Abstract

With recent advances in state-of-the-art spatial light modulators, the optical joint transform correlator (JTC) and the binary joint transform correlator (BJTC) are becoming practical signal-processing tools. The performance of these devices is limited by the difficulty of separating the cross correlation between the reference and the targets in the scene from signals resulting from cross correlations between objects in the target scene. One technique that reduces this problem is to use a sliding window in the Fourier plane as a convolution mask filter to set an adaptive binarization threshold. This suppresses the autocorrelation response and reduces the dynamic range of the Fourier-plane signal. This results in correlation performance improvement by a factor of 2 to 4. A mathematical model is developed to describe the windowing process for both the JTC and BJTC for the case in which the scene contains multiple targets and background clutter. The derivation of the windowing process is general and includes any spatial high-pass or bandpass filtering in the Fourier plane. The results are supported with experimental data.

© 1995 Optical Society of America

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  1. 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); “Space-variant joint transform image correlation using a binary spatial light modulator at the Fourier plane,” J. Opt. Soc. Am. A 4, 86 (1987).
    [CrossRef] [PubMed]
  2. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  3. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  4. S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
    [CrossRef]
  5. W. B. Hahn, D. L. Flannery, “Basic design elements of binary joint-transform correlation and selected optimization techniques,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1347, 344–356 (1990).
  6. B. Javidi, J. Wang, “Binary transform correlation with median and subset median thresholding,” Appl. Opt. 30, 967–976 (1991).
    [CrossRef] [PubMed]
  7. B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple level threshold crossing,” Appl. Opt. 30, 4234–4244 (1991).
    [CrossRef] [PubMed]
  8. W. B. Hahn, D. L. Flannery, “Basic design elements of binary joint-transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  9. B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).
  10. T. J. Grycewicz, “Binarization of the joint transform correlator Fourier plane based on local window processing,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2237, 260–266 (1994).
  11. J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1805 (1994).
    [CrossRef]
  12. B. Javidi, J. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [CrossRef]
  13. K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
    [CrossRef]
  14. K. L. Schehrer, M. G. Roe, R. A. Dobson, “Rapid tracking of a human retina using a nonlinear joint transform correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 381–390 (1993).
  15. R. Juday, J. Knopp, C. Soutar, “Partial rotation invariance in retinal pattern recognition,” in Visual Information Processing III, F. O. Huck, R. D. Juday, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2239, 157–166 (1994).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), pp. 233–234.
  17. B. Javidi, J. L. Horner, “Single SLM joint transform correlator,” Appl. Opt. 28, 1027–1032 (1989).
    [CrossRef] [PubMed]
  18. J. C. Russ, The Image Processing Handbook (CRC, Boca Raton, Fla., 1992), p. 143.
  19. K. A. Bauchert, “Real time hardware implementation of an optical correlation image preprocessing algorithm using an off-the-shelf image processing board,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 44–47 (1993).

1994 (2)

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1805 (1994).
[CrossRef]

B. Javidi, J. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

1992 (1)

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

1991 (3)

1990 (1)

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

1989 (2)

1988 (1)

1966 (1)

Bauchert, K. A.

K. A. Bauchert, “Real time hardware implementation of an optical correlation image preprocessing algorithm using an off-the-shelf image processing board,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 44–47 (1993).

Cline, J. D.

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Dobson, R. A.

K. L. Schehrer, M. G. Roe, R. A. Dobson, “Rapid tracking of a human retina using a nonlinear joint transform correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 381–390 (1993).

Fazlollahi, A.

B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).

Fielding, K. H.

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Flannery, D. L.

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

W. B. Hahn, D. L. Flannery, “Basic design elements of binary joint-transform correlation and selected optimization techniques,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1347, 344–356 (1990).

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), pp. 233–234.

Grycewicz, T. J.

T. J. Grycewicz, “Binarization of the joint transform correlator Fourier plane based on local window processing,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2237, 260–266 (1994).

Hahn, W. B.

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

W. B. Hahn, D. L. Flannery, “Basic design elements of binary joint-transform correlation and selected optimization techniques,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1347, 344–356 (1990).

Horner, J.

B. Javidi, J. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).

Horner, J. L.

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

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

Javidi, B.

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1805 (1994).
[CrossRef]

B. Javidi, J. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

B. Javidi, J. Wang, “Binary transform correlation with median and subset median thresholding,” Appl. Opt. 30, 967–976 (1991).
[CrossRef] [PubMed]

B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple level threshold crossing,” Appl. Opt. 30, 4234–4244 (1991).
[CrossRef] [PubMed]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

B. Javidi, J. L. Horner, “Single SLM joint transform correlator,” Appl. Opt. 28, 1027–1032 (1989).
[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); “Space-variant joint transform image correlation using a binary spatial light modulator at the Fourier plane,” J. Opt. Soc. Am. A 4, 86 (1987).
[CrossRef] [PubMed]

B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).

Juday, R.

R. Juday, J. Knopp, C. Soutar, “Partial rotation invariance in retinal pattern recognition,” in Visual Information Processing III, F. O. Huck, R. D. Juday, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2239, 157–166 (1994).

Kabrisky, M.

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Knopp, J.

R. Juday, J. Knopp, C. Soutar, “Partial rotation invariance in retinal pattern recognition,” in Visual Information Processing III, F. O. Huck, R. D. Juday, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2239, 157–166 (1994).

Kuo, C. J.

Li, J.

B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).

Makekau, C. K.

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Mills, J. P.

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Roe, M. G.

K. L. Schehrer, M. G. Roe, R. A. Dobson, “Rapid tracking of a human retina using a nonlinear joint transform correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 381–390 (1993).

Rogers, S. K.

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Russ, J. C.

J. C. Russ, The Image Processing Handbook (CRC, Boca Raton, Fla., 1992), p. 143.

Schehrer, K. L.

K. L. Schehrer, M. G. Roe, R. A. Dobson, “Rapid tracking of a human retina using a nonlinear joint transform correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 381–390 (1993).

Soutar, C.

R. Juday, J. Knopp, C. Soutar, “Partial rotation invariance in retinal pattern recognition,” in Visual Information Processing III, F. O. Huck, R. D. Juday, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2239, 157–166 (1994).

Tang, Q.

Wang, J.

Weaver, C. S.

Appl. Opt. (6)

Opt. Eng. (5)

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1805 (1994).
[CrossRef]

B. Javidi, J. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

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

S. K. Rogers, J. D. Cline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlation,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Other (8)

W. B. Hahn, D. L. Flannery, “Basic design elements of binary joint-transform correlation and selected optimization techniques,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1347, 344–356 (1990).

B. Javidi, J. Horner, A. Fazlollahi, J. Li, “Illumination-invariant pattern recognition with a binary nonlinear joint transform correlator using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 100–106 (1993).

T. J. Grycewicz, “Binarization of the joint transform correlator Fourier plane based on local window processing,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2237, 260–266 (1994).

K. L. Schehrer, M. G. Roe, R. A. Dobson, “Rapid tracking of a human retina using a nonlinear joint transform correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 381–390 (1993).

R. Juday, J. Knopp, C. Soutar, “Partial rotation invariance in retinal pattern recognition,” in Visual Information Processing III, F. O. Huck, R. D. Juday, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2239, 157–166 (1994).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), pp. 233–234.

J. C. Russ, The Image Processing Handbook (CRC, Boca Raton, Fla., 1992), p. 143.

K. A. Bauchert, “Real time hardware implementation of an optical correlation image preprocessing algorithm using an off-the-shelf image processing board,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 44–47 (1993).

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Figures (6)

Fig. 1
Fig. 1

Block diagram of the BJTC.

Fig. 2
Fig. 2

Window masks and their associated Fourier transforms. The weights at all the pixels except for the center pixel is −1. The center value is the number of mask pixels minus 1: this yields a mean of 0. The brightness of the output plane represents the magnitude of the Fourier transform of the mask.

Fig. 3
Fig. 3

Block diagram of the single SLM BJTC setup used for the experiments. MOSLM, magneto-optic SLM.

Fig. 4
Fig. 4

Input image: (a) original scene, (b) after binarization by the diff-3 algorithm.

Fig. 5
Fig. 5

Transform-plane signals: (a) detected joint power spectrum, (b) spectrum binarized by the use of the global image median as a threshold, (c) spectrum binarized by the use of a 5 × 1 Fourier-plane window.

Fig. 6
Fig. 6

Correlation-plane outputs. The contrast and the brightness have been adjusted to show the detail of the noise structure, but are the same for all four images. (a) Output with the global median used as a threshold, (b) output with a 5 × 1 Fourier-plane window, (c) output with a 5 × 5 Fourier-plane window, (d) output with a 1 × 5 Fourier-plane window.

Tables (2)

Tables Icon

Table 1 Correlation-Peak Height for Different Binarization Algorithms

Tables Icon

Table 2 Expected Ratio of the Correlation-Peak Heights Based on the Fourier Transform of the Window Function

Equations (24)

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in ( x , y ) = r ( x x 0 , y y 0 ) + s ( x x 1 , y y 1 ) .
F [ r ( x , y ) ] = R ( α , β ) exp [ j ϕ 0 ( α , β ) ] , F [ s ( x , y ) ] = S ( α , β ) exp [ j ϕ 1 ( α , β ) ] ,
T ( α , β ) = | R ( α , β ) exp [ j ϕ 0 ( α , β ) ] + S ( α , β ) ] × exp [ j ϕ 1 ( α , β ) ] | 2 = R 2 ( α , β ) + S 2 ( α , β ) + 2 R ( α , β ) S ( α , β ) × cos [ θ 01 ( α , β ) ] ,
o ( x , y ) r ( x , y ) r ( x , y ) + s ( x , y ) s ( x , y ) + r ( x x 0 , y y 0 ) s ( x x 1 , y y 1 ) + s ( x x 1 , y y 1 ) r ( x x 0 , y y 0 ) .
T w ( α , β ) = T ( α , β ) w ( α , β ) .
o w ( x , y ) = o ( x , y ) W ( x , y ) .
g ( E ) = sgn ( E ) = sgn [ T ( α , β ) ] , G ( ω ) = g ( E ) exp ( j ω E ) d E = 2 j ω .
T W B ( α , β ) 1 2 π G ( ω ) exp { j ω [ T W ( α , β ) ] } d ω = 1 j π 1 ω exp ( j ω { [ R 2 ( α , β ) + S 2 ( α , β ) ] w ( α , β ) } ) + exp ( j 2 ω { R ( α , β ) S ( α , β ) cos [ θ 01 ( α , β ) ] } w ( α , β ) ) d ω .
exp { j ω [ S 2 ( α , β ) + R 2 ( α , β ) ] w ( α , β ) } = exp [ j ω ( F 1 { [ s ( x , y ) s ( x , y ) + r ( x , y ) r ( x , y ) ] × W ( x , y ) } ) ] exp { j ω [ S 2 ( α , β ) + R 2 ( α , β ) ] W ( 0 , 0 ) } = exp ( 0 ) = 1 .
exp ( j 2 ω { R ( α , β ) S ( α , β ) cos [ θ 01 ( α , β ) ] } w ( α , β ) ) = exp ( j ω F 1 { [ r ( x x 0 , y y 0 ) s ( x x 1 , y y 1 ) + s ( x x 1 , y y 1 ) r ( x x 0 , y y 0 ) ] W ( x , y ) } ) = exp ( j 2 ω { R ( α , β ) S ( α , β ) cos [ θ 01 ( α , β ) ] } × W ( x 0 x 1 , y 0 y 1 ) ) .
T W B ( α , β ) 1 j π 1 ω × exp ( j 2 ω { R ( α , β ) S ( α , β ) cos [ θ 01 ( α , β ) ] × W ( x 0 x 1 , y 0 y 1 ) } ) d ω .
T W B ( α , β ) ν = 0 ν j π j ν 1 ω J ν × [ 2 ω R ( α , β ) S ( α , β ) W ( x 0 x 1 , y 0 y 1 ) ] × cos [ ν θ 01 ( α , β ) ] d ω,
ν = { 1 , ν = 0 2 , ν > 0 .
T W B ( α , β ) 4 π cos [ θ 01 ( α , β ) ] .
o W B ( x , y ) 2 π δ ( x + x 0 x 1 , y + y 0 y 1 ) + 2 π δ ( x x 0 + x 1 , y y 0 + y 1 ) .
F [ s 0 ( x , y ) ] = S 0 ( α , β ) exp { j [ ϕ 0 ( α , β ) ] } , F [ s n ( x x n , y y n ) ] = S n ( α , β ) exp { j [ ϕ n ( α , β ) α x n β y n ] } .
T ( α , β ) = | n = 0 N S n ( α , β ) exp { j [ ϕ n ( α , β ) α x n β y n ] } | 2 = p = 0 N q = 0 N S p ( α , β ) S q ( α , β ) exp [ j θ p q ( α , β ) ] = n = 0 N S n 2 ( α , β ) + 2 p = 1 N q = 0 p 1 S p ( α , β ) × S q ( α , β ) cos [ θ p q ( α , β ) ] ,
θ p q = [ ϕ p ( α , β ) α x p β y p ϕ q ( α , β ) + α x q + β y q ] .
o ( x , y ) p = 0 N q = 0 N s p ( x x p , y y p ) s q ( x x q , y y q ) .
T W B ( α , β ) sgn ( { n = 0 N S n 2 ( α , β ) + 2 p = 0 N q = 1 p 1 S p ( α , β ) × S q ( α , β ) cos [ θ p q ( α , β ) ] } w ( α , β ) ) sgn { p = 0 N q = 1 p 1 S p ( α , β ) S q ( α , β ) × W ( x p x q , y p y q ) cos [ θ p q ( α , β ) ] } .
T W B ( α , β ) sgn { A sin [ Ω ( α , β ) ] } = sgn { sin [ Ω ( α , β ) ] } ,
A max j = 1 N k = 0 j 1 S j ( α , β ) S k ( α , β ) × W ( x j x k , y j y k ) over all ( α , β ) , sin [ Ω ( α , β ) ] = 1 A p = 1 N q = 0 p 1 S p ( α , β ) S q ( α , β ) × W ( x p x q , y p y q ) cos [ θ p q ( α , β ) ] .
T W B ( α , β ) 1 j π 1 ω exp { j ω sin [ Ω ( α , β ) ] } d ω = ν = 0 ν j π j ν 1 ω J ν ( ω ) sin [ ν Ω ( α , β ) ] d ω 4 π sin [ Ω ( α , β ) ] .
o W B ( x , y ) p = 0 N q = 1 p 1 [ s p ( x , y ) s q ( x , y ) ] [ δ ( x + x p x q ) + δ ( x x p + x q ) ] × W ( x p x q , y p y q ) .

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