [ h ˜ ( x , y ) c ]
= α [ c h ˜ ( x + τ x , y + τ y ) d τ x d τ y ]
α { c [ h ˜ ( x , y ) ] } ,
C 2 α { c [ s ˜ ( x , y ) ] } ,
C 2 α c 2 ,

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