Abstract

The correlation performance of the nonlinear joint transform correlator for input images containing a target embedded in input noise is investigated. We focus on a class of images for the target, the input image, and the input noise for which nonlinear joint transform correlators perform well. The target and the input noise have low-pass characteristics; that is, they are not band limited but have most of their energies in the low spatial frequency domain. The analytical results show that for this class of low-pass (but not band-limited) images, the binary joint transform correlator outperforms other types of kth-law nonlinear joint transform correlators with k ≠ 0 when the input-noise bandwidth is less than the target bandwidth. We show that binary nonlinear joint transform correlators perform well when the input noise is signallike; that is, the input noise or the objects to be rejected have energy spectra that are similar to the target energy spectrum. Correlation tests with computer simulations are presented. The nonlinear joint transform correlator performance is determined for various degrees of nonlinear transformations and for different input-noise parameters. For the nonlinear joint transform correlator the correlation-signal term and the output-noise term arising from the input noise are determined for various degrees of nonlinearity for the target embedded in input noise.

© 1994 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 9, 215–222 (1970).
    [CrossRef]
  4. A. Kozma, “Photographic recording of spatially modulated coherent light,” J. Opt. Soc. Am. 56, 428–432 (1966).
    [CrossRef]
  5. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  6. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  7. K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  8. S. K. Rogers, J. D. Kline, M. Kabrisky, J. P. Mills, “New binarization techniques for joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
    [CrossRef]
  9. B. Javidi, J. L. Horner, “Single spatial light modulator joint transform correlator,” Appl. Opt. 28, 1027–1032 (1989).
    [CrossRef] [PubMed]
  10. B. Javidi, J. Wang, “Limitation of the classic 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]
  11. B. Javidi, J. Wang, “Quantization and truncation effects on binary joint transform correlation,” Opt. Commun. 84, 374–382 (1991).
    [CrossRef]
  12. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  13. J. L. Horner, “Metrics for assessing pattern recognition performance,” Appl. Opt. 31, 165–166 (1992). In personal communications, Horner suggested to us the term PNR.
    [CrossRef] [PubMed]
  14. B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple level thresholding crossing,” Appl. Opt. 30, 4234–4244 (1991).
    [CrossRef] [PubMed]
  15. B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).
  16. I. N. Bronshtein, K. A. SemendyayevHandbook of Mathematics (Van Nostrand Reinhold, New York, 1985).

1992 (3)

1991 (2)

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

B. Javidi, J. Wang, “Quantization and truncation effects on binary joint transform correlation,” Opt. Commun. 84, 374–382 (1991).
[CrossRef]

1990 (3)

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

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

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

1989 (2)

1970 (1)

1966 (2)

1964 (1)

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

Bronshtein, I. N.

I. N. Bronshtein, K. A. SemendyayevHandbook of Mathematics (Van Nostrand Reinhold, New York, 1985).

Fazlollahi, A. H.

B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).

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]

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). In personal communications, Horner suggested to us the term PNR.
[CrossRef] [PubMed]

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

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

B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).

Javidi, B.

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

B. Javidi, J. Wang, “Quantization and truncation effects on binary joint transform correlation,” Opt. Commun. 84, 374–382 (1991).
[CrossRef]

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

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

B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).

Kabrisky, M.

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

Kline, J. D.

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

Kozma, A.

Li, G.

B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).

Mills, J. P.

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

Rogers, S. K.

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

Rotz, F. B.

Semendyayev, K. A.

I. N. Bronshtein, K. A. SemendyayevHandbook of Mathematics (Van Nostrand Reinhold, New York, 1985).

Tang, Q.

VanderLugt, A.

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

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

Vijaya Kumar, B. V. K.

Wang, J.

Weaver, C. S.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

B. Javidi, J. Wang, “Quantization and truncation effects on binary joint transform correlation,” Opt. Commun. 84, 374–382 (1991).
[CrossRef]

Opt. Eng. (3)

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]

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

Other (2)

B. Javidi, J. L. Horner, G. Li, A. H. Fazlollahi, “Illumination-invariant pattern recognition with a binary nonlinear JTC using spatial frequency dependent threshold function,” in Photonics for Processors, Neural Networks, and Memories, W. J. Miceli, J. L. Horner, S. T. Kowel, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 10–16 (1993).

I. N. Bronshtein, K. A. SemendyayevHandbook of Mathematics (Van Nostrand Reinhold, New York, 1985).

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Figures (12)

Fig. 1
Fig. 1

Nonlinear joint transform correlator: FTL1, FTL2, Fourier-transform lenses; CCD1, charge-coupled device; SLM, spatial light modulator.

Fig. 2
Fig. 2

Target image used in the tests.

Fig. 3
Fig. 3

(a) Fourier amplitude of the target (tank image in Fig. 2); (b) Fourier amplitude of a Gaussian-distributed colored input noise with bandwidth of 10 pixels, standard deviation of 0.2, and mean of zero.

Fig. 4
Fig. 4

Some examples of the target in the additive nonoverlapping input noise. The input noise has bandwidth Bn, mean value mn, and standard deviation σn: (a) Bn, = 5 pixels; mn = 0.3, σn = 0.1; (b) Bn = 10 pixels, mn = 0. 3, σn = 0.2; (c) Bn = 20 pixels, mn = 0.5, σn = 0.2; (d) Bn = 100 pixels, mn = 0.5, σn = 0.3.

Fig. 5
Fig. 5

Computer-simulation results of output PNR [please see Eq. (21)] versus input-noise bandwidth Bn for different values of input-noise standard deviation σn. The input-noise mean value mn is 0.3. The plots are presented for different values of the severity of the kth-law nonlinearity in the Fourier plane: (a) k = 0,(b) k = 0.2, (c) k = 0.5, (d) k = 0.8, (e) k = 1.

Fig. 6
Fig. 6

Same as Fig. 5 except that input noise mean value mn is 0.5.

Fig. 7
Fig. 7

Computer-simulation results for output PNR [please see Eq. (21)] versus input-noise bandwidth Bn for different degrees of Fourier-plane nonlinearity k [k = 0 (upper solid curve), k = 0.2 (dashed curve), k = 0.5 (dotted curve), k = 0.8 (dashed–dotted curve), k = 1 (lower solid curve)]. The input-noise mean value mn is 0.3. The plots are presented for different values of input-noise standard deviation σn: (a) σn = 0.3, (b) σn = 0.2, (c) σn = 0.1.

Fig. 8
Fig. 8

Same as Fig. 7 except that input-noise mean value mn, is 0.5.

Fig. 9
Fig. 9

Computer-simulation results for output PNR [please see Eq. (22)] versus the degree of nonlinearity k when input-noise bandwidth Bn is 10pixels. The input noise has a mean value mn of 0.5. The plots are presented for input-noise standard deviation σn of 0.1, 0.2, and 0.3.

Fig. 10
Fig. 10

Analytical plots of the normalized output PNR [please see Eq. (C3)] versus the degree of nonlinearity k when the input noise has an energy spectrum similar to that of the target. Various models of the input-noise energy spectrum are presented: (a) exponential, (b) Gaussian, (c) Cauchy.

Fig. 11
Fig. 11

Analytical plots of the normalized output PNR [please see Eq. (C7)] versus the input-noise bandwidth for different degrees of the Fourier-plane nonlinearity k. It is assumed that the input noise and the target have the energy spectrum given by Eq. (C6). The plots are presented for various models of the input-noise energy spectrum: (a) exponential, (b) Gaussian, (c) Cauchy.

Fig. 12
Fig. 12

Number of failures for the plots of Figs. 5 and 6: k is the severity of the kth-law nonlinearity, k = 1 corresponds to a linear joint transform correlator, and k = 0 corresponds to a binary joint transform correlator. The input noise has mean mn, standard deviation σn, and bandwidth Bn. The number of simulations for each case is 30 times. For k = 0 and k = 0.2 nonlinearities the correlation peak was larger than the output sidelobes, and no failures were observed.

Equations (55)

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E ( β ) = S 2 ( β ) + R 2 ( β ) + 2 S ( β ) R ( β ) × cos [ 2 y 0 β + ϕ S ( β ) - ϕ R ( β ) ] ,
s ( y + y 0 ) = r ( y + y 0 ) + n ( y + y 0 ) .
E ( β ) = R 2 ( β ) + S 2 ( β ) + 2 R 2 ( β ) cos ( 2 y 0 β ) + 2 N ( β ) R ( β ) cos [ 2 y 0 β + ϕ N ( β ) - ϕ R ( β ) ] .
PNR = I ( 2 y 0 ) E n 0 ,
( PNR ) c = | R ( β ) 2 d β | 2 2 π R ( β ) 2 N ( β ) 2 d β / W .
g ( β ) = v = 0 g v ( β ) = v = 0 γ ( v ) R ( β ) k S ( β ) k × cos { 2 v y 0 β + v [ ϕ S ( β ) - ϕ R ( β ) ] } ,
γ ( v ) = v Γ ( k + 1 ) Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) ,
g 1 ( β ) = g 1 s ( β ) + g 1 n ( β ) = γ ( 1 ) R ( β ) k + 1 S ( β ) k - 1 cos ( 2 y 0 β ) + γ ( 1 ) R ( β ) k N ( β ) S ( β ) k - 1 × cos [ 2 y 0 β + ϕ N ( β ) - ϕ R ( β ) ] ,
g 1 s ( β ) = γ ( 1 ) R ( β ) k + 1 S ( β ) k - 1 cos ( 2 y 0 β ) ,
g 1 n ( β ) = γ ( 1 ) R ( β ) k N ( β ) S ( β ) k - 1 × cos [ 2 y 0 β + ϕ N ( β ) - ϕ R ( β ) ] .
PNR = I 1 ( 2 y 0 ) E n 0 ,
( PNR ) k = | R ( β ) k + 1 S ( β ) k - 1 d β | 2 2 π R ( β ) 2 k N ( β ) 2 S ( β ) 2 k - 2 d β / W .
( PNR ) B = | R ( β ) S ( β ) - 1 d β | 2 2 π N ( β ) 2 S ( β ) - 2 d β / W .
( PNR ) k = lim a 2 | 0 a [ R ( β ) S ( β ) ] k R ( β ) S ( β ) d β | 2 2 π W 0 a [ R ( β ) S ( β ) ] 2 k [ N ( β ) S ( β ) ] 2 d β .
( PNR ) B = lim a 2 | 0 a R ( β ) S ( β ) - 1 d β | 2 2 π W 0 a N ( β ) 2 S ( β ) - 2 d β .
β [ R ( β ) S ( β ) ] 0 ,             β [ N ( β ) S ( β ) ] 0 ,             β [ R ( β ) S ( β ) ] 0.
1 a 0 a [ R ( β ) S ( β ) ] k R ( β ) S ( β ) d β 1 a 2 0 a [ R ( β ) S ( β ) ] k d β 0 a R ( β ) S ( β ) d β ,
1 a 0 a [ R ( β ) S ( β ) ] 2 k [ N ( β ) S ( β ) ] 2 d β 1 a 2 0 a [ R ( β ) S ( β ) ] 2 k d β 0 a [ N ( β ) S ( β ) ] 2 d β .
( PNR ) k lim a 2 1 a | 0 a [ R ( β ) S ( β ) ] k d β | 2 | 0 a R ( β ) / S ( β ) d β | 2 2 π W 0 a [ R ( β ) S ( β ) ] 2 k d β 0 a [ N ( β ) / S ( β ) ] 2 d β .
1 a | 0 a [ R ( β ) S ( β ) ] 2 d β | 2 0 a [ R ( β ) S ( β ) ] 2 k d β .
( PNR ) k lim a 2 | 0 a R ( β ) S ( β ) - 1 d β | 2 2 π W 0 a N ( β ) 2 S ( β ) - 2 d β = ( PNR ) B .
H ( α / B x , β / β y , B x , B y ) 2 = 4 π ln 2 B x B y exp ( - ln 2 ) ( α 2 B x 2 + β 2 B y 2 ) ,
PNR = E { [ I ( x 0 , y 0 ) ] } E { [ j R j R n 0 ( x i , y i ) / N ] } ,
s ( y + y 0 ) = r ( y + y 0 ) + n ( y + y 0 ) .
S ( β ) exp j ϕ S ( β ) = R ( β ) exp j ϕ R ( β ) + N ( β ) exp j ϕ N ( β ) .
tan [ ϕ S ( β ) ] = R ( β ) sin [ ϕ R ( β ) ] + N ( β ) sin [ ϕ N ( β ) ] R ( β ) cos [ ϕ R ( β ) ] + N ( β ) cos [ ϕ N ( β ) ] .
tan [ 2 β y 0 + ϕ S ( β ) - ϕ R ( β ) ] = R ( β ) sin ( 2 β y 0 ) + N ( β ) sin [ 2 β y 0 + ϕ N ( β ) - ϕ R ( β ) ] R ( β ) cos ( 2 β y 0 ) + N ( β ) cos [ 2 β y 0 + ϕ N ( β ) - ϕ R ( β ) ]
cos [ 2 β y 0 + ϕ S ( β ) - ϕ R ( β ) ] = R ( β ) S ( β ) cos ( 2 β y 0 ) + N ( β ) S ( β ) cos [ 2 β y 0 + ϕ N ( β ) - ϕ R ( β ) ] .
R ( β ) 2 = c r f ( β / B r , B r ) ,
N ( β ) 2 = c n f ( β / B n , B n ) ,
S ( β ) 2 = c s f ( β / B s , B s ) ,
f ( 0 , B ) = const . ,
f ( , B ) = 0 ,
f ( 1 , B ) = ( 1 / 2 ) f ( 0 , B ) ,
f ( β / B , B ) β 0 ,
f ( β / B , B ) d β = 1.
β [ f ( β / B 1 , B 1 ) f ( β / B 2 , B 2 ) ] 0 ,             B 1 B 2 ,             or β [ f ( β / B 1 , B 1 ) f ( β / B 2 , B 2 ) ] 0 ,             B 1 B 2 .
S ( B s ) 2 R ( B s ) 2 + N ( B s ) 2 + 2 R ( B s ) N ( B s ) .
B n B r .
B n B s B r .
β [ R ( β ) 2 S ( β ) 2 ] 0.
β [ R ( β ) 2 S ( β ) 2 ] = c R c S β [ f ( β / B r , B r ) f ( β / B s , B s ) ] .
β [ R ( β ) 2 S ( β ) 2 ] 0.
β [ N ( β ) 2 S ( β ) 2 ] 0.
β [ R ( β ) S ( β ) ] 0 ,             β [ R ( β ) S ( β ) ] 0 ,             β [ N ( β ) S ( β ) ] 0.
N ( β ) = a ( β ) R ( β ) ,
N ( β ) 2 a 0 R ( β ) 2 ,             a 0 1 ,
PNR 2 | 0 a R ( β ) 2 k d β | 2 2 π a 0 0 a R ( β ) 4 k d β / W .
f ( β / B , B ) = ln 2 π B exp ( - ln 2 β B ) .
f ( β / B , B ) = 2 ( π ln 2 ) 1 / 2 B exp ( - ln 2 β 2 B 2 ) .
f ( β / B , B ) = 2 B 1 1 + β 2 B 2 .
PNR = { 1 2 π a 0 / ( W B r ) 4 ( ln 2 ) k 1 - exp - ( ln 2 ) k a / B r 1 + exp - ( ln 2 ) k a / B r k 0 2 a 2 π a 0 / W k = 0 .
M ( β ) 2 ( 2 π m ) 2 δ ( β ) + σ 2 f ( β / B , B ) ,
( PNR ) k = | R ( β ) k + 1 { R ( β ) 2 + N ( β ) 2 + 2 R ( β ) N ( β ) cos [ ϕ R ( β ) - ϕ N ( β ) ] } ( k - 1 ) / 2 d β | 2 2 π R ( β ) 2 k N ( β ) 2 { R ( β ) 2 + N ( β ) 2 + 2 R ( β ) N ( β ) cos [ ϕ R ( β ) - ϕ N ( β ) ] } k - 1 d β / W .
PSR = R 0 2 SL 2 ,

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