Abstract

We design receivers to detect a known pattern or a reference signal in the presence of very general and non-Gaussian types of noise. Three sources of input-noise degradation are considered: additive, multiplicative, and disjoint background. The detection process involves two steps: (1) estimation of the relevant noise parameters within the framework of hypothesis testing and (2) maximizing a certain metric that measures the likelihood of the target being at a given location. The parameter estimation portion is carried out by moment-matching techniques. Because of the number of unknown parameters and the fact that various types of input-noise processes are non-Gaussian, the methods that are used to estimate these parameters differ from the standard methods of maximizing the likelihood function. To verify the existence of the target at a certain location, we use lp-norm metric for p0 to measure the likelihood of the target being present at the location of interest. Computer simulations are used to show that for the images tested here, the receivers designed herein perform better than some 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]
  2. N. Towghi, B. Javidi, “Optimum receivers for pattern recognition in the presence of Gaussian noise with unknown statistics,” J. Opt. Soc. Am. A 18, 1844–1852 (2001).
    [CrossRef]
  3. S. Venkatesh, D. Psaltis, “Binary filters for pattern-classification,” IEEE Trans. Acoust. Speech Signal Process. 37, 604–611 (1989).
    [CrossRef]
  4. M. Fan, J. W. Goodman, “Optimal binary phase only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef]
  5. D. Casasent, “Unified synthetic function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  6. D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  7. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  8. D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  9. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef]
  10. 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]
  11. Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [CrossRef]
  12. P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
    [CrossRef]
  13. S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
    [CrossRef]
  14. N. M. Nasrabadi, W. Li, “Object recognition by a Hopefield neural network,” IEEE Trans. Syst. Man Cybern. 21, 1–13 (1991).
    [CrossRef]
  15. 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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  16. 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]
  17. 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).
    [CrossRef]
  18. 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]

2001

1998

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]

P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
[CrossRef]

1997

S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
[CrossRef]

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).
[CrossRef]

1996

1993

1992

1991

N. M. Nasrabadi, W. Li, “Object recognition by a Hopefield neural network,” IEEE Trans. Syst. Man Cybern. 21, 1–13 (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, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

1990

1989

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

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

1988

1984

1976

Casasent, D.

Fan, M.

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.

Goodman, J. W.

Goudail, F.

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.

Li, J.

Li, W.

N. M. Nasrabadi, W. Li, “Object recognition by a Hopefield neural network,” IEEE Trans. Syst. Man Cybern. 21, 1–13 (1991).
[CrossRef]

Nasrabadi, N.

P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
[CrossRef]

S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
[CrossRef]

Nasrabadi, N. M.

N. M. Nasrabadi, W. Li, “Object recognition by a Hopefield neural network,” IEEE Trans. Syst. Man Cybern. 21, 1–13 (1991).
[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]

Réfrégier, Ph.

Scott, P. D.

P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
[CrossRef]

S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
[CrossRef]

Towghi, N.

Venkatesh, S.

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

Wang, J.

Willet, P.

Willett, P.

Yang, S. S.

S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
[CrossRef]

Young, S.

P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

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

IEEE Trans. Image Process.

P. D. Scott, S. Young, N. Nasrabadi, “Foveal automatic target recognition using a multiresolution neural network,” IEEE Trans. Image Process. 7, 1122–1135 (1998).
[CrossRef]

S. S. Yang, P. D. Scott, N. Nasrabadi, “Object recognition using a multilayer Hopefield neural net,” IEEE Trans. Image Process. 6, 357–372 (1997).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

N. M. Nasrabadi, W. Li, “Object recognition by a Hopefield neural network,” IEEE Trans. Syst. Man Cybern. 21, 1–13 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Lett.

Proc. IEEE

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

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Figures (6)

Fig. 1
Fig. 1

Scene containing two true targets (tanks) and a false target (helicopter).

Fig. 2
Fig. 2

Performance of the p-norm receivers [Eq. (29)] and the general receiver [Eq. (30)]. 2(a) Noisy scene with additive noise that is white and non-Gaussian with mean 0.2 and standard deviation 0.1039, disjoint background noise that is colored non-Gaussian with mean 0.268 and standard deviation 0.098, and bandwidth of 50×50 pixels. There is multiplicative noise on the center target that is white non-Gaussian with mean 0.8 and standard deviation 0.226. There is no multiplicative noise on the true target in the upper left-hand corner and on the false target. 2(b) Output of the general receiver (p=0) [Eq. (30)]. 2(c) Output of the p-norm receiver when p=0.25 [Eq. (29)]. 2(d) Output of the p-norm receiver when p=0.5 [Eq. (29)]. 2(e) Output of the p-norm receiver when p=1 [Eq. (29)]. 2(f) Output of the lp -norm receiver when p=2 [Eq. (29)].

Fig. 3
Fig. 3

Performance of the p-norm receivers [Eq. (29)] and the general receiver [Eq. (30)]. 3(a) Noisy scene with additive noise that is white Gaussian with mean 0 and standard deviation 0.2, disjoint background noise that is colored Gaussian with mean 0 and standard deviation 0.2, and bandwidth of 50×50 pixels. There is multiplicative noise on the center target that is white Gaussian with mean 0.82 and standard deviation 0.4. The true target in the upper left-hand corner is multiplied by an illumination constant of 0.95. There is no multiplicative noise on the false target. 3(b) Output of the general receiver (p=0) [Eq. (30)]. 3(c) Output of the p-norm receiver when p=0.25 [Eq. (29)]. 3(d) Output of the p-norm receiver when p=0.5 [Eq. (29)]. 3(e) Output of the p-norm receiver when p=1 [Eq. (29)]. 3(f) Output of the p-norm receiver when p=2 [Eq. (29)].

Fig. 4
Fig. 4

Performance of the p-norm receivers [Eq. (29)] and general receiver [Eq. (30)]. 4(a) Noisy scene with additive noise that is white and uniformly distributed on [0.0,0.25] with mean 0.125 and standard deviation 0.072, disjoint background noise that is colored Gaussian with mean 0.7 and standard deviation 1.2, and bandwidth of 50×50 pixels. There is multiplicative noise on the center target that is white non-Gaussian with mean 1.1928 and standard deviation 0.447. There is no multiplicative noise on the true target in the upper left-hand corner. The false target is multiplied by an illumination constant of 0.95. 4(b) Output of the general receiver (p=0) [Eq. (30)]. 4(c) Output of the p-norm receiver when p=0.25 [Eq. (29)]. 4(d) Output of the p-norm receiver when p=0.5 [Eq. (29)]. 4(e) Output of the p-norm receiver when p=1 [Eq. (29)].

Fig. 5
Fig. 5

Performance of the p-norm receivers [Eq. (29)] and general receiver [Eq. (30)]. 5(a) Noisy scene with additive noise that is white Gaussian with mean 0 and standard deviation 0.2, disjoint background noise that is colored Gaussian with mean 0 and standard deviation 0.2, and bandwidth of 50×50 pixels. There is multiplicative noise on the center target that is white Gaussian with mean 0.8 and standard deviation 0.2. The true target in the upper left-hand corner is multiplied by an illumination constant of 0.95. There is no multiplicative noise on the false target. 5(b) Output of the general receiver (p=0) [Eq. (30)]. 5(c) Output of the p-norm receiver when p=0.25 [Eq. (29)]. 5(d) Output of the p-norm receiver when p=0.5 [Eq. (29)]. 5(e) Output of the p-norm receiver when p=1 [Eq. (29)].

Fig. 6
Fig. 6

Performance of the p-norm receivers [Eq. (29)] and general receiver [Eq. (30)]. 6(a) A realistic scene with additive noise that is white and square of zero mean Gaussian with standard deviation of 0.3. 6(b) Output of the general receiver (p=0) [Eq. (30)]. 6(c) Output of the p-norm receiver when p=1 [Eq. (29)]. 6(d) Output of the p-norm receiver when p=3 [Eq. (29)]. 6(e) Output of the p-norm receiver when p=6 [Eq. (29)].

Equations (36)

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s(t)=g[r(t-tj)w(t-tj)],
s(t)=nr(t-tj)r(t-tj)w(t-tj)+nb(t)×[1-w(t-tj)]+nd(t).
s(i)=[nr(i-j)r(i-j)+nd(i)]w(i-j)+[nb(i)+nd(i)][1-w(i-j)].
gj(i)=[nr(i-j)r(i-j)+nd(i)]w(i-j)+[nb(i)+nd(i)][1-w(i-j)],
ej=Gj-E[Gj|S].
E|ej(i)|2=E[|Gj(i)-s(i)|2].
Cp=1J j|cj|p1/p.
C0=exp1J j=0J-1 log(|cj|).
λj=i=1M{E[|Gj(i)-s(i)|2]}p,
λj=i=1M log{E[|Gj(i)-s(i)|2]}.
s(i)=[nr(i-j)r(i-j)+nd(i)]w(i-j)+[nb(i)+nd(i)][1-w(i-j)]=Gj(i),
n1(i)=nr(i)-1.
s(i)=[n1(i-j)r(i-j)+r(i-j)+nd(i)]w(i-j)+[nb(i)+nd(i)][1-w(i-j)].
SWj={s(i)}[i:w(i-j)=1],
SWj2={s2(i)}[i:w(i-j)=1],
XWj={r(i-j)+r(i-j)w(i-j)n1+nd}[i:w(i-j)=1],
E(XW)j={r(i-j)+r(i-j)w(i-j)m1+md}[i:w(i-j)=1],
E[(XWj)2]={[r(i-j)+m1r(i-j)+md]2+σ12r2(i-j)+σd2}[i:w(i-j)=1],
r22r¯r¯nwmˆ1(j)mˆd(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)0r(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).
E[(XWj)2]={[r(i-j)+mˆ1(j)r(i-j)+mˆ(j)]2+σ12r2(i-j)+σd2}[i:w(i-j)=1].
r44r22r22nwσˆ12(j)σˆd2(j)=A(j)B(j),
A(j)=i:w(i-j)=1[s˜j(i)-mˆ1(j)r(i-j)+mˆd(j)]2r2(i-j),
B(j)=i:w(i-j)=1[s˜j(i)-mˆ1(j)r(i-j)+mˆd(j)]2.
mˆ(j)=1n0 i:w(i-j)=0s(i),
σˆ2(j)=1n0 i:w(i-j)=0[s(i)-mˆ(j)]2,
λj=-n0[σˆ2(j)]p-i:w(i-j)=1|r2(i-j)w(i-j)σˆ12(j)+σˆd2(j)|p,
λj=-n0 log[σˆ2(j)]-i:w(i-j)=1 log[r2(i-j)σˆ12(j)+σˆd2(j)].
Yb=0.06X12+0.123X2+0.07,
Ym=0.64+0.16X32,
Ya=0.2+0.06(X42-1)+0.123X5,
Ym=0.5+0.2X1+0.83X2,
Ya=0.25X,

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