Abstract

Two types of filter are proposed to detect a noisy target embedded in nonoverlapping background noise by optimization of two proposed criteria that are used in the assessment of filter design and performance. Criterion 1 is defined as the ratio of the square of the expected value of the correlation-peak amplitude to the expected value of the output-signal energy. Criterion 2 is defined as the ratio of the square of the expected value of the correlation-peak amplitude to the average output-signal variance. It is shown that, for the nonoverlapping target and scene noise models, the target window and the scene noise window affect the filter functions significantly. Computer-simulation tests of the generalized optimum filter for various kinds of noisy input image are provided to investigate filter performance in terms of peak-to-output-energy ratio, discrimination against undesired objects, and tolerance to target distortion (for example, target rotation and scaling). We compare the results with those of other filters to verify the performance of the optimum filters.

© 1994 Optical Society of America

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References

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  1. J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
    [CrossRef]
  2. A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  3. D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  4. H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef]
  5. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  6. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  7. D. L. Flannery, J. L. Homer, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  8. B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
    [CrossRef] [PubMed]
  9. 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]
  10. B. Javidi, P. Réfrégier, P. K. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
    [CrossRef]
  11. P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  14. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  15. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  16. G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
    [CrossRef] [PubMed]
  17. B. Javidi, J. Wang, A. H. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
    [CrossRef] [PubMed]

1994

1993

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

B. Javidi, P. Réfrégier, P. K. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

1992

1990

1989

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

1987

1986

1985

1984

1976

1969

1964

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

1960

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Casasent, D.

Caufield, H. J.

Fazlollahi, A. H.

Flannery, D. L.

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

Gianino, P. D.

Hassebrook, L.

Homer, J. L.

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

Horner, J. L.

Javidi, B.

Leger, J. R.

Mahalanobis, A.

Maloney, W. T.

Psaltis, D.

Ravichandran, G.

Réfrégier, P.

B. Javidi, P. Réfrégier, P. K. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

Turin, J. L.

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Vijaya Kumar, V. K.

Wang, J.

Willett, P. K.

B. Javidi, P. Réfrégier, P. K. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

Zhang, G.

Appl. Opt.

D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

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

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

B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
[CrossRef] [PubMed]

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

G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
[CrossRef] [PubMed]

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

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

J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

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]

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

IEEE Trans. Inf. Theory

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

IRE Trans. Inf. Theory

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

B. Javidi, P. Réfrégier, P. K. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

Proc. IEEE

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

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Figures (7)

Fig. 1
Fig. 1

Input image with a tank as the target in spatially nonoverlapping background noise. The target noise nr(t) is white and Gaussian distributed with a mean of mr = 0 and a standard deviation of σr = 0.05. The background noise nb(t) is white and Gaussian distributed with a mean of mb = 0.2 and a standard deviation of σb = 0.1.

Fig. 2
Fig. 2

Performance of the optimum filter Hopt (solid curves), the phase-only filter Hpof (long-dashed curves), and the conventional matched filter Hcmf (short-dashed curves) when the mean mb of the background input noise nb(t) is varied. The target noise nr(t) is white and Gaussian distributed with a mean of mr = 0 and a standard deviation of σr = 0.1. The background noise nb(t) is white and Gaussian distributed with a standard deviation of σb = 0.2. (a) POE versus mb, (b) expected value of the output correlation-peak intensity versus mb.

Fig. 3
Fig. 3

Performance of the optimum filter Hopt (solid curves), the phase-only filter Hpof (long-dashed curves), and the conventional matched filter Hcmf (short-dashed curves) when the standard deviation σb of the background input noise nb(t) is varied. The target noise nr(t) is white and Gaussian distributed with a mean of mr = 0 and a standard deviation of σr = 0.1. The background noise nb(t) is white and Gaussian distributed with a mean of mb = 0.4. (a) POE versus σb, (b) expected value of the output correlation-peak intensity versus σb.

Fig. 4
Fig. 4

Performance of the optimum filter Hopt (solid curves), the phase-only filter Hpof (long-dashed curves), and the conventional matched filter Hcmf (short-dashed curves) when the bandwidth of the background noise BWb is varied. The target noise nr(t) is white and Gaussian distributed with a mean of mr = 0 and a standard deviation of σr = 0.1. The background noise nb(t) is colored and Gaussian distributed with a mean of mb = 0.4 and a standard deviation of σb = 0.3. (a) POE versus BWb, (b) expected value of the output correlation-peak intensity versus BWb.

Fig. 5
Fig. 5

Discrimination test of the optimum filter, the phase-only filter, and the conventional matched filter when a noisy target tank and three undesired objects (a plane, a car, and a vehicle) are in the nonoverlapping background noise. The target noise nr(t) is white and Gaussian distributed with a mean of mr = 0 and a standard deviation of σr = 0.1. The background noise nb(t) is white and Gaussian distributed with a mean of mb = 0.4 and a standard deviation of σb = 0.3. (a) Input image, (b) output of the optimum filter, (c) output of the phase-only filter, (d) output of the conventional matched filter.

Fig. 6
Fig. 6

Tolerance test of the optimum filter, the phase-only filter, and the conventional matched filter to target rotation and scaling. Five target tanks are placed in nonoverlapping background noise. Target tank 1 is identical to the reference tank used in the filter design. Target tanks 2 and 3 are rotated by 2 and 4 deg, respectively. Target tanks 4 and 5 are scaled up by 8% and 10%, respectively. The background noise nb(t) is white and Gaussian distributed with a mean of mb = 0.4 and a standard deviation of σb = 0.3. In the filter design the input-noise parameters are chosen as mb = 0.4, σb = 0.3, mr = 0, and σr = 0.2. (a) Input image, (b) output of the optimum filter, (c) output of the phase-only filter, (d) output of the conventional matched filter.

Fig. 7
Fig. 7

Output of the optimum filter, the phase-only filter, and the conventional matched filter when multiple targets and objects are in the nonoverlapping background noise. Target tank 1 is identical to the reference tank used in the filter design. Target tank 2 is rotated by 4 deg, and target tank 3 is scaled up by 10%. Two undesired objects (a car and a vehicle) are in the input image. The input noise nb(t) is white and Gaussian distributed with a mean of mb = 0.4 and a standard deviation of σb = 0.3. In the filter design the input-noise parameters are chosen as mb = 0.4, σb = 0.3, mr = 0, and σr = 0.2. (a) Input image, (b) output of the optimum filter, (c) output of the phase-only filter, (d) output of the conventional matched filter.

Equations (33)

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POE = E [ y ( τ , τ ) ] 2 / E { [ y ( t , τ ) ] 2 ¯ } ,
S Ñ R = E [ y ( τ , τ ) ] 2 / Var [ y ( t , τ ) ] ¯ ,
s ( t , τ ) = r ( t - τ ) + ñ ( t ) .
y ( t , τ ) = 1 2 π H ( ω ) S ( ω , τ ) exp ( j ω t ) d ω ,
E [ y ( τ , τ ) ] = 1 2 π H ( ω ) E [ S ( ω , τ ) exp ( j ω τ ) ] d ω .
E { [ y ( t , τ ) ] 2 ¯ } = 1 2 π L H ( ω ) 2 E [ S ( ω , τ ) 2 ] d ω .
Var [ y ( t , τ ) ] ¯ = 1 2 π L H ( ω ) 2 Var [ S ( ω , τ ) ] d ω .
POE = | H ( ω ) E [ S ( ω , τ ) exp ( j ω τ ) ] d ω | 2 2 π / L H ( ω ) 2 E [ S ( ω , τ ) 2 ] d ω .
POE L 2 π | E [ S ( ω , τ ) exp ( j ω τ ) ] { E [ S ( ω , τ ) 2 ] } 1 / 2 | 2 d ω .
H opt * ( ω ) = E [ S ( ω , τ ) exp ( j ω τ ) ] E [ S ( ω , τ ) 2 ] ,
S Ñ R = | H ( ω ) E [ S ( ω , τ ) exp ( j ω τ ) ] d ω | 2 2 π / L H ( ω ) 2 Var [ S ( ω , τ ) ] d ω .
S Ñ R | E [ S ( ω , τ ) exp ( j ω τ ) ] { Var [ S ( ω , τ ) ] } 1 / 2 | 2 d ω .
H gmf * ( ω ) = E [ S ( ω , τ ) exp ( j ω τ ) ] Var [ S ( ω , τ ) ] .
ñ ( t ) = ñ b ( t ) + ñ r ( t ) ,
ñ b ( t ) = n b ( t ) [ w 0 ( t ) - w r ( t - τ ) ] ,
ñ r ( t ) = n r ( t ) w r ( t - τ ) ,
f ( τ ) = { 1 / d τ scene area 0 elsewhere ,
E [ S ( ω , τ ) exp ( j ω τ ) ] = F { E [ s ( t + τ , t ) ] } = F { E [ r ( t ) + n b ( t + τ ) × [ w 0 ( t + τ ) - w r ( t ) ] + n r ( t + τ ) w r ( t ) ] } = R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) ,
W 1 ( ω ) = W 0 ( ω ) 2 / d - W r ( ω ) .
E [ S ( ω , τ ) 2 ] = R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) 2 + 1 2 π W 2 ( ω ) * N b 0 ( ω ) + 1 2 π W r ( ω ) 2 * N r 0 ( ω ) + m b 2 [ W 2 ( ω ) - W 1 ( ω ) 2 ] ,
W 2 ( ω ) = W 0 ( ω ) 2 + W r ( ω ) 2 - 2 W 0 ( ω ) 2 real [ W r ( ω ) ] / d .
H opt * ( ω ) = R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) 2 + 1 2 π W 2 ( ω ) * N b 0 ( ω ) + 1 2 π W r ( ω ) 2 * N r 0 ( ω ) + m b 2 [ W 2 ( ω ) - W 1 ( ω ) 2 .
E [ S ( ω , τ ) ] = R ( ω ) W 0 ( ω ) / d + m b [ W 0 ( ω ) - W r ( ω ) W 0 ( ω ) / d ] + m r W r ( ω ) W 0 ( ω ) / d .
H gmf * ( ω ) = E [ S ( ω , τ ) exp ( j ω τ ) ] E [ S ( ω , τ ) 2 ] - E [ S ( ω , τ ) ] 2 = { R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) } / { R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) 2 + 1 2 π W 2 ( ω ) * N b 0 ( ω ) + 1 2 π W r ( ω ) 2 * N r 0 ( ω ) + m b 2 [ W 2 ( ω ) - W 1 ( ω ) 2 ] - R ( ω ) W 0 ( ω ) / d + m b [ W 0 ( ω ) - W r ( ω ) W 0 ( ω ) / d ] + m r W r ( ω ) W 0 ( ω ) / d 2 } .
w ( t , τ ) = 1 - w r ( t - τ ) .
W 1 ( ω ) = δ ( ω ) - W r ( ω ) ,
W 2 ( ω ) = W 1 ( ω ) 2 .
H opt * ( ω ) = R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) 2 + 1 2 π W 1 ( ω ) 2 * N b 0 ( ω ) + 1 2 π W r ( ω ) 2 * N r 0 ( ω ) .
H opt ( ω ) = 1 R ( ω ) + m b W 1 ( ω ) + m r W r ( ω ) .
H opt * ( ω ) = R ( ω ) + m b δ ( ω ) R ( ω ) + m b δ ( ω ) 2 + 1 2 π N b 0 ( ω ) .
H opt * ( ω ) = R ( ω ) R ( ω ) 2 + 1 2 π N b 0 ( ω )             for ω 0.
H gmf ( ω ) = R * ( ω ) 1 2 π W 1 ( ω ) 2 * N b 0 ( ω ) + 1 2 π W r ( ω ) 2 * N r 0 ( ω ) .
H gmf ( ω ) = R * ( ω ) / 1 2 π N b 0 ( ω ) .

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