Abstract

An important tool in optical pattern recognition, the joint fractional transform correlator (JFTC), was introduced recently. We analyze the peak properties of fractional correlation (FC) by symbolic derivation and computer simulation. We show that the FC has a maximum correlation peak when the second fractional Fourier transform is reduced to the conventional Fourier transform. We introduce nonlinear operations in a joint fractional transform power spectrum and propose a differential JFTC and a binary differential JFTC. Numerical simulations show that such nonlinear JFTCs exhibit remarkable improvement in correlation peak intensity, discrimination capability, and signal-to-noise ratio. An optoelectronic setup that can implement such nonlinear JFTCs is also proposed.

© 2001 Optical Society of America

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
    [CrossRef]
  4. F. T. S. Yu, S. Jutamulia, T. W. Lin, D. A. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
    [CrossRef] [PubMed]
  5. M. S. Alam, M. A. Karim, “Improved correlation discrimination in a multiobject bipolar joint transform correlator,” Opt. Laser Technol. 24, 45–50 (1992).
    [CrossRef]
  6. B. Javidi, “Synthetic discriminant function-based binary nonlinear optical correlator,” Appl. Opt. 28, 2490–2493 (1989).
    [CrossRef] [PubMed]
  7. M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
    [CrossRef]
  8. Q. Tang, B. Javidi, “Multiple-object detection with a chirp-encoded joint transform correlator,” Appl. Opt. 32, 4344–4350 (1993).
    [CrossRef]
  9. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  12. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  13. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  14. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in the fractional Fourier domain and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  15. M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.
  16. M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
    [CrossRef]
  17. A. W. Lohmann, D. Mendlovic, “Fractional joint transform correlator,” Appl. Opt. 36, 7402–7407 (1997).
    [CrossRef]
  18. C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
    [CrossRef]
  19. M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
    [CrossRef]
  20. M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
    [CrossRef]
  21. S. Zhong, J. Jiang, S. Liu, C. Li, “Binary joint transform correlator based on differential processing of the joint transform power spectrum,” Appl. Opt. 36, 1776–1780 (1997).
    [CrossRef] [PubMed]
  22. G. W. Lu, Z. Zhang, F. T. S. Yu, “Phase-encoded input joint transform correlation with improved pattern discriminability,” Opt. Lett. 20, 1307–1309 (1995).
    [CrossRef] [PubMed]
  23. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–221 (1996).
    [CrossRef] [PubMed]

1998 (1)

1997 (2)

1996 (1)

1995 (4)

1994 (2)

1993 (3)

1992 (1)

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

1990 (2)

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

1989 (1)

1987 (1)

1984 (1)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

1966 (1)

1964 (1)

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

Alam, M. S.

M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
[CrossRef]

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

Arikan, O.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

Barshan, B.

Bitran, Y.

Candan, Ç.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

Dorsch, R. G.

Erden, M. F.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

Goodman, J. W.

Gregory, D. A.

Güleryüz, Ö.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

Javidi, B.

Jiang, J.

Jutamulia, S.

Karim, M. A.

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

Kuo, C. J.

Kutay, M. A.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

Li, C.

Liang, M.

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

Lin, T. W.

Liu, L.

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

Liu, S.

Lohmann, A.

Lohmann, A. W.

Lu, G. W.

Lu, X. J.

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

Luo, Y.

Mendlovic, D.

Onural, L.

Ozaktas, H. M.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in the fractional Fourier domain and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

Özaktas, H.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

Painchaud, D.

Tang, Q.

VanderLugt, A.

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

Wang, Z.

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

Weaver, C. S.

Wu, S.

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

Yu, F. T. S.

Zhang, Z.

Zhong, S.

Appl. Opt. (10)

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

F. T. S. Yu, S. Jutamulia, T. W. Lin, D. A. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
[CrossRef] [PubMed]

B. Javidi, “Synthetic discriminant function-based binary nonlinear optical correlator,” Appl. Opt. 28, 2490–2493 (1989).
[CrossRef] [PubMed]

Q. Tang, B. Javidi, “Multiple-object detection with a chirp-encoded joint transform correlator,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, “Fractional joint transform correlator,” Appl. Opt. 36, 7402–7407 (1997).
[CrossRef]

C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

S. Zhong, J. Jiang, S. Liu, C. Li, “Binary joint transform correlator based on differential processing of the joint transform power spectrum,” Appl. Opt. 36, 1776–1780 (1997).
[CrossRef] [PubMed]

B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–221 (1996).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Comparison of discrimination capabilities of four types of correlation,” Opt. Commun. 75, 225–230 (1990).
[CrossRef]

M. Liang, L. Liu, S. Wu, Z. Wang, “Discrimination capabilities of a phase-only matched filter made from outline features,” Opt. Commun. 75, 231–234 (1990).
[CrossRef]

Opt. Eng. (1)

M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
[CrossRef]

Opt. Laser Technol. (1)

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

Opt. Lett. (1)

Other (2)

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arıkan, Ö. Güleryüz, Ç. Candan, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configuration,” in Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arıkan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the IEEE Symposium on Time-Frequency and Time Scale Analysis (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 481–484.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Block diagram of the fractional correlation: C.C., complex conjugation.

Fig. 2
Fig. 2

Optoelectronic hybrid setup that implements the NJFTC. Nonlinear operations can be performed electronically with the aid of a computer. P, plane; L, lens; other notation defined in text.

Fig. 3
Fig. 3

Properties of a FC peak. The functions u(x 1) and v(x 2) used here are windowed functions. The FC has a maximum value at α2 = 1.

Fig. 4
Fig. 4

Input (left) and reference (right) images.

Fig. 5
Fig. 5

Simulation results and comparisons. (a) Correlation output of the CJTC. (b) Correlation output of the JFTC with α1 = 0.552 and α2 = 1. (c) Correlation output of the JFTC with α1 = 0.552 and α2 = 0.5. (d) Correlation output of the DJFTC with α1 = 0.552 and α2 = 1. (e) Correlation output of the BDJFTC with α1 = 0.552 and α2 = 1. (f) Correlation output of the BDJFTC with α1 = 0.552 and α2 = 0.8. The choice of fractional order α1 is calculated from Eq. (18).

Fig. 6
Fig. 6

Images used in analysis of the performance of the correlators: target image letter F (bottom left), nontarget image letter E, and reference image letter F (bottom right).

Tables (1)

Tables Icon

Table 1 Numerical Simulations of Performances of the CJTC, JFTC, DJFTC, and BDJFTC with α1 = 0.552 and α2 = 1

Equations (30)

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αfx=c -+ fxKαx; wdx,Kαx; w=expiπx2+w2cot ϕα-i2π xwsin ϕα,
Cα1,α2x=α2α1ux1α1*vx2.
Cα1,α2x=-+-+-+ ux1v*x2×expiπΨ1x, x1, x2, y×exp-i2πΨ2x, x1, x2, ydx1dx2dy,
Ψ1x, x1, x2, y=x2+y2tanϕ2+x12+y2tanϕ1-x22+y2tanϕ1,Ψ2x, x1, x2, y=yx1-x2sinϕ1+xsinϕ2,ϕ1=π2 α1,  ϕ2=π2 α2.
|C1,10|2=-+-+-+ ux1v*x2×exp-i2πyx1-x2dx1dx2dy2.
-+exp-i2πyx1-x2dy=δx1-x2,
|C1,10|2=-+ ux1v*x1dx12.
|Cα1,10|2=-+-+-+ ux1v*x2×expiπx12tanϕ1-x22tanϕ1×exp-i2πyx1sinϕ1-x2sinϕ1dx1dx2dy2=-+-+ ux1v*x2×expiπx12tanϕ1-x22tanϕ1×δx1sinϕ1-x2sinϕ1dx1dx22=-+ ux1v*x1dx12.
|C1,α20|2=-+-+-+ ux1v*x2×expiπy2tanϕ2exp-i2πyx1-x2dx1dx2dy2=-+-+-+ ux1v*x2expiπy2×tanϕ2exp-i2πytanϕ2x1-x2dx1dx2dy2=-+-+ ux1v*x2tanϕ2×exp-iπ tanϕ2x1-x22dx1dx22.
|Cα1,α20|2=-+-+-+ ux1v*x2×expiπx12tanϕ1-x22tanϕ1+y2tanϕ2exp-i2πyx1sinϕ1-x2sinϕ1dx1dx2dy2=-+-+ ux1v*x2tanϕ2×expiπ cotϕ1x12-x22×exp-iπ tanϕ2csc2ϕ1x1-x22dx1dx22.
fx, y=tx+a, y+rx-a, y.
Fwx, wy=Twx, wyexpi2πap+Rwx, wyexp-i2πap.
wx, wy=|Fwx, wy|2=|Twx, wyexpi2πap+Rwx, wyexp-i2πap|2=|Twx, wy|2+|Rwx, wy|2+Twx, wyR*wx, wyexpi4πap+T*wx, wyRwx, wyexp-i4πap.
Ex, y=tx, ytx, y+rx, yrx, y+tx+2a, yrx+2a, y+rx-2a, ytx-2a, y,
fx, y=tx+a, y+rx-a, yexpi2πpx
α1tx+a, y=expiπα2wx+a cosπα12sinπα12×α1tx, ywx+a cosπα12, wy,
α1rx-a, yexpi2πpx=expiπp2wx-p sinπα12cosπα12-a2wx-a cosπα12sinπα12×α1rx, ywx-a cosπα12-p sinπα12, wy.
-a cosπα12-p sinπα12=a cosπα12,
wx, wy=expiπa2wx+a cosπα12sinπα12α1tx, ywx+a cosπα12, wy+expiπa2-p22sin πα1+wxp cosπα12-a sinπα12×α1rx, ywx+a cosπα12, wy2=α1tx, ywx+a cosπα12, wyα1*tx, ywx+a cosπα12, wy+α1rx, ywx+a cosπα12, wyα1*rx, ywx+a cosπα12, wy+expiπp2 sinπα12cosπα12+2wx2a sinπα12-p cosπα12×α1tx, ywx+a cosπα12, wyα1*rx, ywx+a cosπα12, wy+expiπ-p2 sinπα12cosπα12+2wx-2a sinπα12+p cosπα12×α1*tx, ywx+a cosπα12, wyα1rx, ywx+a cosπα12, wy.
expiπa cosπα122x+a cosπα12cosπα22sinπα22×Cα1α2x+a cosπα12cosπα22, y.
expiπ2a sinπα12-p cosπα12cosπα22×x+2a sinπα12-p cosπα12sinπα12×expiπp2 sinπα12cosπα12expiπa cosπα12×x+a cosπα12cosπα22sinπα22×Cα1α2x+a cosπα12cosπα22-2a sinπα12-p cosπα12sinπα22, y.
exp-iπ2a sinπα12-p cosπα12cosπα22×x-2a sinπα12-p cosπα12sinπα12×exp-iπp2 sinπα12cosπα12×expiπa cosπα12x+a cosπα12cosπα22×sinπα22Cα1α2x+a cosπα12cosπα22+2a sinπα12-p cosπα12sinπα22, y.
-a cosπα12cosπα22+2a sinπα12-p cosπα12sinπα22, 0,-a cosπα12cosπα22-2a sinπα12-p cosπα12sinπα22, 0,
α2wx, wywx=i2π cscπα22 xα2wx, wy=i2π cscπα22 xEx, y.
i2π cscπα22 xEx, y2=4π2cscπα222x2|Ex, y|2.
gwx, wy=sgnwx, wywx,
sgnx=+1x0-1x<0.
gm, n=sgnm+1, n-m, n=+1m+1, nm, n-1m+1, n<m, n,
SCR=APICPI,
SNR=EAPI(EAPI-EAPI2)1/2,

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