Abstract

We propose a new decision theory approach adapted to practical target detection and location tasks in which the spectral density of Gaussian additive noise is unknown. We determine the maximum likelihood and the maximum a posteriori solutions for that problem. We demonstrate that the nonlinear joint-transform correlation, which is frequently used in optical correlators, can be considered an approximation of these optimal processors. This new result constitutes a theoretical support in the context of detection theory for the use of nonlinearities in optical correlators.

© 1998 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. 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]
  4. L. Pichon, J.-P. Huignard, “Dynamic joint-Fourier transform correlator by Bragg diffraction in photorefractive BSO crystal,” Opt. Commun. 36, 277–280 (1981).
    [CrossRef]
  5. F. Turon, E. Ahouzi, J. Campos, K. Chalasinska-Macukow, M. J. Yzuel, “Nonlinearity effects in the pure phase correlation method in multiobject scenes,” Appl. Opt. 33, 2188–2191 (1994).
    [CrossRef] [PubMed]
  6. S. Vallmitjana, A. Carnicer, E. Martin-Badosa, I. Juvells, “Nonlinear filtering in object and Fourier space in a joint-transform optical correlator: comparison and experimental realization,” Appl. Opt. 34, 3942–3949 (1995).
    [CrossRef] [PubMed]
  7. 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]
  8. A. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  9. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  10. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  11. 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]
  12. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  13. M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993).

1995 (2)

1994 (2)

1992 (1)

1990 (2)

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]

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.

Buchkremer, H. St.

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).

Goodman, J. W.

Hart, P. E.

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

Hassebrook, L.

Hastings, N.

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

Huignard, J.-P.

Javidi, B.

Jensen, A. S.

Joffre, P.

Juvells, I.

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.

Turon, F.

Vallmitjana, S.

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Weaver, C. S.

Yzuel, M. J.

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

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

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. Lett. (2)

Other (2)

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

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

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Figures (3)

Fig. 1
Fig. 1

Plot of the function P(x)=(α2/x2)exp(-α2/x) for α =2.

Fig. 2
Fig. 2

Reference object used in the simulations.

Fig. 3
Fig. 3

(a) Scene containing the object represented in Fig. 2 corrupted with additive white noise. (b) Scene containing the object represented in Fig. 2 corrupted with additive correlated noise with a power spectrum in 1/f2. (c) Result of processing (a) by using the optimal MAP estimate [Eq. (35)]. (d) Result of processing (b) by using the optimal MAP estimate [Eq. (35)]. (e) Result of processing (a) by using the first-order approximation of the optimal processor [Eq. (36)]. (f) Result of processing (b) by using the first-order approximation of the optimal processor [Eq. (36)]. Note that (c)–(f) are plots of the maximum of each line of the correlation plane. The reference used is Fig. 2.

Equations (63)

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si=rij+ni,
sˆk=rˆkj+nˆk.
P(nˆ|Γ, j)=k=0N-112πΓkexp-|nˆk|22Γk.
P(sˆ|Γ, j)=k=0N-112πΓkexp-12Γk|sˆk-rˆkj|2.
P(sˆ|Γ, j)Γk=0.
Γk=|sˆk-rˆkj|2.
lML(j)=-k=0N-112log(|sˆk-rˆkj|2)+K,
Δkj=|sˆk-rˆkj|2,
Ukj=(sˆk)*rˆkj+(rˆkj)*sˆk,
Dk=|sˆk|2+|rˆk|2,
lML(j)=K-12k=0N-1log(Dk)+log1-UkjDk.
l˜ML(j)=K+12k=0N-1UkjDk,
Ukj=(sˆk)*rˆk exp(-i2πjk)+(rˆk)*sˆk exp(i2πjk),
l˜ML(j)=K+12k=0N-1(sˆk)*rˆk|sˆk|2+|rˆk|2exp(-i2πjk)+(rˆk)*sˆk|sˆk|2+|rˆk|2exp(i2πjk).
l˜ML(j)=K+12k=0N-1[Fˆk exp(-i2πjk)+Fˆk* exp(i2πjk)]
Fˆk=(sˆk)*rˆk|sˆk|2+|rˆk|2.
l˜ML(j)=K+k=0N-1Fˆk exp(-i2πjk)=K+F(j),
Cˆk=(sˆk)*rˆka+μ|sˆk|2+(1-μ)|rˆk|2.
P(Γ, j|sˆ)=P(sˆ|Γ, j)P(Γ, j)P(sˆ),
lMAP(j, Γ)=log[P(sˆ|Γ, j)Pp(Γ)].
lMAP(j)=lMAP[j, ΓMAP(j)].
jMAP=argmaxj lMAP[j, ΓMAP(j)],
ΓMAP(j)=G(Δj),
G(y)=argmaxx M(x, y),
M[x, y]=k=0N-1-yk2xk-12log(xk)+log[Pp(x)].
lMAP(j)l˜MAP(j)=KM+k=0N-1Ukj2Gk(D),
Fˆk=(sˆk)*rˆkGk(|sˆ0|2+|rˆ0|2, |sˆ1|2+|rˆ1|2, , |sˆN-1|2+|rˆN-1|2).
ΓkMAP(j)=Gk(Δkj),
Gk(y)=argmaxx mk(x, y),
mk(x, y)=-y2x-12log(x)+log[Pk(x)].
lMAP(j)l˜MAP(j)=KM+k=0N-1Ukj2Gk(Dk),
Fˆk=(sˆk)*rˆkGk(|sˆk|2+|rˆk|2).
hˆk=rˆkGk(|sˆk|2+|rˆk|2).
Fˆk=(sˆk)*rˆkG(|sˆk|2+|rˆk|2).
Fˆk=(sˆk)*rˆka+|sˆk|2+|rˆk|2,
P(Γk)=α2(Γk)2exp-α2Γk,Γk0.
lMAP(j)=-12k=0N-1 log(2α2+|sˆk-rˆkj|2)+K,
Fˆk=(sˆk)*rˆk2α2+|sˆk|2+|rˆk|2.
lMAP(j, Γ)=KM+k=0N-1-Δkj2Γk-1/2 log(Γk)+log[Pp(Γ0, Γ1, , ΓN-1)],
M(x, y)=k=0N-1-yk2xk-12log(xk)+log[Pp(x)],
M(x, y)xk=M(xk)(x, y),M(x, y)yk=M(yk)(x, y).
M(xk)[ΓMAP(j), Δj]=0.
ΓMAP(j)=G(Δj).
M(xk)[G(y), y]=0
lMAP(j)=lMAP[j, ΓMAP(j)],
lMAP(j)=KM+M[G(Δj), Δj].
M[G(Δj), Δj]M[G(D), D]+k=0N-1dM[G(D), D]dDk(-Ukj).
dM[G(D), D]dDk0=k=0N-1M(xk)[G(D), D]dGk(D)dDk0+M(yk0)[G(D), D].
M(yk)[G(D), D]=-12Gk(D),
M[G(Δj), Δj]M[G(D), D]+k=0N-1Ukj2Gk(D).
δlMAP(j)=k=0N-1[sˆk]*rˆkGk(|sˆ0|2+|rˆ0|2, |sˆ1|2+|rˆ1|2,,|sˆN-1|2+|rˆN-1|2)exp(-i2πjk),
Fˆk=[sˆk]*rˆkGk(|sˆ0|2+|rˆ0|2, |sˆ1|2+|rˆ1|2, , |sˆN-1|2+|rˆN-1|2).
log[Pp(Γ0, Γ1, , ΓN-1)]=k=0N-1 log[Pk(Γk)].
lMAP(j, Γ)=KM+k=0N-1-Δkj2Γk-12log(Γk)+log[Pk(Γk)],
mk(x, y)=-y2x-1/2 log(x)+log[Pk(x)],
mk(x, y)x=(mk)x(x, y),mk(x, y)y=(mk)y(x, y).
(mk)x[ΓkMAP(j), Δkj]=0.
ΓkMAP(j)=Gk(Δkj).
G(Δj)k=Gk(Δkj).
lMAP(j)=KM+k=0N-1mk[Gk(Δkj), Δkj].
l˜MAP(j)=KM+k=0N-1mk[Gk(Dk), Dk]+Ukj2Gk[Dk].
δlMAP(j)=k=0N-1(sˆk)*rˆkGk(|sˆk|2+|rˆk|2)exp(-i2πjk),
Fˆk=(sˆk)*rˆkGk(|sˆk|2+|rˆk|2).

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