Abstract

A Bayesian approach adapted to practical detection and location tasks of a target in additive Gaussian noise with unknown spectral density is developed and studied. The relevance of this theory is first discussed in comparison with a maximum a posteriori solution that has been recently developed [J. Opt. Soc. Am. A 15, 61 (1998)]. The analysis is first performed without considering a particular prior for the spectral density of the noise. General results of the Bayesian approach are thus provided as well as properties of its first-order development, which corresponds to the so-called nonlinear joint-transform correlation frequently used in optical correlators. It is demonstrated that the kernel of the nonlinear filtering is an increasing function of the sum of the spectral density of the reference object and of the input image. Furthermore, it is shown that a power-law mathematical form of the nonlinear filtering is directly related to assumptions on the asymptotic behavior of the prior density probabilities of the unknown spectral density of the noise. These properties constitute new theoretical results in the context of statistical theory concerning the use of nonlinearities in optical correlators.

© 1999 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. R. Kotynski, F. Goudail, Ph. Réfrégier, “Comparison of the performance of linear and nonlinear filters in the presence of nonergodic noise,” J. Opt. Soc. Am. A 14, 2162–2167 (1997).
    [CrossRef]
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  13. V. Laude, S. Formont, “Bayesian target location in images,” Opt. Eng. 36, 2649–2659 (1997).
    [CrossRef]
  14. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  15. B. Javidi, Ph. Réfrégier, P. Willet, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
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    [PubMed]
  17. Ph. Réfrégier, V. Laude, B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
    [CrossRef] [PubMed]
  18. Ph. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1456 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

1998 (1)

1997 (2)

1996 (1)

1995 (3)

1994 (4)

1993 (2)

1992 (2)

1990 (1)

1989 (1)

1981 (1)

L. Pichon, J.-P. Huignard, “Dynamic joint-Fourier transform correlator by Bragg diffraction in photorefractive bso crystal,” Opt. Commun. 36, 277–280 (1981).
[CrossRef]

1968 (1)

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227–241 (1968).
[CrossRef]

1966 (1)

1964 (1)

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

Ahouzi, E.

Bann, S.

Brenner, K. H.

Campos, J.

Carnicer, A.

Chalasinska-Macukow, K.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Evans, M.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Formont, S.

V. Laude, S. Formont, “Bayesian target location in images,” Opt. Eng. 36, 2649–2659 (1997).
[CrossRef]

Garthwaite, P. H.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Goodman, J. W.

Goudail, F.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Hastings, N.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Huignard, J.-P.

Javidi, B.

Jaynes, E. T.

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227–241 (1968).
[CrossRef]

Jensen, A. S.

Joffre, P.

Jolliffe, I. T.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Jones, B.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Juvells, I.

Kober, V.

Kotynski, R.

Laude, V.

Lohman, G.

Martin-Badosa, E.

Peacock, B.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Pichon, L.

L. Pichon, J.-P. Huignard, “Dynamic joint-Fourier transform correlator by Bragg diffraction in photorefractive bso crystal,” Opt. Commun. 36, 277–280 (1981).
[CrossRef]

Rajbenbach, H.

Rasmussen, E.

Réfrégier, Ph.

Ph. Réfrégier, F. Goudail, “A decision theoretical approach to nonlinear joint-transform correlation,” J. Opt. Soc. Am. A 15, 61–67 (1998).
[CrossRef]

R. Kotynski, F. Goudail, Ph. Réfrégier, “Comparison of the performance of linear and nonlinear filters in the presence of nonergodic noise,” J. Opt. Soc. Am. A 14, 2162–2167 (1997).
[CrossRef]

Ph. Réfrégier, V. Laude, B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
[CrossRef] [PubMed]

F. Goudail, V. Laude, Ph. Réfrégier, “Influence of non-overlapping noise on regularized linear filters for pattern recognition,” Opt. Lett. 20, 2237–2239 (1995).
[CrossRef]

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

Ph. Réfrégier, “Application of the stabilizing functional approach to pattern recognition filters,” J. Opt. Soc. Am. A 11, 1243–1251 (1994).
[CrossRef]

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

Ph. Réfrégier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1456 (1993).
[CrossRef] [PubMed]

H. Rajbenbach, S. Bann, Ph. Réfrégier, P. Joffre, J.-P. Huignard, H. St. Buchkremer, A. S. Jensen, E. Rasmussen, K. H. Brenner, G. Lohman, “Compact photorefractive correlator for robotic applications,” Appl. Opt. 31, 5666–5674 (1992).
[CrossRef] [PubMed]

Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef] [PubMed]

Robert, C. P.

C. P. Robert, The Bayesian Choice—A Decision-Theoretic Motivation (Springer-Verlag, New York, 1996).

St. Buchkremer, H.

Therrien, C. W.

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).

Turon, F.

Vallmitjana, S.

VanderLugt, A.

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

Wang, J.

Weaver, C. S.

Willet, P.

Woodward, P. M.

P. M. Woodward, Probabilités Analyse Fréquentielle, Information, Théorie du Radar (Eyrolles, Paris, 1960).

Yzuel, M. J.

Zhang, G.

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

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

IEEE Trans. Syst. Sci. Cybern. (1)

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227–241 (1968).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

L. Pichon, J.-P. Huignard, “Dynamic joint-Fourier transform correlator by Bragg diffraction in photorefractive bso crystal,” Opt. Commun. 36, 277–280 (1981).
[CrossRef]

Opt. Eng. (1)

V. Laude, S. Formont, “Bayesian target location in images,” Opt. Eng. 36, 2649–2659 (1997).
[CrossRef]

Opt. Lett. (5)

Other (6)

P. M. Woodward, Probabilités Analyse Fréquentielle, Information, Théorie du Radar (Eyrolles, Paris, 1960).

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

C. P. Robert, The Bayesian Choice—A Decision-Theoretic Motivation (Springer-Verlag, New York, 1996).

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

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Equations (71)

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si=rij+ni,
sˆk=rˆkj+nˆk.
Pn(nˆ|Γ)=k=0N-1 12πΓkexp-|nˆk|22Γk.
Pn(sˆ|Γ, j)=k=0N-1 12πΓkexp-12Γk|sˆk-rˆkj|2.
R(je)=Γj0sˆC[je(sˆ), j0]Pn(sˆ|Γ, j0)PΓ(Γ)×Pj(j0)dΓdj0dsˆ,
Γ f(Γ)dΓ=Γ0Γ1ΓN-1f(Γ)dΓ0dΓ1dΓN-1,
sˆ f(sˆ)dsˆ=sˆ0sˆ1sˆN-1f(sˆ)dsˆ0dsˆ1dsˆN-1.
δ(0)=1,δ(i)=0,i0.
P(sˆ, j0)=Γ Pn(sˆ|Γ, j0)PΓ(Γ)Pj(j0)dΓ.
PCL=j0sˆδ(je-j0)P(sˆ, j0)dj0dsˆ.
jeopt=arg minjeR(je).
ρ[je(sˆ)]=ΓjC[je(sˆ), j]P(Γ, j|sˆ)dΓdj.
P(j|sˆ)=ΓP(Γ, j|sˆ)dΓ.
lMAP(j, Γ)=log[P(sˆ|Γ, j)Pp(Γ)Pj(j)].
lMAP(j)=lMAP[j, ΓMAP(j)].
jeopt=arg maxjP(j|sˆ).
ρ[je(sˆ)]=jC[je(sˆ), j]P(j|sˆ)dj.
Pn(sˆ|Γ, j)=k=0N-1 αk2πexp-αk2Δkj.
l(sˆ|j)=ln[P(sˆ|j)]=k=0N-1fk(Δkj),
fk(z)=ln0 α2πexp-α z2πk(α)dα.
Δkj=|sˆk-rˆkj|2,
Ukj=(sˆk)*rˆkj+(rˆkj)*sˆk,
Dk=|sˆk|2+|rˆk|2,
|nˆk|2|rˆk|2;
Dk|nˆk|2,
Ukrˆkjnˆk*(rˆkj)*nˆk.
DkUk.
l1(sˆ|j)=k=0N-1[fk(|sˆk|2+|rˆk|2)+Fˆk exp(-i2πjk)+(Fˆk)*exp(i2πjk)],
Fˆk=-fk(|sˆk|2+|rˆk|2)(sˆk)*rˆk.
l1(sˆ|j)=K+k=0N-12Fˆk exp(-i2πjk)=K+2F(j),
Fˆk=(sˆk)*rˆkGk(|sˆk|2+|rˆk|2),
Pk(Γk)=A(Γk)-m exp-Γkβaβ,
Gk(z)(z)1/(1+β);
Gk(z)z;
Gk(z)2Γk,max.
πk(α)=A exp(λα1/2-μα3/2),
Pk(Γk)=A(Γk)2expλΓk1/2-μΓk3/2.
Gk(z)>aifza,
Gk(z)zifza.
Fˆk=(sˆk)*rˆka+|sˆk|2+|rˆk|2,
πk(α)=a exp(-aα).
Pk(Γk)=a(Γk)2exp-aΓk,Γk0,a>0.
Ik(β; z)=0αβπk(α)exp-α z2dα.
ddzIk(β; z)=-12Ik(β+1; z).
fk(z)=ln[Ik(1/2; z)]-12ln(2π).
ddzfk(z)=-Ik(3/2; z)2Ik(1/2; z).
d2(dz)2fk(z)=14Ik(1/2; z)Ik(5/2; z)-[Ik(3/2; z)]2[Ik(1/2; z)]2.
Ik(3/2; z)=0α1/4 exp-α z4×a5/4 exp-α z4πk(α)dα.
[Ik(3/2; z)]2Ik(1/2; z)Ik(5/2; z),
fk(z)=-Ik(3/2; z)2Ik(1/2; z),
[Gk(z)]-1=-fk(z),
π(αk)=A(αk)m-2 exp-1αkβaβ,
I(n; z)=0A(α)n+m-2 exp-1(aα)β-αz2dα.
γ=1/(β+1),μ=βγ,
g(x)=1(ax)β+x2.
I(n; z)=A0(α)n+m-2 exp[-zμg(zγα)]dα.
I(n; z)=Azγ(m+n-1)J(n; z),
J(n; z)=0xn+m-2 exp[-zμg(x)]dx.
J(n; z)x0n+m-20 exp[-zμg(x)]dx,
I(n; z)Ax0n+m-2zγ(m+n-1)Jμ(z).
I(1/2; z)I(3/2; z)zγx0
G(z)(aβz)1/(1+β).
I(n; z)=0A(α)n+m-2 exp-αz2dα,
I(n; z)=Azn+m-1Bn+m-2,
Bn+m-2=0xn+m-2 exp-x2dx.
G(z)z.
I(n; z)=αMA(α)n+m-2 exp-αz2dα.
I(n; z)A(αM)n+m-2C,
C=αM exp-αz2dx.
I(12; z)I(32; z)1αM
G(z)2Γk,max.

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