Abstract

We develop algorithms to detect a known pattern or a reference signal in the presence of additive, disjoint background, and multiplicative white Gaussian noise with unknown statistics. The presence of three different types of noise processes with unknown statistics presents difficulties in estimating the unknown parameters. The standard methods such as expected-maximization-type algorithms are iterative, and in the framework of hypothesis testing they are time-consuming, because corresponding to each hypothesis one must estimate a set of parameters. Other standard methods such as setting the gradient of the likelihood function with respect to the unknown parameters will lead to a nonlinear system of equations that do not have a closed-form solution and require iterative methods. We develop an approach to overcome these handicaps and derive algorithms to detect a known object. We present new methods to estimate unknown parameters within the framework of hypothesis testing. The methods that we present are direct and provide closed-form estimates of the unknown parameters. Computer simulations are used to show that for the images tested, the receivers that we have designed perform better than existing receivers.

© 2001 Optical Society of America

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References

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  1. N. Towghi, B. Javidi, J. Li, “Generalized optimum receiver for pattern recognition with multiplicative, additive, and nonoverlapping background noise,” J. Opt. Soc. Am. A 15, 1557–1565 (1998).
    [CrossRef]
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    [CrossRef]
  3. S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
    [CrossRef]
  4. D. Casasent, “Unified synthetic function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  5. D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  6. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  7. H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef]
  8. D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  9. 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]
  10. Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).
  11. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
  15. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  16. Ph. Réfrégier, “Optimal pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [CrossRef]
  17. N. Towghi, B. Javidi, “lp-norm optimum filters for image recognition. Part I. Algorithms,” J. Opt. Soc. Am. A 16, 1928–1935 (1999).
    [CrossRef]
  18. L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

1999 (1)

1998 (2)

N. Towghi, B. Javidi, J. Li, “Generalized optimum receiver for pattern recognition with multiplicative, additive, and nonoverlapping background noise,” J. Opt. Soc. Am. A 15, 1557–1565 (1998).
[CrossRef]

L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
[CrossRef]

1997 (1)

1996 (1)

1994 (1)

1993 (1)

1992 (1)

1991 (3)

Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Ph. Réfrégier, “Optimal pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

1990 (1)

1989 (3)

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

S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
[CrossRef]

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

1987 (1)

1984 (2)

1976 (1)

1969 (1)

1960 (1)

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

Casasent, D.

Caufield, H. J.

Der, S. Z.

L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
[CrossRef]

Fazlollahi, A.

Figue, J.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Flannery, D. L.

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

Gianino, P. D.

Goudail, F.

F. Goudail, Ph. Réfrégier, “Optimal detection of a target with random gray levels on a spatial disjoint background noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

Ph. Réfrégier, F. Goudail, “Decision theory applied to nonlinear joint transform correlators,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–167.

Horner, J. L.

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

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

Javidi, B.

Laude, V.

Li, J.

Mahalanobis, A.

Maloney, W. T.

Nasrabadi, N. M.

L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
[CrossRef]

Psaltis, D.

S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
[CrossRef]

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

Refregier, Ph.

Réfrégier, Ph.

F. Goudail, Ph. Réfrégier, “Optimal detection of a target with random gray levels on a spatial disjoint background noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

Ph. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
[PubMed]

B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
[CrossRef] [PubMed]

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Ph. Réfrégier, “Optimal pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Ph. Réfrégier, F. Goudail, “Decision theory applied to nonlinear joint transform correlators,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–167.

Towghi, N.

Turin, J. L.

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

Venkatesh, S.

S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
[CrossRef]

Vijaya Kumar, B. V. K.

Wang, J.

Wang, L. C.

L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
[CrossRef]

Willet, P.

Willett, P.

Appl. Opt. (7)

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
[CrossRef]

IEEE Trans. Image Process. (1)

L. C. Wang, S. Z. Der, N. M. Nasrabadi, “Automatic target recognition using feature-decomposition and data-decomposition modular neural networks,” IEEE Trans. Image Process. 7, 1113–1121 (1998).
[CrossRef]

IRE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

Ph. Réfrégier, “Optimal pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Opt. Comput. Process. (1)

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Opt. Lett. (5)

Proc. IEEE (1)

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

Other (1)

Ph. Réfrégier, F. Goudail, “Decision theory applied to nonlinear joint transform correlators,” in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–167.

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Figures (4)

Fig. 1
Fig. 1

Performance of optimum receivers. (a) Target to be detected. (b) Target in a natural scene. (c) Scene containing the target distorted by additive, disjoint background, and multiplicative white stationary Gaussian noise. Additive noise has mean 0 and standard deviation 0.3; disjoint background noise has mean 0.2 and standard deviation 0.2; multiplicative noise on the target has mean 0.75 and standard deviation 0.25. (d) Output of the optimum receiver of Eq. (27). (e) Output of the optimum receiver of Eq. (28).

Fig. 2
Fig. 2

Performance of optimum receivers. (a) Scene containing two true targets (tanks) and a false target (helicopter). (b) Noisy scene with additive noise that is white Gaussian with mean 0.1 and standard deviation 0.3 and disjoint background noise that is white Gaussian with mean 0.2 and standard deviation 0.2. There is multiplicative noise on the central target that is white Gaussian with mean 0.95 and standard deviation 0.2. (c) Output of the optimum receiver of Eq. (27). (d) Output of the optimum receiver of Eq. (28).

Fig. 3
Fig. 3

Performance of optimum receiver. (a) Scene containing two true targets (tanks) and a false target (helicopter). (b) Noisy scene with additive noise that is white Gaussian with mean 0.1 and standard deviation 0.4 and disjoint background noise that is white Gaussian with mean 0.2 and standard deviation 0.2. There is multiplicative noise on the central target that is white Gaussian with mean 0.95 and standard deviation 0.25. (c) Output of the optimum receiver of Eq. (27).

Fig. 4
Fig. 4

Comparison of the performance of the optimum receiver with other known algorithms. (a) Scene containing a true target located in the center. (b) Noisy scene with additive white Gaussian noise of mean 0 and standard deviation 0.2 and disjoint background noise of mean 0.2 and and standard deviation 0.2. There is multiplicative noise on the target. The mean and standard deviation of the multiplicative noise are 0.8 and 0.2, respectively. (c) Output of the classical matched filter. (d) Output of the receiver designed to handle unknown uniform illumination on the target in the presence of unknown background noise.19 (e) Output of the receiver designed to handle a target that has random gray levels.22 (f) Output of the optimum receiver designed in this paper [Eq. (27)].

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

Hj : s(t)=[nr(t-tj)r(t-tj)]w(t-tj)nb(t)×[1-w(t-tj)]+nd(t).
Hj : s(i)=[nr(i-j)r(i-j)]w(i-j)+nb(i)×[1-w(i-j)]+nd(i).
s(i)w(i-j)=[n1(i-j)r(i-j)+r(i-j)]w(i-j)+nb(i)[1-w(i-j)]+nd(i).
P[s|Hj]
=1(2π)n/2i:w(i-j)=1[σ12r2(i-j)+σd2]-1/2
×exp-i:w(i-j)=1×[s(i)-r(i-j)m1-r(i-j)-md]2w(i-j)2[r2(i-j)σ12+σd2]×i:w(i-j)=0(σb2+σd2)-1/2×exp-i:w(i-j)=0[s(i)-mb-md]22(σb2+σd2),
s(i)w(i-j)=[r(i-j)n1(i-j)+r(i-j)+nd(i)]×w(i-j),
w(i-j)=1.
Xj={r(i-j)n1(i-j)+r(i-j)+nd(i)}[i:w(i-j)=1].
s(i)=nb(i)+nd(i),w(i-j)=0.
Yj={nb(i)+nd(i)}[i:w(i-j)=0].
X^j=Sj={s(i)}[i:w(i-j)=1],
E(Xj)={r(i-j)+r(i-j)×w(i-j)m1+md}[i:w(i-j)=1],
r22r¯r¯nw m^1(j)m^d(j)=s˜j*wr(j)s˜j*w(j),
s˜j(i)=[s(i)-r(i-j)]w(i-j),
r¯=i:r(i)0r(i),rp=i:r(i)0|r(i)|p1/p,
s˜j*wr(j)=i:w(i-j)=1s˜j(i)r(i-j)w(i-j),
s˜j*w(j)=i:w(i-j)=1s˜j(i)w(i-j).
mˆ(j)=1noi:w(i-j)=0s(i).
Var^(xi)=[s˜j(i)-m^1(j)r(i-j)+m^d(j)]2w(i-j),
i : w(i-j)=1,
Var(Xj)={σ12r2(i-j)w(i-j)+σd2}[i:w(i-j)=1].
r44r22r22nw σ^12(j)σ^d2(j)=A(j)B(j),
A(j)=i:w(i-j)=1[s˜j(i)-m^1(j)r(i-j)+m˜d(j)]2×r2(i-j),
B(j)=i:w(i-j)=1[s˜j(i)-m^1(j)r(i-j)+m^d(j)]2.
σ^2(j)=1noi:w(i-j)=0[s(i)-mˆ(j)]2,
log P(s|Hj)=-(Cj+Dj+Ej),
Cj=i:w(i-j)=1log[r2(i-j)σ^12(j)+σ^d2(j)],
Dj=i:w(i-j)=1×[s(i)-r(i-j)m^1(j)-r(i-j)-m^d(j)]2r2(i-j)σ^12(j)+σ^d2(j),
Ej=no2log[σ^2(j)].
λj=-Cj-Dj-Ej
λj=-Cj-Ej,

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