Abstract

The projection-slice synthetic discriminant function (PSDF) filter is introduced and proposed for distortion-invariant pattern-recognition applications. The projection-slice theorem, often used in tomographic applications for medical imaging, is utilized to implement a distortion-invariant filter. Taking M projections from one training image and combining them with (N - 1) M projections taken from another N - 1 training image accomplishes this. With the projection-slice theorem, each set of these M projections can be represented as M one-dimensional slices of the two-dimensional Fourier transform of the particular training image. Therefore, the PSDF filter has the advantage of matching each of the training images with at least M slices of their respective Fourier transforms. This filter is theoretically analyzed, numerically simulated, and experimentally implemented and tested to verify the simulation results. These tests show that the PSDF filter significantly outperforms the matched-filter and the basic synthetic discriminant function technique for the particular images used.

© 1997 Optical Society of America

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References

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  1. A. Vanderlugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
    [CrossRef]
  2. H. J. Caulfield, V. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  3. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  4. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  5. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1979).
    [CrossRef]
  6. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  7. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  8. V. B. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  9. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
    [CrossRef]
  10. D. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  11. H. Stark, P. Indraneel, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
    [CrossRef]
  12. V. R. Riasati, “Projection-slice synthetic discriminant functions for pattern recognition,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Ala., 1996).
  13. M. L. James, G. M. Smith, J. C. Wolford, Applied Numerical Methods for Digital Computation (Harper & Row, New York, 1985).
  14. R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
    [CrossRef]
  15. R. M. Mersereau, “Direct Fourier transform techniques in 3-D image reconstruction,” Comput. Biol. Med. 6, 247–258 (1976).
    [CrossRef] [PubMed]
  16. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  17. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  18. J. L. Horner, ed., Optical Signal Processing (Academic, New York, 1987).
  19. D. Casasent, “Optical feature extraction,” in Optical Signal Processing, J. H. Horner, ed. (Academic, Orlando, Fla., 1987), pp. 175–195.
  20. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Boston, 1990).
  21. A. Vanderlugt, Optical Signal Processing (Wiley, New York, 1992).
  22. F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  23. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  25. F. Yu, Optical Information Processing (Krieger, Malabar, Fla., 1990).

1992 (2)

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

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

1990 (1)

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

1989 (1)

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).

1988 (1)

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

1987 (1)

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

1985 (1)

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

1984 (1)

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

1981 (1)

H. Stark, P. Indraneel, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

1980 (1)

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

1979 (1)

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1979).
[CrossRef]

1976 (1)

R. M. Mersereau, “Direct Fourier transform techniques in 3-D image reconstruction,” Comput. Biol. Med. 6, 247–258 (1976).
[CrossRef] [PubMed]

1974 (1)

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

1969 (1)

H. J. Caulfield, V. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

1964 (1)

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

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Bahri, Z.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Casasent, D.

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

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

D. Casasent, “Optical feature extraction,” in Optical Signal Processing, J. H. Horner, ed. (Academic, Orlando, Fla., 1987), pp. 175–195.

Caulfield, H. J.

H. J. Caulfield, V. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

Dickey, F. M.

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Dudgeon, D.

D. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Boston, 1990).

Gianino, P. D.

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hassebrook, L.

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

Hester, C. F.

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

Horner, J. L.

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

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

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

J. L. Horner, ed., Optical Signal Processing (Academic, New York, 1987).

Indraneel, P.

H. Stark, P. Indraneel, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

James, M. L.

M. L. James, G. M. Smith, J. C. Wolford, Applied Numerical Methods for Digital Computation (Harper & Row, New York, 1985).

Mahalanobis, A.

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

Maloney, V. T.

H. J. Caulfield, V. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

Mason, J. J.

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Mersereau, R. M.

R. M. Mersereau, “Direct Fourier transform techniques in 3-D image reconstruction,” Comput. Biol. Med. 6, 247–258 (1976).
[CrossRef] [PubMed]

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

D. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Oppenheim, A. V.

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Riasati, V. R.

V. R. Riasati, “Projection-slice synthetic discriminant functions for pattern recognition,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Ala., 1996).

Smith, G. M.

M. L. James, G. M. Smith, J. C. Wolford, Applied Numerical Methods for Digital Computation (Harper & Row, New York, 1985).

Stalker, K. T.

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Stark, H.

H. Stark, P. Indraneel, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

Vanderlugt, A.

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

A. Vanderlugt, Optical Signal Processing (Wiley, New York, 1992).

Vijaya Kumar, B. V. K.

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

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1979).
[CrossRef]

Vijaya Kumar, V. B. K.

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

Wolford, J. C.

M. L. James, G. M. Smith, J. C. Wolford, Applied Numerical Methods for Digital Computation (Harper & Row, New York, 1985).

Yu, F.

F. Yu, Optical Information Processing (Krieger, Malabar, Fla., 1990).

Appl. Opt. (10)

H. J. Caulfield, V. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

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

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

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

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

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

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

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Comput. Biol. Med. (1)

R. M. Mersereau, “Direct Fourier transform techniques in 3-D image reconstruction,” Comput. Biol. Med. 6, 247–258 (1976).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

H. Stark, P. Indraneel, “An investigation of computerized tomography by direct Fourier inversion and optimum interpolation,” IEEE Trans. Biomed. Eng. BME-28, 496–505 (1981).
[CrossRef]

IEEE Trans. Inf. Theory (1)

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

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

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1979).
[CrossRef]

Proc. IEEE (1)

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Other (9)

V. R. Riasati, “Projection-slice synthetic discriminant functions for pattern recognition,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Ala., 1996).

M. L. James, G. M. Smith, J. C. Wolford, Applied Numerical Methods for Digital Computation (Harper & Row, New York, 1985).

J. L. Horner, ed., Optical Signal Processing (Academic, New York, 1987).

D. Casasent, “Optical feature extraction,” in Optical Signal Processing, J. H. Horner, ed. (Academic, Orlando, Fla., 1987), pp. 175–195.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Boston, 1990).

A. Vanderlugt, Optical Signal Processing (Wiley, New York, 1992).

D. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

F. Yu, Optical Information Processing (Krieger, Malabar, Fla., 1990).

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Figures (18)

Fig. 1
Fig. 1

Training images for the target for the rotation distortion of the image: (a) 0°, (b) 10°, (c) 20°, and (d) 30°, in-plane rotation.

Fig. 2
Fig. 2

Training images for the target for scaling distortion of the image. The scaling factors used are (a) 0.7, (b) 0.8, (c) 0.9, and (d) 1.0.

Fig. 3
Fig. 3

Reconstruction of the tank image shown in Fig. 2(a) by use of 32 slices for different orders of |ω|: (a) slices taken with the order γ = 0.005, (b) reconstruction of (a), (c) slices taken with the order γ = 0.5, (d) reconstruction of (c), (e) slices taken with the order γ = 1.0, (f) reconstruction of (e), (g) slices taken with the order γ = 1.5, (h) reconstruction of (g), (i) slices taken with the order γ = 2.0, and (j) reconstruction of (i).

Fig. 4
Fig. 4

Ratio of (a) the correlation output peak of the filter with the tank image relative to the correlation output peak of the filter with the reference image, and (b) the PCE found by use of the tank image relative to the PCE found by use of the reference image.

Fig. 5
Fig. 5

PSDF (a) impulse response and (b) transfer function for rotation distortion and (c) impulse response and (d) transfer function for scaling distortion.

Fig. 6
Fig. 6

SDF (a) impulse response of the filter for rotation invariance, (b) transfer function of the filter for rotation invariance, (c) the impulse response of the filter for scale invariance, and (d) transfer function of the filter for scale invariance.

Fig. 7
Fig. 7

Different images used for testing the filters: (a) the target, a generic tank, (b) a B2 Stealth bomber, (c) a propeller plane, and (d) a 929 Porsche.

Fig. 8
Fig. 8

(a) 20° discrimination test scene and the output generated by use of the (b) PSDF, (c) SDF, and (d) MF.

Fig. 9
Fig. 9

(a) 0.8 scale discrimination test scene and the output generated by use of the (b) PSDF, (c) SDF, and (d) MF.

Fig. 10
Fig. 10

Output PC for the target relative to the output PC for the other images as a function of rotation for (a) the PSDF filter, (b) the SDF filter, and (c) the MF.

Fig. 11
Fig. 11

Output PC for the target relative to the output PC for the other images as a function of scale for (a) the PSDF filter, (b) the SDF filter, and (c) the MF.

Fig. 12
Fig. 12

PCE values for each of the filters as a function of rotation.

Fig. 13
Fig. 13

PCE values for each of the filters as a function of scaling.

Fig. 14
Fig. 14

(a) Schematic diagram for the correlation system and (b) the laboratory implementation of the correlation system used in the experimental tests. The input plane moves with the output correlation plane to produce the necessary scale-search capability.

Fig. 15
Fig. 15

Input scenes (a) with the SLM and (b) without the SLM in the system.

Fig. 16
Fig. 16

Output correlation results for the experimental tests performed with the 20° rotated input with the (a) PSDF filter, (b) SDF filter, and (c) MF.

Fig. 17
Fig. 17

Experimental PC results for the (a) PSDF, (b) SDF, and (c) MF.

Fig. 18
Fig. 18

Experimental PCE results for the PSDF, SDF, and MF.

Tables (2)

Tables Icon

Table 1 Fisher Discriminant Measure Used to Evaluate the Discrimination Performance of Each Filter under Consideration

Tables Icon

Table 2 Ratio of the Minimum Target Correlation Peak to the Maximum False-Target Correlation Peak

Equations (24)

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

θmn=mπM+nπMN,n=0, 1, , N-1, m=0, 1, , M-1,
Hu, v=n=0N-1m=0M-1anSnω cos θ, ω sin θδθ-θmnωγ,
ω=u2+v2, θ=tan-1vu,
u=ω cosθ, v=ω sinθ,
hx, y  six, y|τx=τy=0=h*x, y six, ydxdy=14π2H* u, vSiu, vdudv=ci.
hx, y  six, y|τx=0,τy=0=14π2-0πn=0N-1m=0M-1 an*×Sn*ω cos θ, ω sin θ×δθ-θmn×Siω cos θ, ω sin θ×ωγ+1dθdω.
hx, y  six, y|τx=0,τy=0=14π2-n=0N-1m=0M-1 an*×Sn*ω cos θmn, ω sin θmn×Siω cos θmn,ω sin θmn×ωγ+1dω.
hx, y  six, y|τx=0, τy=0=n=1Nan*Rin=ci,
Rin=14π2-m=0M-1 Sn*ω cos θmn, ω sin θmn×Siω cosθmn, ω sin θmnωγ+1dω.
Ra=c,
hx, y=14π2--Hu, vexpixu+jyvdudv.
hx, y=14π2-0πHω cos θ, ω sin θ×expjxω cos θ+jyω sin θ×ωγ+1dθdω=14π2-0πn=0N-1m=0M-1 anSnω cos θ, ω sin θ×ξθ-θmnexpjxω cos θ+jyω sin θ×ωγ+1dθdω.
hx, y=14π2-n=0N-1m=0M-1anSnω cos θmn, ω sin θmn×expjxω cos θmn+jyω sin θmnωdω=14π2-n=0N-1m=0M-1anSθmnω×expjxω cos θmn+jyω sin θmnωdω,
Sθmnω=Snω cos θmn, ω sin θmn,
hx, y=14π2n=0N-1m=0M-1- anSθmnω×expjxω cos θmn+jyω sin θmnωdω.
hx, y=12πn=0N-1m=0M-1angnx cos θmn+y sin θmn=12πn=0N-1m=0M-1angθmnt,
t=x cos θmn+y sin θmn,
gθmnt=12π- Sθmnωexpjωtωdω=ddt-pθmnτt- τdτ=pθmnt*kt.
hx, y=12πn=0N-1m=0M-1 anĝθmnt,
ĝθmnt=12π-Sθmnωexpjωtωγ+1dω=pθmnt*kˆt.
rτx, τy=h x, y  six, y,
rτx, τy=F-1n=0N-1m=0M-1 anSnω cos θmn, ω sin θmn×Siω cos θmn, ω sin θmnωγ+1.
r0,0=-n=0N-1m=0M-1anSnω cos θmn, ω sin θmn×Siω cos θmn, ω sin θmnωγ+1dω=-m=0M-1aiSi2ω cos θmn, ω sin θmn+n=0 niN-1m=0M-1anSnω cos θmn, ω sin θmn×Siω cos θmn, ω sin θmnωγ+1dω.
r0, 0=-m=0M-1aiSiω cos θmn, ω sin θmn×Si+1/2ω cos θmn, ω sin θmnωγ+1+m=0M-1 ai+1Si+1ω cos θmn, ω sin θmn×Si+1/2ω cos θmn, ω sin θmn+n=0nini+1N-1m=0M-1anSnω cos θmn, ω sin θmn×Si+1/2ω cos θmn, ωsin θmnωγ+1dω.

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