Abstract

We derive an optimum filter function to detect a target degraded by multiplicative noise and additive overlapping noise and placed in background noise. The filter is designed to maximize the metric peak-to-output energy, which is the ratio of the expected value squared of the output peak at the target position to the expected value of the average output energy. The optimum filter provides improved discrimination as well as robustness to input noise. One advantage of the filter described here over the homomorphic filters is that the additional preprocess on the input image, that is, the input logarithmic operation, is not required for reducing the effects of multiplicative noise. The performance of the filter is examined in terms of discrimination against background noise and robustness to multiplicative and additive input noise. Both multiplicative amplitude noise and multiplicative complex noise are considered.

© 1997 Optical Society of America

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References

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1996 (2)

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

Ph. Réfrégier, F. Goudail, T. Gaidon, “Optimal location of random targets in random background: random Markov fields modelization,” Opt. Commun. 128, 211–215 (1996).
[CrossRef]

1994 (2)

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)

1981 (1)

1976 (1)

1969 (1)

1964 (1)

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

Arsenault, H. H.

Casasent, D.

Caulifield, H. J.

Denis, M.

Downie, J. D.

Flannery, D. L.

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

Gaidon, T.

Ph. Réfrégier, F. Goudail, T. Gaidon, “Optimal location of random targets in random background: random Markov fields modelization,” Opt. Commun. 128, 211–215 (1996).
[CrossRef]

Gianino, P. D.

Goudail, F.

Ph. Réfrégier, F. Goudail, T. Gaidon, “Optimal location of random targets in random background: random Markov fields modelization,” Opt. Commun. 128, 211–215 (1996).
[CrossRef]

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

Horner, J. L.

Javidi, B.

Maloney, W. T.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Psaltis, D.

Refregier, P.

Réfrégier, Ph.

Ph. Réfrégier, F. Goudail, T. Gaidon, “Optimal location of random targets in random background: random Markov fields modelization,” Opt. Commun. 128, 211–215 (1996).
[CrossRef]

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

Réfrégier, R.

Refreigier, P.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

VanderLugt, A.

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

Walkup, J. F.

Wang, J.

Willett, P. K.

Zhang, G.

Appl. Opt. (4)

IEEE Trans. Inf. Theory (1)

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

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

Opt. Commun. (1)

Ph. Réfrégier, F. Goudail, T. Gaidon, “Optimal location of random targets in random background: random Markov fields modelization,” Opt. Commun. 128, 211–215 (1996).
[CrossRef]

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)

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (6)

Fig. 1
Fig. 1

Target used in the computer simulation.

Fig. 2
Fig. 2

Measurement of the filters’ output for a target placed in background noise and degraded by multiplicative amplitude noise and additive noise. (a) Peak-to-output-energy ratio (POE) versus multiplicative amplitude noise standard deviation, (b) average (PSR) versus multiplicative amplitude noise standard deviation.

Fig. 3
Fig. 3

(a) Image containing a target in background noise, (b) the input image applied to the filters. The input image is Fig. 3(a) degraded by multiplicative amplitude noise and additive noise, (c) output of the optimum filter, (d) output of the logarithmic matched filter, (e) output of the phase only filter, (f) output of the conventional matched filter.

Fig. 4
Fig. 4

Measurement of the filters’ output for a target placed in background noise and degraded by multiplicative complex noise and additive noise. (a) POE versus multiplicative complex noise standard deviation, (b) average PSR versus multiplicative complex noise standard deviation.

Fig. 5
Fig. 5

(a) Image containing a target in background noise, (b) input image applied to the filters. The input image is Fig. 5(a) degraded by multiplicative complex noise and additive noise, (c) output of the optimum filter, (d) output of the logarithmic matched filter, (e) output of the phase only filter, (f) output of the conventional matched filter.

Fig. 6
Fig. 6

Tests of the filters for a target placed in background noise and deteriorated by multiplicative noise and additive noise. (a) Scene image containing a target in background noise, (b) input image applied to the filters containing the scene image degraded by multiplicative amplitude noise and additive noise, (c) output of the optimum filter, (d) output of the logarithmic matched filter, (e) output of the phase only filter, (f) output of the conventional matched filter.

Tables (2)

Equations (29)

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s1(t, τ)=r(t-τ)+nb(t)[w0(t)-wr(t-τ)],
s(t, τ)=nm(t)s1(t, τ)+na(t)w0(t)={r(t-τ)+nb(t)[w0(t)-wr(t-τ)]}×nm(t)+na(t)w0(t),
POE=|E[y(τ, τ)]|2/E{[y(t, τ)]2¯},
Hopt*(ω)=E[S(ω, τ)exp(jωτ)]/E[|S(ω, τ)|2],
Hopt*(ω)=mmE[S1(ω, τ)exp(jωτ)]+ma|W0(ω)|2/d12πNm(ω)*E[|S1(ω, τ)|2]+12π|W0(ω)|2*Na(ω)+2mmmaRe{W0*(ω)E[S1(ω, τ)]},
Hopt*(ω)=mm[R(ω)+mbW1(ω)]+ma |W0(ω)|2d
12πNm(ω)*|R(ω)+mbW1(ω)|2+12πNb(ω)*W2(ω)-mb2|W1(ω)|×12π|W0(ω)|2*Na(ω)+2mmma|W0(ω)|2×ReR(ω)d+mb1-Wrωd,
d=W00=w0tdt,
W1(ω)=|W0(ω)|2/d-Wr(ω),
W2(ω)=|W0(ω)|2+|Wr(ω)|2-2|W0(ω)|2Rel[Wr(ω)]/d,
Hopt*(ω)=mmES1(ω, τ)exp(jωτ)12πNm(ω)*E|S1(ω, τ)|2=mmR(ω)+mbW1(ω)12πNm(ω)*|R(ω)+mbW1(ω)|2+12πNb(ω)*W2(ω)-mb2|W1(ω)|.
Hopt*(ω)=mmR(ω)/Nm(ω)*|R(ω)|2.
Hopt*(ω)=E[S1(ω, τ)exp(jωτ)]E[|S1(ω, τ)|2]=R(ω)+mbW1(ω)|R(ω)+mbW1(ω)|2+12πW2(ω)*Nb(ω)-mb2|W1(ω)|2.
Hopt*(ω)=R(ω)+mbW1(ω)12πNm(ω)*|R(ω)+mbW1(ω)|2+12πNb(ω)*W2(ω)-mb2|W1(ω)|+12π|W0(ω)|2*Na(ω),
Nb(ω)=2πmb2δ(ω)+σb2,
Na(ω)=σa2,
Nm(ω)=2πδ(ω)+σm2.
nm(t)=1+σm exp[j2πϕ(t)],
y(t, τ)=12πH(ω)S(ω, τ)exp(jωt)dω,
E[y(τ, τ)]=12πH(ω)E[S(ω, τ)exp(jωτ)]dω.
E[[y(t, τ)]2¯]=12πL|H(ω)|2E[|S(ω, τ)|2]dω,
POE=H(ω)E[S(ω, τ)exp(jωτ)]dω22π/L|H(ω)|2E[|S(ω, τ)|2]dω.
POEL2πE[S(ω, τ)exp(jωτ)]E[|S(ω, τ)|2]2dω.
Hopt*(ω)=E[S(ω, τ)exp(jωτ)]E[|S(ω, τ)|2].
s(t)=nm(t)s1(t)+na(t),
ms1=ms-mamm
σs12=(σs2+ms2)-(σa2+ma2)-2mmms1maσm2+mm2-ms12.
mbm˜s-mamm
σb2(σ˜s2+m˜s2)-(σa2+ma2)-2mmmbmaσm2+mm2-mb2.

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