Abstract

We investigate the use of Fourier plane nonlinear filtering for phase-encoded images. We investigate the performance of the nonlinear joint transform correlator and the nonlinearly transformed matched filter for phase-encoded images with different types of input noise. We use the peak-to-output-energy ratio, peak-to-sidelobe ratio, and discrimination ratio as the metrics for measuring the performances. We mathematically analyze the peak-to-output-energy ratio of the nonlinearly transformed matched filter for phase-encoded images with spatially nonoverlapping white noise. Computer simulations are provided to show the performance improvements of the nonlinear filtering techniques for the phase-encoded images. In comparison with linear filtering techniques, we find that the nonlinear filtering techniques substantially improve the performance metrics. From the computer-simulation results it can be seen that the nonlinear joint transform correlator performs better than the nonlinearly transformed matched filter in detecting phase-encoded targets in the presence of different types of noise, such as additive overlapping white noise, spatially nonoverlapping white background noise, spatially nonoverlapping colored background noise, and nontarget objects.

© 1998 Optical Society of America

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References

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  1. R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
    [CrossRef]
  2. S. Maze, Ph. Réfrégier, “Optical correlation: influence of the coding of the input images,” Appl. Opt. 33, 6788–6796 (1994).
    [CrossRef] [PubMed]
  3. B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [CrossRef]
  4. N. Konftorti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 13, 251–253 (1988).
    [CrossRef]
  5. T. H. Barnes, T. Eiju, K. Matsuda, N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
    [CrossRef] [PubMed]
  6. L. J. Hornbeck, “Deformable-mirror spatial light modulators,” in Spatial Light Modulators and Applications III, U. Efron, ed., Proc. SPIE1150, 86–102 (1990).
    [CrossRef]
  7. L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
    [CrossRef] [PubMed]
  8. C. Hester, M. Temmen, “Phase-phase implementation of optical correlators,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).
    [CrossRef]
  9. B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
    [CrossRef] [PubMed]
  10. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  11. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  12. B. Javidi, J. Wang, “Design of filters to detect a noisy target in non-overlapping background noise,” J. Opt. Soc. Am. A 11, 2604–2612 (1994).
    [CrossRef]
  13. Ph. Réfrégier, F. Goudail, “Decision theory applied to object location and nonlinear joint-transform correlation,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997).

1995 (1)

1994 (4)

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

S. Maze, Ph. Réfrégier, “Optical correlation: influence of the coding of the input images,” Appl. Opt. 33, 6788–6796 (1994).
[CrossRef] [PubMed]

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

B. Javidi, J. Wang, “Design of filters to detect a noisy target in non-overlapping background noise,” J. Opt. Soc. Am. A 11, 2604–2612 (1994).
[CrossRef]

1990 (1)

1989 (2)

1988 (1)

1984 (1)

Barnes, T. H.

Eiju, T.

Gianino, P. D.

Goldstein, D. H.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Goudail, F.

Ph. Réfrégier, F. Goudail, “Decision theory applied to object location and nonlinear joint-transform correlation,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997).

Hester, C.

C. Hester, M. Temmen, “Phase-phase implementation of optical correlators,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).
[CrossRef]

Hornbeck, L. J.

L. J. Hornbeck, “Deformable-mirror spatial light modulators,” in Spatial Light Modulators and Applications III, U. Efron, ed., Proc. SPIE1150, 86–102 (1990).
[CrossRef]

Horner, J. L.

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

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

Javidi, B.

Kallman, R. R.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Konftorti, N.

Marom, E.

Matsuda, K.

Maze, S.

Neto, L. G.

Ooyama, N.

Réfrégier, Ph.

S. Maze, Ph. Réfrégier, “Optical correlation: influence of the coding of the input images,” Appl. Opt. 33, 6788–6796 (1994).
[CrossRef] [PubMed]

Ph. Réfrégier, F. Goudail, “Decision theory applied to object location and nonlinear joint-transform correlation,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997).

Roberge, D.

Sheng, Y.

Temmen, M.

C. Hester, M. Temmen, “Phase-phase implementation of optical correlators,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).
[CrossRef]

Wang, J.

Wu, S.-T.

Appl. Opt. (6)

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

Opt. Eng. (2)

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

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Opt. Lett. (1)

Other (3)

L. J. Hornbeck, “Deformable-mirror spatial light modulators,” in Spatial Light Modulators and Applications III, U. Efron, ed., Proc. SPIE1150, 86–102 (1990).
[CrossRef]

Ph. Réfrégier, F. Goudail, “Decision theory applied to object location and nonlinear joint-transform correlation,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997).

C. Hester, M. Temmen, “Phase-phase implementation of optical correlators,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Helicopter in spatially nonoverlapping background noise. The background noise n 0(t) is white Gaussian with mean 0 and standard deviation σ0 = 0.3.

Fig. 2
Fig. 2

Comparison of the analytical and simulated POE ratio of the nonlinearly transformed matched filter for the phase-encoded image with spatially nonoverlapping background noise. The background noise n 0(t) is white Gaussian with mean 0 and standard deviation σ0 = 0.3.

Fig. 3
Fig. 3

Comparison of the analytical and simulated POE ratio of the nonlinearly transformed matched filter for the phase-encoded image with additive overlapping white Gaussian noise with mean 0 and standard deviation σ0 = 0.3.

Fig. 4
Fig. 4

POE ratio and the PSR of the nonlinear JTC (dashed–dotted curve) and nonlinearly transformed matched filter (solid curve) for the phase-encoded image with spatially nonoverlapping white Gaussian noise with mean 0 and standard deviation σ0 = 0.3: (a) POE ratio. (b) PSR.

Fig. 5
Fig. 5

POE ratio and the PSR of the nonlinear JTC (dashed–dotted curve) and nonlinearly transformed matched filter (solid curve) for the phase-encoded image with spatially nonoverlapping background noise and additive noise. The spatially nonoverlapping background noise is white Gaussian with mean m 0 = 0.1 and standard deviation σ0 = 0.1. The additive overlapping noise is also white Gaussian with mean 0 and standard deviation σ d = 0.15: (a) POE ratio. (b) PSR.

Fig. 6
Fig. 6

POE ratio and the PSR of the nonlinear JTC (dashed–dotted curve) and nonlinearly transformed matched filter (solid curve) for the phase-encoded image with spatially nonoverlapping colored background noise with mean m 0= 0.1 and standard deviation σ0 = 0.1 and additive overlapping white noise with mean 0 and standard deviation σ d = 0.15: (a) POE ratio. (b) PSR.

Fig. 7
Fig. 7

POE ratio and the PSR of the nonlinear JTC (dashed–dotted curve) and nonlinearly transformed matched filter (solid curve) for the phase-encoded image with additive white Gaussian noise with mean 0 and standard deviation σ0 = 0.3: (a) POE ratio. (b) PSR.

Fig. 8
Fig. 8

Discrimination test of the nonlinear JTC, the nonlinearly transformed matched filter, and the conventional matched filter when a target helicopter and three nontarget objects are in the nonoverlapping background noise and distorted by the additive overlapping white noise: (a) Input image. (b) Output of the nonlinear JTC. (c) Output of the nonlinearly transformed matched filter. (d) Output of the conventional matched filter.

Equations (29)

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r p t = exp jar t ,
s t = r t + n t ,
n t = n 0 t 1 - w t .
s p t = exp { ja r t + n t } = exp { jar t } exp { jar t } .
POE = | E y t 0 | 2 E { | y t | 2 ¯ } ,
PSR = E | y t 0 | 2 max [ | y t | sl 2 ] ,
y p t = s p t + τ + r p t - τ .
| Y P f | 2 = | S P f | 2 + | R P f | 2 + 2 | S P f | | R P f | cos ( 4 π f τ + ϕ S P - ϕ R P ) ,
R P ( f ) = | R P ( f ) | exp [ j ϕ R P ( f ) ] = F { exp [ jar ( t ) ] } , S P ( f ) = | S P ( f ) | exp [ j ϕ S P ( f ) ] = F { exp { ja [ r ( t ] + n ( t ) ] } } = R P ( f )   *   N P ( f ) .
g f = | E f | k sgn E f ,   0 k 1 ,
E f = | Y p f | 2 - | S p f | 2 - | R p f | 2
g 1 f = γ | R P f | k | S P f | k cos [ 4 π f τ + ϕ S P f - ϕ R P f ] ,
POE kjtc = | E { y 1 ( 2 τ ) } | 2 E { | y 1 ( t ) | 2 ¯ } ,
H k f = | R P f | k exp [ - j ϕ R P f ] .
y ( t ) = F - 1 { S P ( f ) H k ( f ) } =   S P ( f ) H k ( f ) exp ( j 2 π ft ) d f .
N 0 ( a ) = E { exp [ jan 0 ( t ) ] } =   exp ( jan 0 ) p ( n 0 ) d n 0 = F - 1 { p ( n 0 ) } | α = a ,
E y 0 = E   S P f H k f d f = N 0 a     R P f H k f d f + 1 - N 0 a ×   R P f   *   W f H k f d f ,
E { | y ( t ) | 2 ¯ } = E 1 L     | S P ( f ) | 2 | H k ( f ) | 2 d f = 1 L   | N 0 ( a ) R P ( f ) + [ 1 - N 0 ( a ) ] × R P ( f )   *   W ( f ) | 2 | R P ( f ) | 2 k d f + [ 1 - | N 0 ( a ) | 2 ]     { | R P ( f ) | 2 - R P ( f )   *   W ( f ) R P * ( f ) } | R P ( f ) | 2 k d f ,
POE k filter = L |   N 0 ( a ) R P ( f ) + 1 - N 0 ( a ) R P ( f )   *   W ( f ) H k ( f ) d f |   2 | N 0 ( a ) R P ( f ) + 1 - N 0 ( a ) R P ( f )   *   W ( f ) | 2 | R P ( f ) | 2 k d f + 1 - | N 0 ( a ) | 2     { | R P ( f ) | 2 - R P ( f )   *   W ( f ) R P * ( f ) } | R P ( f ) | 2 k d f .
POE k filter = L N 0 ( a )     R P ( f ) H k ( f ) d f 2 | N 0 ( a ) | 2     | R P ( f ) | 2 + 2 k d f + [ 1 - | N 0 ( a ) | 2 ]   | R P ( f ) | 2 | R P ( f ) | 2 k d f .
E n 0 t 1 n 0 t 2 = σ 0 2 0 for   t 1 = t 2 for   t 1 t 2 .
E { exp { ja [ n 0 ( t 1 ) - n 0 ( t 2 ) ] } } = | N 0 ( a ) | 2 + δ t 1 , t 2 [ 1 - | N 0 ( a ) | 2 ] ,
δ t 1 , t 2 = 1 0 for   t 1 = t 2 for   t 1 t 2 .
E exp jan ( t ) = N 0 ( a ) + 1 - N 0 ( a ) w ( t ) ,
E N P ( f ) = N 0 ( a ) δ ( f ) + 1 - N 0 ( a ) W ( f ) ,
E { exp { ja [ n ( t 1 ) - n ( t 2 ) ] } } = { | N 0 ( a ) | 2 + δ t 1 , t 2 × [ 1 - | N 0 ( a ) | 2 ] } [ 1 - w ( t 1 ) ] × [ 1 - w ( t 2 ) ] + N 0 * ( a ) w ( t 1 ) × [ 1 - w ( t 2 ) ] + N 0 ( a ) w ( t 2 ) × [ 1 - w ( t 1 ) ] + w ( t 1 ) w ( t 2 ) .
E { N P ( f 1 ) N P * ( f 2 ) } =   E { exp { ja [ n ( t 1 - n ( t 2 ) ] } } × exp [ - j 2 π ( f 1 t 1 - f 2 t 2 ) ] d t 1 d t 2 = | N 0 ( a ) | 2 δ ( f 1 ) δ ( f 2 ) + [ N 0 ( a ) - | N 0 ( a ) | 2 ] δ ( f 1 ) W * ( f 2 ) + [ N 0 * ( a ) - | N 0 ( a ) | 2 ] δ ( f 2 ) W ( f 1 ) + | 1 - N 0 ( a ) | 2 W ( f 1 ) W * ( f 2 ) + [ 1 - | N 0 ( a ) | 2 ] [ δ ( f 1 - f 2 ) - W ( f 1 - f 2 ) ] ,
E y 0 = E   S P f H k f d f =   [ R P f   *   E N P f ] H k f d f = N 0 a     R P f H k f d f + [ 1 - N 0 a ] ×   [ R P f   *   W f ] H k f d f ,
E | y ( t ) | 2 ¯ = E 1 L     | S P ( f ) | 2 | H k ( f ) | 2 d f = 1 L   E | R P ( f )   *   N P ( f ) | 2 | H k ( f ) | 2 d f = 1 L     d f 1 d f 2 d f R P f - f 1 R P * f - f 2 × E N P f 1 N P * f 2 | H k ( f ) | 2 .

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