Abstract

The correlation performance of binary joint transform correlators with unknown input-image light illumination is investigated for different thresholding methods used in the Fourier plane. It is shown that a binary joint transform correlator that uses a spatial frequency dependent threshold function for binarization of the joint power spectrum is invariant to uniform input-image illumination. Computer simulations and optical experimental results are provided.

© 1995 Optical Society of America

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References

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  1. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  2. 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).
    [CrossRef] [PubMed]
  3. B. Javidi, J. L. Horner, “Single spatial light modulator joint transform correlator,” Appl. Opt. 28, 1027–1032 (1989).
    [CrossRef] [PubMed]
  4. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  5. 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]
  6. J. Wang, “Multi-object detection using binary optical joint transform correlator,” Ph. D. dissertation (University of Connecticut, Storrs, Conn., 1993).
  7. K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  8. D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  9. E. Marom, “Error diffusion binarization for joint transform correlators,”Appl. Opt. 32, 707–714 (1993).
    [CrossRef] [PubMed]
  10. B. Javidi, J. Wang, “Binary nonlinear joint transform correlation with median and subset median thresholding,” Appl. Opt. 30, 967–976 (1991).
    [CrossRef] [PubMed]
  11. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  12. A. Vanderlugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 9, 215–222 (1970).
    [CrossRef]
  13. B. Javidi, J. Wang, “Limitation of the classical definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
    [CrossRef] [PubMed]
  14. B. Javidi, J. Wang, A. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
    [CrossRef] [PubMed]
  15. J. L. Horner, “Metrics for assessing pattern recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  16. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

1994 (1)

1993 (1)

1992 (3)

1991 (2)

1990 (2)

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

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

1989 (3)

1988 (1)

1970 (1)

1966 (1)

Fazlollahi, A.

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Flannery, D. L.

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

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Goodman, J. W.

Hahn, W. B.

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

Hassebrook, L.

Horner, J. L.

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

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

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

Javidi, B.

Kuo, C. J.

Marom, E.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

Rotz, F. B.

Tang, Q.

Vanderlugt, A.

Vijaya Kumar, B. V. K.

Wang, J.

Weaver, C. S.

Appl. Opt. (12)

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

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

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

E. Marom, “Error diffusion binarization for joint transform correlators,”Appl. Opt. 32, 707–714 (1993).
[CrossRef] [PubMed]

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

A. Vanderlugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 9, 215–222 (1970).
[CrossRef]

B. Javidi, J. Wang, “Limitation of the classical definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
[CrossRef] [PubMed]

B. Javidi, J. Wang, A. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
[CrossRef] [PubMed]

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]

Opt. Eng. (2)

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

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Proc. IEEE (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Other (2)

J. Wang, “Multi-object detection using binary optical joint transform correlator,” Ph. D. dissertation (University of Connecticut, Storrs, Conn., 1993).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

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Figures (14)

Fig. 1
Fig. 1

Target used in the computer simulations.

Fig. 2
Fig. 2

(a) Target in colored scene noise, (b) target in white scene noise. The noise mean value is 0.5, and the noise standard deviation is 0.2. The maximum value of the target is 1.

Fig. 3
Fig. 3

Normalized correlation peak of the binary JTC versus input-image illumination a, for different thresholding methods, for a noiseless target.

Fig. 4
Fig. 4

Peak-to-noise ratio versus input-image illumination a for (a) colored input scene noise and (b) white input scene noise. The peak-to-noise ratio is measured for a equal to 0.1, 0.25, 0.5, 1, 2, and 4. The target is placed at the center of the input image. The asterisk represents the failure of the system to detect the target by the provision of a correlation peak smaller than the sidelobe. The curves represent the sliding-window local median (short-dashed curves), the threshold function (solid curves), and median thresholding (short–long-dashed curves).

Fig. 5
Fig. 5

Same as Fig. 4, but the target is placed at the upper-left corner of the input image.

Fig. 6
Fig. 6

Same as Fig. 4, but the target is placed at the lower-right corner of the input image.

Fig. 7
Fig. 7

Optical implementation of a hybrid electro-optical binary joint transform correlator: SLM, spatial light modulator; NDF, neutral-density filter; BS, beam splitter; FTL, Fourier-transform lens.

Fig. 8
Fig. 8

Input image (target in scene noise) used in the experiments.

Fig. 9
Fig. 9

Experimental data for a = 1: (a) joint power spectrum, (b) binarized joint power spectrum with the median-thresholding method, (c) binarized joint spectrum with the sliding-window local-median thresholding method, (d) binarized joint power spectrum with the threshold-function method.

Fig. 10
Fig. 10

Experimental results: correlation signals for three binarization methods for different input-image illumination a. Figure 1 was used as the input image.

Fig. 11
Fig. 11

Experimental results: normalized correlation peak versus input-image illumination a for three thresholding methods.

Fig. 12
Fig. 12

Same as Fig. 10, but Fig. 8 was used as the input image.

Fig. 13
Fig. 13

One-dimensional joint power spectrum E(α) = R2(α)[a2 + 2a cos(2x0α) + 1)] for a low-pass R2(α), when the input image is the same as the reference image. R is the Fourier magnitude of the reference image, and a is the input-image illumination coefficient.

Fig. 14
Fig. 14

Median of the histogram of the joint power spectrum of the tank image used in the computer simulations of Section 3 (Fig. 1) versus a, along with its upper and lower bounds found in Eq. (11).

Equations (34)

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t ( x , y ) = r ( x x 0 , y ) + a s ( x + x 0 , y ) .
E ( α , β ) = R 2 ( α , β ) + a 2 S 2 ( α , β ) + 2 a R ( α , β ) S ( α , β ) × cos [ 2 x 0 α + ϕ S ( α , β ) ϕ R ( α , β ) ] ,
g ( α , β ) = { 1 E ( α , β ) E T 0 E ( α , β ) < E T ,
g 1 , a ( α , β ) = 2 π { 1 [ R 2 ( α , β ) + a 2 S 2 ( α , β ) E T 2 a R ( α , β ) S ( α , β ) ] 2 } 1 / 2 × cos [ 2 x 0 α + ϕ S ( α , β ) ϕ R ( α , β ) ]
E T ( α , β ) = R 2 ( α , β ) + a 2 S 2 ( α , β ) .
g 1 , a ( α , β ) = ( 1 / π ) exp [ j ϕ S ( α , β ) ] × exp [ j ϕ R ( α , β ) ] exp ( j 2 x 0 α ) .
g 1 , a ( α ) = 2 π { 1 [ ( 1 + a 2 ) R 2 ( α ) E T , a 2 a R 2 ( α ) ] 2 } 1 / 2 cos ( 2 x 0 α ) ,
E T , a = median ( hist { R 2 ( α ) [ a 2 + 2 a cos ( 2 x 0 α ) + 1 ] } ) .
h ( E ) = 2 π α min E α max E d α [ E 2 + 2 ( a 2 + 1 ) R 2 ( α ) E ( a 2 1 ) 2 R 4 ( α ) ] 1 / 2
0 E T , α h ( E ) d E = E T , a E max h ( E ) d E ,
( a 2 + 1 2 ) E T , 1 E T , a ( a 2 a + 1 ) E T , 1 .
g 1 , a ( α ) 2 π { 1 ( 1 + a 2 2 a ) 2 × [ R 2 ( α ) a 2 a + 1 1 + a 2 E T , 1 R 2 ( α ) ] 2 } 1 / 2 cos ( 2 x 0 α ) 2 π { 1 ( 1 + a 2 2 a ) 2 [ R 2 ( α ) E T , 1 R 2 ( α ) ] 2 } 1 / 2 cos ( 2 x 0 α ) .
PNR = I ( x 0 , y 0 ) i R j R | n ( x i , y j ) | / N ,
E ( α ) = a 2 + 2 a cos ( 2 x 0 α ) + 1
h ( E ) = 2 d α d E = 1 x 0 [ E 2 + 2 E ( a 2 + 1 ) ( a 2 1 ) 2 ] 1 / 2 ,
E ( α ) = R 2 ( α i ) [ a 2 + 2 a cos ( 2 x 0 α ) + 1 ] , α i Δ α / 2 < α α i + Δ α / 2 ,
h ( E , α i ) = 2 2 x 0 [ E 2 + 2 ( a 2 + 1 ) R 2 ( α i ) E ( a 2 1 ) 2 R 4 ( α i ) ] 1 / 2 .
h ( E ) = α = α min E α max E 2 x 0 [ E 2 + 2 ( a 2 + 1 ) R 2 ( α i ) E ( a 2 1 ) 2 R 4 ( α i ) ] 1 / 2 ,
2 π 0 α min E T , a ( a 1 ) 2 R 2 ( α ) ( a + 1 ) 2 R 2 ( α ) d α d E [ E 2 + 2 ( a 2 + 1 ) R 2 ( α ) E ( a 2 1 ) 2 R 4 ( α ) ] 1 / 2 + 2 π α min E T , a α max E T , a E T , a ( a + 1 ) 2 R 2 ( α ) d α d E [ E 2 + 2 ( a 2 + 1 ) R 2 ( α ) E ( a 2 1 ) 2 R 4 ( α ) ] 1 / 2 = 2 π α min E T , a α max E T , a ( a 1 ) 2 R 2 ( α ) E T , a d α d E [ E 2 + 2 ( a 2 + 1 ) R 2 ( α ) E ( a 2 1 ) 2 R 4 ( α ) ] 1 / 2 + 2 π α max E T , a L ( a 1 ) 2 R 2 ( α ) ( a + 1 ) 2 R 2 ( α ) d α d E [ E 2 + 2 ( a 2 + 1 ) R 2 ( α ) E ( a 2 1 ) 2 R 4 ( α ) ] 1 / 2
2 π α min E T , a α max E T , a arcsin [ 1 E T , a ( a 1 ) 2 R 2 ( α ) 2 a R 2 ( α ) ] d α = L α max E T , a α min E T , a .
E T , a = ( a 1 ) 2 R 2 ( a min E T , a ) ,
E T , a = ( a + 1 ) 2 R 2 ( α max E T , a ) ,
2 π 0 α max E T , a arcsin [ 1 E T , a ( a 1 ) 2 R 2 ( a ) 2 a R 2 ( a ) ] d a = L a max E T , a .
arcsin [ 1 E T , a ( a 1 ) 2 R 2 ( α ) 2 a R 2 ( α ) ] π 2 ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) .
0 α max E T , a ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α L α max E T , a .
0 α max E T , a ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α = 0 α max E T , 1 ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α + α max E T , 1 α max E T , a ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α .
α max E T , 1 α max E T , a ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α ( a 2 + 1 ) R 2 ( α max E T , a ) ( a + 1 ) 2 R 2 ( α max E T , a ) 2 a R 2 ( α max E T , a ) ( α max E T , a α max E T , 1 ) = ( α max E T , 1 α max E T , a ) .
0 α max E T , a ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α 0 α max E T , 1 ( a 2 + 1 ) R 2 ( α ) E T , a 2 a R 2 ( α ) d α + α max E T , 1 α max E T , a .
E T , a a E T , 1 + ( a 1 ) 2 α max E T , 1 0 α max E T , 1 d α R 2 ( α ) .
0 α max E T , 1 2 R 2 ( α ) E T , 1 2 R 2 ( α ) d α L α max E T , 1 > 0 .
E T , 1 2 < α max E T , 1 0 α max E T , 1 d α R 2 ( α )
E T , a ( a 2 + 1 ) 2 E T , 1 .
l < 3 α max E T , 1 / 4 .
L α max E T , 1 < α max E T , 1 / 2 .

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