Abstract

We design an optimum receiver to detect a pattern or a reference signal. We design a receiver that detects the signal distorted by a multiplicative noise on the signal itself, as well as by additive noise and by nonoverlapping scene noise. We design the optimum receiver under the condition in which the statistics of the multiplicative and nonoverlapping scene noise are not available. In the case in which additive noise is present and the statistics of the multiplicative noise are not known, the usual method of maximizing the likelihood function to estimate the statistics of stationary noise fails. We overcome this problem by viewing the noise processes as vector random variables and describe two different schemes to estimate the statistics of the multiplicative noise. Using computer simulations we show that, for the images tested here, the optimum receiver performs better than some of the existing receivers.

© 1998 Optical Society of America

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References

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  7. 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).
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  8. B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
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  9. A. Fazlollahi, B. Javidi, P. Willet, “Minimum-error-probability receiver for detecting a noisy target in colored background noise,” J. Opt. Soc. Am. A 14, 845–852 (1997).
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  10. 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).
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  11. Ph. Réfrégier, F. Goudail, “Decision theory applied to nonlinear joint transform correlator, in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE, Bellingham, Wash., 1997).

1997

1996

1993

1992

1989

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

1984

1976

1969

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.

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 correlator, in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE, Bellingham, Wash., 1997).

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.

Maloney, W. T.

Psaltis, D.

Réfrégier, Ph.

Turin, J. L.

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

Wang, J.

Willet, P.

Willett, P.

Appl. Opt.

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.

Proc. IEEE

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

Other

Ph. Réfrégier, F. Goudail, “Decision theory applied to nonlinear joint transform correlator, in Optoelectronic Information Processing, B. Javidi, Ph. Réfrégier, eds. (SPIE, Bellingham, Wash., 1997).

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Figures (3)

Fig. 1
Fig. 1

Performance of the receivers of Eqs. (28), (30), (32), (36), and (43) when the target is buried in disjoint background, additive, and multiplicative noise. Here mr=0.8, σr=0.2, mb=0.2, σb=0.2, md=0, and σd=0.3. (a) The target r is an image of a tank located in the center of the scene; (b) scene of (a) with additive, disjoint background, and multiplicative noise added to the scene. (c) and (d) output of the receiver designed to handle all three types of noise process on the target itself; (c) output of such a receiver based on Eq. (28); (d) output of the receiver of Eq. (30); (e) output of the receiver of Eq. (32), the receiver designed to handle disjoint background and multiplicative noise on the target; (f) the output of the receiver of Eq. (36); the receiver is designed to handle random gray levels and disjoint background noise; (g) output of the receiver of Eq. (43), the receiver designed to handle an unknown illumination and disjoint background noise.

Fig. 2
Fig. 2

Performance of the receivers of Eqs. (28), (30), and (32) when the target is buried in disjoint background and additive. Here mr=1, σr=0, mb=0.5, σb=0.8, md=0, and σd=0.4. (a) The target r is an image of a tank located in the center of the scene; (b) scene of (a) with additive and disjoint background noise; (c) and (d) output of the receiver designed to handle all three types of noise process on the target itself; (c) output of such a receiver based on Eq. (28); (d) output of the receiver of Eq. (30); (e) output of the receiver of Eq. (32), the receiver designed to handle disjoint background and multiplicative noise on the target.

Fig. 3
Fig. 3

Performance of the receivers of Eqs. (28), (30), (32), (36), and (43) when the scene contains a false target. The target is buried in disjoint background, additive, and multiplicative noise while the false target is buried in additive noise. Here mr=0.8, σr=0.2, mb=0.2, σb=0.2, md=0, and σd=0.3. (a) The target r is an image of a tank located in the center of the scene and a false target is in the upper left corner; (b) scene of (a) with additive, disjoint background, and multiplicative noise on the target, but not on the false target, added to the scene; (c) and (d) output of the receiver designed to handle all three types of noise process on the target itself; (c) output of such a receiver based on Eq. (28); (d) output of the receiver of Eq. (30); (e) output of the receiver of Eq. (32), the receiver designed to handle disjoint background and multiplicative noise on the target; (f) output of the receiver of Eq. (36), the receiver designed to handle random gray levels and disjoint background noise; (g) output of the receiver of Eq. (43), the receiver designed to handle an unknown illumination and disjoint background noise.

Equations (48)

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Hj:s(t)=nr(t)r(t-tj)w(t-tj)+nb(t)[1-w(t-tj)]+nd(t).
s(i)=nr(i)r(i-j)w(i-j)+nb(i)[1-w(i-j)]+nd(i).
P[s|Hj]
=1(2π)m/2(σb2+σd2)no/2σrnw×i: w(i-j)=1[r2(i-j)+(σd/σr)2]-1/2 ×exp-i=1m [s(i)-mb-md]2[1-w(i-j)]2[σb2+σd2]×exp-i=1m[s(i)-r(i-j)mr-md]2w(i-j)2[r2(i-j)σr2+σd2],
C=1(2π)m/2(σb2+σd2)no/2σrnw×i:w(i-j)=1[r2(i-j)+(σd/σr)2]-1/2.
λj=i=1m[s(i)-mb-md]2[1-w(i-j)]σb2+σd2+[s(i)-r(i-j)mr-md]2w(i-j)r2(i-j)σr2+σd2.
s(i)=r(i-j)w(i-j)nr(i)+nd(i).
Xˆ=S=[s(1),, s(nw)],
E(X)=[r(1-j)w(1-j)mr+md,, r(nw-j)w(nw-j)+md],
mˆr(j)=i=1m[s(i)-md]w(i-j)r(i-j)r22,
r2=i=1m|r(i-j)w(i-j)|21/2.
x¯i=r(i-j)w(i-j)mˆr(j)+md.
Varˆ(xi)=(si-x¯i)2.
Var(xi)=r2(i-j)w(i-j)σr2+σd2.
σˆr2(j)=i=1m{[s(i)-md-r(i-j)mˆr(j)]2-σd2}w(i-j)r2(i-j)r44,
r4=i=1m|r(i-j)|4w(i-j)1/4.
mˆb(j)=1noi=1m[s(i)-md][1-w(i-j)],
σˆb2(j)=1noi=1m{[s(i)-md-mˆb(j)]2-σd2}×[1-w(i-j)].
σˆr2(j)=i=1m[s(i)-r(i-j)mˆr(j)]2w(i-j)r2(i-j)r44,
σˆb2(j)=1noi=1m[s(i)-mˆb(j)]2[1-w(i-j)].
E[w(i-j)nr(i)]=Es(i)w(i-j)r(i-j)-mdw(i-j)r(i-j),
mr(j)=E[s(i)-md]w(i-j)r(i-j).
mˆr(j)=1nwi=1m [s(i)-md]w(i-j)r(i-j),
(σˆ)r2(j)=(1/nw)i=1m[s(i)-md]w(i-j)r(i-j)-mr(j)2w(i-j).
log P(s|Hj)=-(1/2)[(m/2)log(2π)+no+Aj+Bj+Cj],
Aj=no log[σˆb2(j),
Bj=i:w(i-j)=1 log[r2(i-j)σˆr2(j)+σd2],
Cj=i=1m [s(i)-r(i-j)mˆr(j)-md]2w(i-j)[r2(i-j)σˆr2(j)+σd2].
λj=Aj+Bj+Cj
log P(s|Hj)=-[(m/2)log(2π)+Kj+Lj+Mj+Nj],
Kj=(no/2)log[σˆb2(j)+σd2]+nw log[σˆr(j)],
Lj=(1/2)i:w(i-j)=1 logr2(i-j)+σd2σˆr2(j),
Mj=i=1m [s(i)-r(i-j)mˆr(j)-md]2w(i-j)2[r2(i-j)σˆr2(j)+σd2],
Nj=i=1m [s(i)-mˆb(j)-md]2[1-w(i-j)]2[σˆb2(j)+σd2]=noσˆb2(j)2[σˆb2(j)+σd2].
λj=Kj+Lj+Mj+Nj
P(s|Hj)=C[σˆb(j)]no[σˆr(j)]nw,
λj=nw log(1/nw)i=1ms(i)r(i-j)-mˆr(j)2w(i-j)+no log(1/no)i=1m[s(i)-mˆb(j)]2×[1-w(i-j)],
mˆr(j)=1nwi=1m s(i)w(i-j)r(i-j),
mˆb(j)=1noi=1ms(i)[1-w(i-j)].
s(i)=nr(i)w(i-j)+nb(i)[1-w(i-j)].
λj=nw log(1/nw)i=1m[s(i)-mˆr(j)]2w(i-j)+no log(1/no)i=1m[s(i)-mˆb(j)]2×[1-w(i-j)],
H(j):s(i)=ar(i-j)w(i-j)+nb(i)[1-w(i-j)]+nd(i).
aˆj=i=1m[s(i)-md]r(i-j)w(i-j)i=1mr2(i-j)w(i-j),
mˆb(j)=1noi=1m[s(i)-md][1-w(i-j)],
σˆb2(j)=1noi=1m[s(i)-md-mˆb(j)]2[1-w(i-j)].
λj=log[σˆb2(j)]+(1/σd2)Ej,
Ej=i=1m[s(i)-md]2w(i-j)-i=1m[s(i)-md]r(i-j)w(i-j)2i=1mr2(i-j).
λj=i=1m [s(i)-mb-md]2[1-w(i-j)][σb2+σd2]+(1/σd2)Ej.

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