Abstract

A minimum-mean-square-error filter is proposed to detect a noisy target in spatially nonoverlapping background noise. In this model, both the background noise that is spatially nonoverlapping with the target and the noise that is additive to the target and the input image are considered. The criterion used to design the filter is to minimize the mean-square-error between the filter output and a delta function located at the target position in the presence of the noise. Computer-simulation results for a number of noisy input images are presented, and the performance of the filter is determined. We also test the filter discrimination against undesired objects and tolerance to target distortions, such as rotation and scaling.

© 1996 Optical Society of America

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  1. A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  3. D. Casasent, D. Psaltis, “Position, rotation, and scale-invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  4. For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [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. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
  8. B. Javidi, J. Wang, “Optimum distortion-invariant filter for detecting a noisy distorted target in nonoverlapping background noise,” J. Opt. Soc. Am. A 12, 2604–2617 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. Ph. Refregier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
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  17. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  18. M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [CrossRef] [PubMed]
  19. B. Javidi, Ph. Refregier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
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  21. B. Javidi, Ph. Refregier, G. Zhang, F. Parchekani, “Performance of minimum mean-square-error filter for spatially nonoverlapping target and input scene noise,” Appl. Opt. 33, 8197–8209 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]

1995 (1)

1994 (1)

1993 (2)

1992 (3)

1991 (2)

1990 (4)

1989 (2)

F. M. Dickey, B. D. Hansche, “Quad-phase correlation filters for pattern recognition,” Appl. Opt. 28, 1611–1613 (1989).
[CrossRef] [PubMed]

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

1987 (1)

1985 (1)

1984 (2)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

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

1976 (1)

1969 (1)

1964 (1)

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

Casasent, D.

Caulfield, H. J.

DeLaurentis, J. M.

Dickey, F. M.

Flannery, D. L.

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Fleisher, M.

Gianino, P. D.

Hansche, B. D.

Horner, J. L.

Javidi, B.

Kaura, M. A.

Kumar, B. V. K. V.

Leger, J. R.

Mahalanobis, A.

Mahlab, U.

Maloney, W. T.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Parchekani, F.

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

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

Refregier, Ph.

Rhodes, W. T.

Romero, L. A.

Shamir, J.

VanderLugt, A. B.

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

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Vijaya Kumar, B. V. K.

Wang, J.

Willett, P.

Zhang, G.

Appl. Opt. (12)

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

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]

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

F. M. Dickey, B. D. Hansche, “Quad-phase correlation filters for pattern recognition,” Appl. Opt. 28, 1611–1613 (1989).
[CrossRef] [PubMed]

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

M. A. Kaura, W. T. Rhodes, “Optical correlator performance using a phase-with-constrained-magnitude complex spatial filter,” Appl. Opt. 29, 2587–2593 (1990).
[CrossRef] [PubMed]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[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 noise,” Appl. Opt. 31, 6826–6829 (1992).
[CrossRef] [PubMed]

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

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

B. Javidi, Ph. Refregier, G. Zhang, F. Parchekani, “Performance of minimum mean-square-error filter for spatially nonoverlapping target and input scene noise,” Appl. Opt. 33, 8197–8209 (1994).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Opt. Lett. (5)

Proc. IEEE (1)

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Car image (62 × 47 pixels) used as the target image in computer simulations.

Fig. 2
Fig. 2

Input image and correlation outputs for the minimum-mean-square-error filter and the minimum-mean-square-error filter designed with no additive noise in the test of detector noise and color background noise: (a) Input image with color background noise (mean, m n = 0.5; standard deviation, σ n = 0.1; bandwidth, B n = 20 pixels) and white detector noise (zero mean; standard deviation, σ a = 0.2). (b) Three-dimensional (3-D) plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter designed with no additive noise for the input image shown in (a).

Fig. 3
Fig. 3

Input image and correlation outputs for the minimum-mean-square-error filter and the minimum-mean-square-error filter designed with no target noise in the test of the target noise and color background noise: (a) Input image with a color background noise (mean, m n = 0.5; standard deviation, σ n = 0.15; bandwidth B n = 35 pixels) and white target noise (zero mean; standard deviation, σ r = 0.35), with no detector noise. (b) A 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (c) A 3-D plot of the correlation output for the minimum-mean-square-error filter designed with no target noise for the input image shown in (a).

Fig. 4
Fig. 4

Input image and correlation outputs for the minimum-mean-square-error filter and the same filter but designed with no additive or target noise for the test of distortion tolerance: (a) Input images with real background noise (mean, m n = 0.57; standard deviation, σ n = 0.12) and white detector noise (zero mean; standard deviation, σ a = 0.05). Clockwise (from upper left) the image comprises a car, a vehicle (carrier), a helicopter, an ambulance, a car image that is scaled up by 10%, and a car image that is rotated by 4°. In addition to scaling and rotation of the target, a zero-mean white target noise with a standard deviation of σ r = 0.05 overlaps with the last two cars. The target to be identified comprises the car images, and the other objects are to be rejected. (b) Image-identification diagram for (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (d) 3-D plot of the correlation output for the same filter but designed with no additive or target noise for the input image shown in (a).

Fig. 5
Fig. 5

Input image and correlation outputs for the minimum-mean-square-error filter and the same filter but designed with no additive or target noise for the discrimination test. (a) Input image with real background-scene noise composed of clouds (mean, m n = 0.45; standard deviation, σ n = 0.135) and white, zero-mean detector noise (standard, deviation, σ a = 0.3), containing (clockwise from upper left image) a lear jet, an Embraer airplane, a Falcon airplane, a Canadian Airline’s jet, and another Falcon airplane. A zero-mean white target noise with a standard deviation of σ r = 0.1 overlaps the second Falcon airplane. The target to be identified is the Falcon airplane, and the other objects are to be rejected. (b) Image-identification diagram for (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (d) 3-D plot of the correlation output for the same filter but designed with no additive or target noise for the input image shown in (a).

Fig. 6
Fig. 6

Block diagram of the minimum-mean-square-error filter. The impulse response of the filter h(· · ·) is optimized to minimize the mean square error between y d (· · ·) and y a (· · ·), where y d (· · ·) is the desired response and y a (· · ·) the actual response for the input x(i). E(· · ·) is the error at each pixel.

Fig. 7
Fig. 7

Spatially nonoverlapping target and input-scene noise, additive detector noise, and distortion-noise model: (a) The reference or target to be detected r(· · ·), (b) window of the target w r (· · ·), (c) complement window of the target, w c (· · ·) = [1 − w r (· · ·)]w x (· · ·), (d) nonoverlapping input-scene noise n(· · ·), (e) additive detector noise n a (· · ·), (f) overlapping distortion noise n r (· · ·)w r (· · ·), (g) input image x(· · ·) = {r(· · ·) + n(· · ·)[1 − w r (· · ·)] + n a (· · ·) + n r (· · ·)w r (· · ·)}w x (· · ·), and (h) window of the input image w x (· · ·).

Tables (4)

Tables Icon

Table 1 Correlation Results for the Minimum-Mean-Square-Error Filter Designed by the Use of Different Detector-Noise Standard Deviationsa

Tables Icon

Table 2 Correlation Results for the Minimum-Mean-Square-Error Filter Designed by Use of Different Target-Noise Standard Deviationsa

Tables Icon

Table 3 Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Tolerance Testa

Tables Icon

Table 4 Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Discrimination Testa

Equations (29)

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x ( i ) = r ( i ) + n ( i ) w c ( i ) + n a ( i ) w x ( i ) + n r ( i ) w r ( i ) , i = , - 1 , 0 , 1 , .
E = i E ( i ) = i E { y d ( i ) - y a ( i ) 2 } = i E { δ ( i ) - h ( i ) * x ( i ) 2 } .
E = 1 + i E { h ( i ) * x ( i ) 2 } - 2 Real { E [ h ( i ) * x ( i ) ] i = 0 } = 1 + p q h ( p ) h ( q ) B p q - 2 Real { p h ( p ) E [ x ( - p ) ] } ,
B p q = B ( q - p ) = C r r ( q - p ) + m n C w c r ( q - p ) + m r C w r r ( q - p ) + m a C w x r ( q - p ) + m n C r w c ( q - p ) + R n n ( q - p ) C w c w c ( q - p ) + m r m n C w r w c ( q - p ) + m a m n C w x w c ( q - p ) + m r C r w r ( q - p ) + m n m r C w c w r ( q - p ) + R n r n r ( q - p ) C w r w r ( q - p ) + m a m r C w x w r ( q - p ) + m a C r w x ( q - p ) + m n m a C w c w x ( q - p ) + m r m a C w r w x ( q - p ) + R n a n a ( q - p ) C w x w x ( q - p ) ,
B ^ ( ν ) = r ^ ( ν ) 2 + m n w ^ c ( ν ) r ^ * ( ν ) + m r w ^ r ( v ) r ^ * ( ν ) + m a w ^ x ( ν ) r ^ * ( ν ) + m n r ^ ( ν ) w ^ c * ( ν ) + S ^ n ( ν ) * w ^ c ( ν ) 2 + m r m n w ^ r ( ν ) w ^ c * ( ν ) + m a m n w ^ x ( ν ) w ^ c * ( ν ) + m r r ^ ( ν ) w ^ r * ( ν ) + m n m r w ^ c ( ν ) w ^ r * ( ν ) + S ^ n r ( ν ) * w ^ r ( v ) 2 + m a m r w ^ x ( ν ) w ^ r * ( ν ) + m a r ^ ( ν ) w ^ x * ( ν ) + m n m a w ^ c ( ν ) w ^ x * ( ν ) + m r m a w ^ r ( ν ) w ^ x * ( ν ) + S ^ n a ( ν ) * w ^ x ( ν ) 2 ,
r ˜ ( ν ) E { x ^ ( ν ) } = r ^ ( ν ) + m n w ^ c ( ν ) + m a w ^ x ( ν ) + m r w ^ r ( ν ) .
E = 1 + 1 2 π 2 π h ^ ( ν ) B ^ ( ν ) h ^ * ( ν ) d ν - 2 ( 1 2 π 2 π Real { h ^ ( ν ) E [ x ^ ( ν ) ] } d ν ) .
h ^ ( ν ) = r ˜ * ( ν ) r ˜ ( ν ) 2 + [ S ^ n ( ν ) - m n 2 δ ˜ ( ν ) ] * w ^ c ( ν ) 2 + [ S ^ n a ( ν ) - m a 2 δ ˜ ( ν ) ] * w ^ x ( ν ) 2 + [ S ^ n r ( ν ) - m r 2 δ ˜ ( ν ) ] * w ^ r ( ν ) 2 ,
S ^ n ( ν ) = S ^ n 0 ( ν ) + m n 2 δ ˜ ( ν ) , S ^ n a ( ν ) = S ^ n a 0 ( ν ) + m a 2 δ ˜ ( ν ) , and S ^ n r ( ν ) = S ^ n r 0 ( ν ) + m r 2 δ ˜ ( ν ) .
h ^ ( ν ) = r ˜ * ( ν ) r ˜ ( ν ) 2 + S ^ n 0 ( ν ) * w ^ c ( ν ) 2 +     S ^ n a 0 ( ν ) * w ^ x ( ν ) 2 + S ^ n r 0 ( ν ) * w ^ r ( ν ) 2 .
h ^ ( ν ) = [ r ( ν ) + m a w ^ x ( ν ) ] * r ( ν ) + m a w ^ x ( ν ) 2 + S ^ n a 0 * w ^ x ( ν ) 2 ,
h ^ ( ν ) = [ r ( ν ) + m n w ^ c ( ν ) ] * r ( ν ) + m n w ^ c ( ν ) 2 + S ^ n 0 * w ^ c ( ν ) 2 ,
x ( i ) = [ r ( i ) + m n w c ( i ) + m a w x ( i ) + m r w r ( i ) ] + n u ( i ) = [ r ( i ) + w m ( i ) ] + n u ( i ) = r ¯ ( i ) + n u ( i ) ,             i = , - 1 , 0 , 1 , ,
PCI = I p [ 1 / ( N 1 N 2 ) ] i = 1 N 1 j = 1 N 2 y a ( i , j ) 2 .
E = i E ( i ) = i E { y d ( i ) - y a ( i ) 2 } = i E { d ( i ) - h ( i ) * x ( i ) 2 }
E ( i ) = E { δ 2 ( i ) + h ( i ) * x ( i ) 2 - 2 δ ( i ) Real [ h ( i ) * x ( i ) ] } .
E = 1 + i E { h ( i ) * x ( i ) 2 } - 2 Real { E [ h ( i ) * x ( i ) ] i = 0 } = 1 + p q h ( p ) h ( q ) B p q - 2 Real { p h ( p ) E [ x ( - p ) ] } ,
B p q i E { x ( i - p ) x ( i - q ) } = B ( q - p ) .
B p q = i E { [ r ( i - p ) + n ( i - p ) w c ( i - p ) + n a ( i - p ) × w x ( i - p ) + n r ( i - p ) w r ( i - p ) ] [ r ( i - q ) + n ( i - q ) w c ( i - q ) + n a ( i - q ) w x ( i - q ) + n r ( i - q ) w r ( i - q ) ] } = i { r ( i - p ) r ( i - q ) + m n w c ( i - p ) r ( i - q ) + m r w r ( i - p ) r ( i - q ) + m a w x ( i - p ) r ( i - q ) + m n r ( i - p ) w c ( i - q ) + R n n ( q - p ) w c ( i - p ) × w c ( i - q ) + m r m n w r ( i - p ) w c ( i - q ) + m a m n w x ( i - p ) w c ( i - q ) + m r r ( i - p ) w r ( i - q ) + m n m r w c ( i - p ) w r ( i - q ) + R n r n r ( q - p ) × w r ( i - p ) w r ( i - q ) + m a m r w x ( i - p ) w r ( i - q ) + m a r ( i - p ) w x ( i - q ) + m n m a w c ( i - p ) w x ( i - q ) + m r m a w r ( i - p ) w x ( i - q ) + R n a n a ( q - p ) × w x ( i - p ) w x ( i - q ) } .
B p q = B ( q - p ) = C r r ( q - p ) + m n C w c r ( q - p ) + m r C w r r ( q - p ) + m a C w x r ( q - p ) + m n C r w c ( q - p ) + R n n ( q - p ) C w c w c ( q - p ) + m r m n C w r w c ( q - p ) + m a m n C w x w c ( q - p ) + m r C r w r ( q - p ) + m n m r C w c w r ( q - p ) + R n r n r ( q - p ) C w r w r ( q - p ) + m a m r C w x w r ( q - p ) + m a C r w x ( q - p ) + m n m a C w c w x ( q - p ) + m r m a C w r w x ( q - p ) + R n a n a ( q - p ) C w x w x ( q - p ) ,
B ^ ( ν ) = r ^ ( ν ) 2 + m n w ^ c ( ν ) r ^ * ( ν ) + m r w ^ r ( ν ) r ^ * ( ν ) + m a w ^ x ( ν ) r ^ * ( ν ) + m n r ^ ( ν ) w ^ c * ( ν ) + S ^ n ( ν ) * w ^ c ( ν ) 2 + m r m n w ^ r ( ν ) w ^ c * ( ν ) + m a m n w ^ x ( ν ) w ^ c * ( ν ) + m r r ^ ( ν ) w ^ r * ( ν ) + m n m r w ^ c ( ν ) w ^ r * ( ν ) + S ^ n r ( ν ) * w ^ r ( ν ) 2 + m a m r w ^ x ( ν ) w ^ r * ( ν ) + m a r ^ ( ν ) w ^ x * ( ν ) + m n m a w ^ c ( ν ) w ^ x * ( ν ) + m r m a w ^ r ( ν ) w ^ x * ( ν ) + S ^ n a ( ν ) * w ^ x ( ν ) 2 ,
r ˜ ( ν ) E { x ^ ( ν ) } = r ^ ( ν ) + m n w ^ c ( ν ) + m a w ^ x ( ν ) + m r w ^ r ( ν ) .
B ^ ( ν ) = r ˜ ( ν ) 2 - m n 2 w ^ c ( ν ) 2 - m a 2 w ^ x ( ν ) 2 - m r 2 w ^ r ( ν ) 2 + S ^ n ( ν ) * w ^ c ( ν ) 2 + S ^ n a ( ν ) * w ^ x ( ν ) 2 + S ^ n r ( ν ) * w ^ r ( ν ) 2 = r ˜ ( ν ) 2 + [ S ^ n ( ν ) - m n 2 δ ˜ ( ν ) ] * w ^ c ( ν ) 2 + [ S ^ n a ( ν ) - m a 2 δ ˜ ( ν ) ] * w ^ x ( ν ) 2 + [ S ^ n r ( ν ) - m r 2 δ ˜ ( ν ) ] * w ^ r ( ν ) 2 ,
p q h ( p ) h ( q ) B p q ( q - p ) = q { [ h ( p ) * B p q ( p ) ] q h ( q ) = 1 2 π 2 π FT { [ h ( p ) * B p q ( p ) ] q } h ^ * ( ν ) d ν = 1 2 π 2 π h ^ ( ν ) B ^ ( ν ) h ^ * ( ν ) d ν ,
Real { p h ( p ) E [ x ( - p ) ] } = Real { 1 2 π 2 π h ^ ( ν ) FT * { E [ x ( - p ) ] } d ν } = Real { 1 2 π 2 π h ^ ( ν ) E [ x ^ ( ν ) ] d ν } .
E = 1 + 1 2 π 2 π h ^ ( ν ) B ^ ( ν ) h ^ * ( ν ) d ν - 1 π 2 π Real { h ^ ( ν ) E [ x ^ ( ν ) ] } d ν .
h ^ ( ν ) = E { x ^ * ( ν ) } B ^ ( ν ) = r ˜ * ( ν ) r ˜ ( ν ) 2 + [ S ^ n ( ν ) - m n 2 δ ˜ ( ν ) ] * w ^ c ( ν ) 2 + [ S ^ n a ( ν ) - m a 2 δ ˜ ( ν ) ] * w ^ x ( ν ) 2 + [ S ^ n r ( ν ) - m r 2 δ ˜ ( ν ) ] * w ^ r ( ν ) 2 .
S ^ n ( ν ) = S ^ n 0 ( ν ) + m n 2 δ ˜ ( ν ) , S ^ n a ( ν ) = S ^ n a 0 ( ν ) + m a 2 δ ˜ ( ν ) ,
h ^ ( ν ) = E { x ^ * ( ν ) } B ^ ( ν ) = r ˜ * ( ν ) r ˜ ( ν ) 2 + S ^ n 0 ( ν ) * w ^ c ( ν ) 2 + S ^ n a 0 ( ν ) * w ^ x ( ν ) 2 + S ^ n r 0 ( ν ) * w ^ r ( ν ) 2 .

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