Abstract

We analyze the influence of different optical coding methods of the input image in optical correlators. The noise robustness and the optical efficiency of the correlator are investigated. We show in particular that the signal-to-noise ratio is greatly dependent on the coding method. It decreases drastically for large phase modulation.

© 1994 Optical Society of America

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  1. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2355 (1969).
    [CrossRef] [PubMed]
  2. D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  3. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  4. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  5. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  6. Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).
  7. J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
    [CrossRef] [PubMed]
  8. H. J. Caulfield, “Role of the Horner efficiency in the optimization of spatial filters for optical pattern recognition,” Appl. Opt. 21, 4391–4392 (1982).
    [CrossRef] [PubMed]
  9. D. Psaltis, E. G. Paek, S. S. Wenkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 668–704 (1984).
  10. F. M. Dickey, L. A. Romero, “Dual optimality of the phase-only filter,” Opt. Lett. 14, 4–5 (1989).
    [CrossRef] [PubMed]
  11. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  12. D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
    [CrossRef] [PubMed]
  13. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  14. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  15. J. L. Horner, P. D. Gianino, “Signal-dependent phase distortion in optical correlators,” Appl. Opt. 26, 2484–2490 (1987).
    [CrossRef] [PubMed]
  16. J. L. Horner, R. A. Soref, “Phase-dominant spatial light modulators,” Electron. Lett. 24, 626–627 (1988).
    [CrossRef]
  17. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  18. S. Mazé, Ph. Réfrégier, “Noise robustness of optical correlation for amplitude or phase modulation of the input image,” Opt. Lett. 17, 426–428 (1992).
    [CrossRef] [PubMed]
  19. S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).
  20. A. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  21. D. R. Pape, L. J. Hornbeck, R. L. Reel, “Characteristics of the deformable mirror device for optical processing,” in Advances in Optical Information Processing I, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.388, 65–74 (1983).
  22. H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
    [CrossRef]
  23. V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
    [CrossRef]
  24. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]

1993

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

1992

1991

Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

1990

1989

1988

1987

1986

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

1984

1982

1976

1969

1964

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

Arsenault, H. H.

Casasent, D.

Caulfield, H. J.

Chavel, P.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Dickey, F. M.

Farn, M. W.

Figue, J.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Flannery, D. L.

Gianino, P. D.

Goodman, J. W.

Hassebrook, L.

Hornbeck, L. J.

D. R. Pape, L. J. Hornbeck, R. L. Reel, “Characteristics of the deformable mirror device for optical processing,” in Advances in Optical Information Processing I, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.388, 65–74 (1983).

Horner, J. L.

Hsu, Y. N.

Joffre, P.

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

Juday, R. D.

Kan, Y.

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

Laude, V.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Loomis, J. S.

Maloney, W. T.

Mazé, S.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

S. Mazé, Ph. Réfrégier, “Noise robustness of optical correlation for amplitude or phase modulation of the input image,” Opt. Lett. 17, 426–428 (1992).
[CrossRef] [PubMed]

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

Milkovich, M. E.

Nagai, H.

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Wenkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 668–704 (1984).

Pape, D. R.

D. R. Pape, L. J. Hornbeck, R. L. Reel, “Characteristics of the deformable mirror device for optical processing,” in Advances in Optical Information Processing I, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.388, 65–74 (1983).

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Wenkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 668–704 (1984).

D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

Reel, R. L.

D. R. Pape, L. J. Hornbeck, R. L. Reel, “Characteristics of the deformable mirror device for optical processing,” in Advances in Optical Information Processing I, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.388, 65–74 (1983).

Réfrégier, Ph.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

S. Mazé, Ph. Réfrégier, “Noise robustness of optical correlation for amplitude or phase modulation of the input image,” Opt. Lett. 17, 426–428 (1992).
[CrossRef] [PubMed]

Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

Romero, L. A.

Soref, R. A.

J. L. Horner, R. A. Soref, “Phase-dominant spatial light modulators,” Electron. Lett. 24, 626–627 (1988).
[CrossRef]

Suemune, I.

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Wenkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Wenkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 668–704 (1984).

Yamanishi, M.

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

Appl. Opt.

D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

H. J. Caulfield, “Role of the Horner efficiency in the optimization of spatial filters for optical pattern recognition,” Appl. Opt. 21, 4391–4392 (1982).
[CrossRef] [PubMed]

J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
[CrossRef] [PubMed]

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

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Signal-dependent phase distortion in optical correlators,” Appl. Opt. 26, 2484–2490 (1987).
[CrossRef] [PubMed]

D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
[CrossRef] [PubMed]

M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2355 (1969).
[CrossRef] [PubMed]

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

Electron. Lett.

J. L. Horner, R. A. Soref, “Phase-dominant spatial light modulators,” Electron. Lett. 24, 626–627 (1988).
[CrossRef]

IEEE Trans. Inf. Theory

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

Jpn. Appl. Phys.

H. Nagai, Y. Kan, M. Yamanishi, I. Suemune, “Electrore-flectance spectra and field-induced variation in refractive index of a GaAs/AlAs quantum well structure at room temperature,” Jpn. Appl. Phys. 22, 640–642 (1986).
[CrossRef]

Opt. Commun.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Opt. Comput. Process.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Opt. Eng.

D. Psaltis, E. G. Paek, S. S. Wenkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 668–704 (1984).

Opt. Lett.

Other

D. R. Pape, L. J. Hornbeck, R. L. Reel, “Characteristics of the deformable mirror device for optical processing,” in Advances in Optical Information Processing I, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.388, 65–74 (1983).

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

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Figures (10)

Fig. 1
Fig. 1

Representation of the codings studied (typical value of K of 75°).

Fig. 2
Fig. 2

Image of truck used for numerical simulations.

Fig. 3
Fig. 3

Ratio of the SNR [SNR(P)/SNR(A)] for pure phase coding and a noncentered filter: curve (a), σ2 = 0.0; curve (b), σ2 = 0.1; curve (c), σ2 = 0.2; curve (d), σ2 = 0.5; curve (e), σ2 = 1.

Fig. 4
Fig. 4

Amplitude of the correlation peakm | C P B ( 0 ) C min B | 2 for pure phase coding and a centered filter: curve (a), σ2 = 0.0; curve (b), σ2 = 0.1; curve (c), σ2 = 0.2; curve (d), σ2 = 0.5; curve (e), σ2 = 1.

Fig. 5
Fig. 5

Same as Fig. 3 but with a centered filter: curve (a), σ2 = 0.0; curve (b), σ2 = 0.2; curve (c), σ2 = 0.5; curve (d), σ2 = 1.

Fig. 6
Fig. 6

Amplitude of the correlation peak | C P B ( 0 ) | 2 for pure phase coding and a centered filter: curve (a), σ2 = 0.0; curve (b), σ2 = 0.1; curve (c), σ2 = 0.2; curve (d), σ2 = 0.5; curve (e), σ2 = 1.

Fig. 7
Fig. 7

Ratio of the SNR[SNR(S)/SNR(A)] for coupled amplitude–phase coding and a noncentered filter: curve (a), σ2 = 0.0; curve (b), σ2 = 0.2; curve (c), σ2 = 0.5; curve (d), σ2 = 1.

Fig. 8
Fig. 8

Same as Fig. 6 but for coupled amplitude–phase coding and a noncentered filter.

Fig. 9
Fig. 9

Same as Fig. 7 but for a centered filter.

Fig. 10
Fig. 10

Same as Fig. 6 but for coupled amplitude–phase coding.

Tables (2)

Tables Icon

Table 1 Characteristics of the Correlation Function for the Gray-Level Image of a Truck a

Tables Icon

Table 2 Characteristics of the Correlation Function for the Gray-Level Image of a Truck a

Equations (36)

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

Ψ A ( x ) = f ( x ) , Ψ P ( x ) = exp [ iKf ( x ) ] , Ψ S ( x ) = f ( x ) exp [ iKf ( x ) ] .
C ( x ) = y h ( x + y ) * Ψ ( y ) .
SNR = | C B ( 0 ) | 2 | C B ( 0 ) C B ( 0 ) | 2 ,
MSE = | C B ( 0 ) C B ( 0 ) | 2 .
SNR = | C B ( 0 ) C min B | 2 | C B ( 0 ) C B ( 0 ) | 2 ,
C min B = Min x | C B ( x ) |
η H = x | C ( x ) | 2 x | Ψ ( x ) | 2 ,
| C ( 0 ) C min 0 | 2 ,
P x 1 , x 2 ( n 1 , n 2 ) = P ( n 1 ) P ( n 2 ) ( 1 δ x 1 , x 2 ) + P ( n 1 ) δ ( n 1 n 2 ) δ x 1 , x 2
P ( n ) = 1 ( 2 Π ) 1 / 2 σ exp ( n 2 2 σ 2 ) .
δ x 1 , x 2 = { 1 for x 1 = x 2 0 otherwise .
E = exp ( K 2 σ 2 / 2 ) , F = K σ 2 exp ( K 2 σ 2 / 2 ) ,
C A B ( 0 ) = I ( f 2 ) ,
MSE ( A ) = σ 2 I ( f 2 ) ,
C P B ( 0 ) C min B = E N E C min B ,
MSE ( P ) = ( 1 E 2 ) N ,
C S B ( 0 ) = EI ( f 2 ) + iFI ( f ) ,
MSE ( S ) = ( 1 E 2 ) I ( f 4 ) + ( σ 2 F 2 ) I ( f 2 ) .
C A B ( 0 ) = I ( f 2 ) I 2 ( f ) N ,
MSE ( A ) = σ 2 [ I ( f 2 ) I 2 ( f ) N ] ,
C P B ( 0 ) = E [ N | I exp ( iKf ) | 2 N ] ,
MSE ( P ) = ( 1 E 2 ) [ N | I exp ( iKf ) | 2 N ] .
C S B ( 0 ) = E { I ( f 2 ) | I [ f exp ( iKf ) | 2 N } + iF { I ( f ) I * [ f exp ( iKf ) ] N I exp ( iKf ) } ,
MSE ( S ) = ( 1 E 2 ) I ( f 4 ) + ( σ 2 F 2 ) I ( f 2 ) + | I [ f exp ( iKf ) ] | 2 N 2 × [ ( 1 E 2 ) I ( f 2 ) N ( σ 2 F 2 ) 2 ( 1 E 2 ) N { I [ f cos ( Kf ) ] I [ f 3 cos ( Kf ) ] + I [ f sin ( Kf ) ] I [ f 3 sin ( Kf ) ] }
SNR SNR ( A ) ( σ = 0.1 )
SNR SNR ( A ) ( σ = 1.0 )
SNR SNR ( A ) ( σ = 0.1 )
SNR SNR ( A ) ( σ = 1.0 )
Ψ ( x ) = g ( x ) exp [ iKf ( x ) ] , g ( x ) = a + bf ( x ) ,
Ψ ¯ = 1 / N x Ψ ( x ) = I ( Ψ ) / N .
Γ B ( 0 ) = C B ( 0 ) I * ( Ψ ) N x g ( x ) exp [ iKf ( x ) ] exp [ iKn ( x ) ] I * ( Ψ ) N x b exp [ iKf ( x ) ] n ( x ) exp [ iKn ( x ) ] .
Γ B ( 0 ) = C B ( 0 ) E | I ( Ψ ) | 2 N ibF I * ( Ψ ) N I exp ( iKf )
| Γ B ( 0 ) | 2 = | C B ( 0 ) | 2 + | I ( Ψ ) | 2 N 2 [ E 2 | I ( Ψ ) | 2 + ( 1 E 2 ) I ( g 2 ) ] + b 2 | I ( Ψ ) | 2 N 2 [ F 2 | I exp ( iKf ) | 2 ( σ 2 F 2 ) N ] 2 ( E 2 I ( g 2 ) | I ( Ψ ) | 2 N + ( 1 E 2 ) N × { I [ g cos ( Kf ) ] I [ g 3 cos ( Kf ) ] + I [ g sin ( Kf ) ] I [ g 3 sin ( Kf ) ] } ) 2 b N FEI ( g 2 ) { I [ g sin ( Kf ) ] I [ cos ( Kf ) ] I [ g cos ( Kf ) ] I [ sin ( Kf ) ] } 2 b 2 N F 2 I ( g ) { I [ g cos ( Kf ) ] I [ cos ( Kf ) ] + I [ g sin ( Kf ) ] I [ sin ( Kf ) ] } + 2 bFE | I ( Ψ ) | 2 N 2 { I [ g sin ( Kf ) ] I [ cos ( Kf ) ] I [ g cos ( Kf ) ] I [ sin ( Kf ) ] } .
n ( x ) n ( y ) = { σ 2 for x = y 0 for x y .
exp [ iKn ( x ) ] = exp ( K 2 σ 2 / 2 ) , n ( x ) exp [ iKn ( x ) = iK σ 2 exp ( K 2 σ 2 / 2 ) , exp { iK [ n ( x 1 ) n ( x 2 ) ] } = exp ( K 2 σ 2 ) + δ x 1 , x 2 × [ 1 exp ( K 2 σ 2 ) ] , n ( x 1 ) exp { iK [ n ( x 1 ) n ( x 2 ) ] } = iK σ 2 exp ( K 2 σ 2 ) × ( 1 δ x 1 , x 2 ) ,
n ( x 1 ) n ( x 2 ) exp { iK [ n ( x 1 ) n ( x 2 ) ] = n 1 n 2 n 1 n 2 exp [ iK ( n 1 n 2 ) ] P ( n 1 , n 2 ) d n 1 d n 2 , = n ( x ) exp [ iKn ( x ) ] n ( x ) exp [ iKn ( x ) ] ( 1 δ x 1 , x 2 ) + n 2 ( x ) δ x 1 , x 2 .

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