Abstract

We describe the design of an optimum receiver to detect a noisy target with unknown illumination in nonoverlapping colored background noise. The optimum receiver is designed on the basis of binary Bayesian hypothesis testing with unknown parameters. Both white noise and colored noise are considered to model the additive noise on the target. We show that the solution for the optimum receiver, when the additive noise is white, consists of three terms: The first corresponds to the energy of the input signal, which is defined as the lexicographically ordered samples of the input image within the window of the reference; the second is the square of the correlation between the input signal and the energy-normalized reference; and the third corresponds to the energy of the whitened input signal. When the additive noise is colored, the third term is the same; however, the first two terms change such that the information in the correlation matrix of the additive noise is utilized to process the input signal.

© 1997 Optical Society of America

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References

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  9. Ph. Refregier, 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).
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  12. H. Vincent Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1994).
  13. S. Benedetto, E. Biglieri, V. Castellani, Digital Transmission Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 370, 587.
  14. K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Aperture optimization for emission imaging: effects of a spatially varying background,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
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  15. J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).
  16. G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE 46, 1603–1619 (1958).
    [CrossRef]

1995 (1)

1994 (1)

1993 (2)

1992 (1)

1990 (1)

1989 (1)

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

1984 (1)

1976 (1)

1969 (1)

1965 (1)

J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).

1964 (1)

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

1960 (1)

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

1958 (1)

G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE 46, 1603–1619 (1958).
[CrossRef]

Barrett, H. H.

Benedetto, S.

S. Benedetto, E. Biglieri, V. Castellani, Digital Transmission Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 370, 587.

Biglieri, E.

S. Benedetto, E. Biglieri, V. Castellani, Digital Transmission Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 370, 587.

Casasent, D.

Castellani, V.

S. Benedetto, E. Biglieri, V. Castellani, Digital Transmission Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 370, 587.

Caulfield, 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.

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.

Mazo, J. E.

J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).

Myers, K. J.

Poor, H. Vincent

H. Vincent Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1994).

Psaltis, D.

Refregier, Ph.

Rolland, J. P.

Salz, J.

J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).

Truin, J. L.

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

Turin, G.

G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE 46, 1603–1619 (1958).
[CrossRef]

Vanderlugt, A.

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

Wagner, R. F.

Wang, J.

Willett, P.

Zhang, G.

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).

IEEE Trans. Inf. Theory (1)

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

IRE Trans. Inf. Theory (1)

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

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

Opt. Lett. (2)

Proc. IEEE (1)

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

Proc. IRE (1)

G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE 46, 1603–1619 (1958).
[CrossRef]

Other (2)

H. Vincent Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1994).

S. Benedetto, E. Biglieri, V. Castellani, Digital Transmission Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 370, 587.

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Figures (5)

Fig. 1
Fig. 1

Probability of error of the receiver designed for a noise-free target (dotted–dashed curve) and the receiver designed for a noisy target (solid curve) versus the background noise correlation length Cl for detecting the target in the example of Subsection 4.A.

Fig. 2
Fig. 2

(a) Two target airplanes and two nontarget objects in nonoverlapping colored background noise. The background noise has a mean of 0.5, a standard deviation of 0.2, and a bandwidth of 20 pixels in each dimension (based on a 256×256-point FFT). The reference image (the airplane) has a maximum value of unity. The upper left target has an illumination of 1.5, and the target at the center has a unity illumination. (b) The same image as in (a), but a zero-mean white noise with standard deviation 0.2 is added over the entire image.

Fig. 3
Fig. 3

Output of the optimum receiver designed for a noisy target for the input image of Fig. 2(b). The reference image is the image of the airplane in Fig. 2(a).

Fig. 4
Fig. 4

Output of the optimum receiver designed for a noise-free target for the input image of Fig. 2(b). The reference image is the image of the airplane in Fig. 2(a).

Fig. 5
Fig. 5

Output of the optimum receiver designed for a noisy target for the input image of Fig. 2(b). The reference image is the image of the helicopter in Fig. 2(a).

Equations (56)

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H0:R=NB+Nd,
H1:R=aS+Nd.
L(R)=P1(R)/P0(R),
τ=π0/π1,
P1(R)=i=1W 1σd2πexp-12σd2[r(i)-as(i)]2,
P1(R)=1σdW(2π)Wexp-12σd2(R-aS)T(R-aS).
M0=MB+σd2IW×W,
P0(R)=1|M0|1/2(2π)W×exp-12(R-NB)TM0-1(R-N¯B),
L(R)=|M0|1/2σdWexp-12σd2(R-aS)T(R-aS)+12(R-N¯B)TM0-1(R-N¯B).
LL(R)=log[L(R)]=-12σd2(R-aS)T(R-aS)+12(R-N¯B)TM0-1(R-N¯B).
τ=-log(|M0|1/2/σdW).
[P1(R)]a=ST(R-aS)+(R-aS)TS=0.
aˆ=RTS/STS=RTS/s,
LL(R, aˆ)=-12σd2[RTR-(RTS)2s]+12(R-N¯B)TM0-1(R-N¯B).
LL(R, aˆ)=-12σd2i=1Wr(i)2-i=1W r(i)s(i)2s+12i=1W [y(i)]2,
y(i)1σd2+λiViT(R-N¯B),i=1, 2, , W.
LL(R, aˆ)WB=-σB2σd2+σB2i=1W [r(i)]2-2mBσd2σd2+σB2i=1W r(i)+i=1W r(i)s(i)2s,
LL(R(k), aˆ)=-12σd2i=1m [r(i)]2w(i-k)-i=1m r(i)s(i-k)2s+12i=1m [y(i)]2w(i-k),
LL(R(k), aˆ)WB=-σB2σd2+σB2i=1m [r(i)]2w(i-k)-2mBσd2σd2+σB2i=1m r(i)w(i-k)+i=1mr(i)s(i-k)2s,
LL(R(k), aˆ)=-i=1m [r(i)]2w(i-k)-i=1mr(i)s(i-k)2s.
M0=MB+MD,
LL(R, aˆ)=-R-RTMD-1SSDST×MD-1R-RTMD-1SSDS+(R-N¯B)TM0-1(R-N¯B),
M0(i, j)=MB(i, j)+σd2δi,j,
FR=R˜,
R˜H=(FR)H=RTFH.
FM0FH=M˜0,
M˜0(i, j)=[SB(i)+σd2]δi,j,
12R0TM0-1R0=12R0T(FHF)M0-1(FHF)R0=12(R0TFH)(FM0-1FH)(FR0)=12R˜0HM˜0-1R˜0,
12R0TM0-1R0=12i j r˜0*(i)M˜0-1(i, j)r˜0(j)=12i r˜0*(i)M˜0-1(i)r˜0(i)=12i |r˜0(i)|2M˜0(i)=12i |r˜0(i)|2SB(i)+σd2,
-12σd2RTR-(RTS)2s
=-12σd2i=1W [r(i)]2-i=1W r(i)s(i)2s=-12σd2i |r˜(i)|2-i r˜(i)s˜*(i)2s.
LL(R; aˆ)=-12σd2i |r˜(i)|2-ir˜(i)s˜*(i)2s+12i |r˜0(i)|2SB(i)+σd2.
LL(r(x, y); aˆ)=-12σd2|r˜(α, β)|2 dαdβ+12σd2sr˜(α, β)s˜*×(α, β)dαdβ2+12|r˜0(α, β)|2SB(α, β)+σd2dαdβ,
r0(x, y)=rw(x, y)-mB
LL(R, aˆ)=RTQR,
Q=1sSST-IW×W+σd2M0-1.
Q=1sSST-IW×W,
LL(r(x), aˆ)=-12σd2w[r(x)]2 dx-wr(x)s(x)dx2w[s(x)]2dx+12ww [r(x)-mB][r(x)-mB]M0-1(x, x)dxdx,
M0(x, x)=E([nB(x)-mB+nd(x)]×[nB(x)-mB+nd(x)]),
w M0(x, u)M0-1(u, x)du=δ(x-x),
M0(x, x)=M0(x-x),
[SB(α)+σd2] w exp(-jαu)M0-1(u, x)du
=exp(-jαx)
SB(α)+σd2=- M0(x)exp(-jαx)dx
12ww[r(x)-mB][r(x)-mB]M0-1(x, x)dxdx=12-ww r˜0*(α)[exp(-jαx)]×[r(x)-mB]M0-1(x, x)dxdxdα=12-w r˜0*(α)SB(α)+σd2×[r(x)-mB]exp(-jax)dxdα=12- |r˜0(α)|2SB(α)+σd2dα,
LL(r(x), aˆ)
=-12σd2-|r(α)|2 dα-1s- r˜(α)s˜*(α)dα2+12- |r˜0(α)|2SB(α)+σd2dα,
q=RTQR,
Pe=π0PF+π1PM,
PF=Pr(q>τ|H0),
PM=Pr(q<τ|H1),
Φq(ω)=E[exp(jωq)],ω,
Φq(ω)=E[exp(jωq)]
=exp(12[R¯TM-1(IW×W-2jωMQ)-1R¯-R¯TM-1R¯])[det(IW×W-2jωMQ)]1/2,
PF=12πτ-[exp(-jωq)][Φq(ω)|H0]dωdq,
PM=12π-τ-[exp(-jωq)][Φq(ω)|H1]dωdq.

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