Abstract

An optimum receiver is designed based on multiple-alternative hypothesis testing with unknown parameters for detecting and locating a noisy or a noise-free target in scene noise. For a noise-free target with unknown parameters, the optimum receiver solution can be described by using operations that are independent of the scene noise statistics. Since, in practice, it is not possible to have a completely noise-free target, we examine how the performance of the optimum receiver designed for a noise-free target is affected, in terms of error probability, when there is some additive overlapping noise on the target. We determine the false-alarm and miss error probabilities of the optimum receiver for this case. The error probability of the optimum receiver will be compared with that of the matched filter.

© 1997 Optical Society of America

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References

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  5. B. V. K. Vijaya Kumar, J. D. Brasher, “Relationship between maximizing the signal-to-noise ratio and minimizing the classification error probability for correlation filters,” Opt. Lett. 17, 940–942 (1992).
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  6. D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  7. B. Javidi, J. Wang, “Design of filters to detect a noisy target in nonoverlapping background noise,” J. Opt. Soc. Am. A 11, 2604–2612 (1994).
    [CrossRef]
  8. B. Javidi, Ph. Refregier, P. Willett, “Optimum receiverdesign for pattern recognition with non-overlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
  9. 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).
    [CrossRef] [PubMed]
  10. B. Javidi, A. Fazlollahi, P. Willett, Ph. Refregier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 8, 3858–3868 (1995).
    [CrossRef]
  11. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).
  12. H. Vincent Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1994).
  13. S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982).
  14. J. E. Mazo, J. Salz, “Probability of error for quadratic detectors,” Bell Syst. Tech. J. 44, 2165–2186 (1965).
    [CrossRef]
  15. G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE 46, 1603–1619 (1958).
    [CrossRef]
  16. Y. Barshalom, X. Li, Estimation and Tracking: Principles, Techniques, and Softwares (Artech House, Boston, 1993).

1995 (1)

B. Javidi, A. Fazlollahi, P. Willett, Ph. Refregier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 8, 3858–3868 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (2)

1991 (2)

1990 (2)

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

1989 (1)

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

1965 (1)

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

Barshalom, Y.

Y. Barshalom, X. Li, Estimation and Tracking: Principles, Techniques, and Softwares (Artech House, Boston, 1993).

Brasher, J. D.

Dickey, F. M.

Fazlollahi, A.

B. Javidi, A. Fazlollahi, P. Willett, Ph. Refregier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 8, 3858–3868 (1995).
[CrossRef]

Flannery, D. L.

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

Fleisher, M.

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

Horner, J. L.

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

Javidi, B.

Li, X.

Y. Barshalom, X. Li, Estimation and Tracking: Principles, Techniques, and Softwares (Artech House, Boston, 1993).

Mahlab, U.

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

Mazo, J. E.

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

Refregier, Ph.

Romero, L.

Salz, J.

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

Searle, S. R.

S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982).

Shamir, J.

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

Turin, G.

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

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

Vijaya Kumar, B. V. K.

Vincent Poor, H.

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

Wang, J.

Willett, P.

B. Javidi, A. Fazlollahi, P. Willett, Ph. Refregier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 8, 3858–3868 (1995).
[CrossRef]

B. Javidi, Ph. Refregier, P. Willett, “Optimum receiverdesign for pattern recognition with non-overlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
[CrossRef] [PubMed]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

Opt. Lett. (4)

Proc. IEEE (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” 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 (4)

Y. Barshalom, X. Li, Estimation and Tracking: Principles, Techniques, and Softwares (Artech House, Boston, 1993).

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

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

S. R. Searle, Matrix Algebra Useful for Statistics (Wiley, New York, 1982).

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Figures (6)

Fig. 1
Fig. 1

Error probability Pe of the optimum receiver (with equal priors π0=π1=0.5) versus the threshold τ for the example of Subsection 7.A.1.

Fig. 2
Fig. 2

Error probability Pe of the optimum receiver versus the target window size W for the example of Subsection 7.A.1.

Fig. 3
Fig. 3

(a) Reference (airplane) image used in the example of Subsection 7.B. (b) Reference image in a nonoverlapping colored background with mean mB=0.5 and standard deviation σB =0.2. The maximum value of the reference is normalized to 1. (c) Image in (b) with overlapping white noise added to the entire scene. The additive white-noise standard deviation is σd =0.1.

Fig. 4
Fig. 4

(a) Error probability Pe of the optimum receiver versus the background noise standard deviation σB for the example of Subsection 7.B. The additive noise standard deviation σd is a parameter. The mean value of the background noise is mB =0.5. (b) Probability-density function (PDF) of q under different hypotheses.

Fig. 5
Fig. 5

Error probability Pe of the optimum receiver versus the background noise mean value mB for the example of Subsection 7.B. The additive noise standard deviation σd is a parameter.

Fig. 6
Fig. 6

(a) Error probability Pe of the optimum receiver versus the background noise bandwidth B for the example of Subsection 7.B. The background noise mean and standard deviation are mB=0.5 and σB=0.2, respectively. The additive noise standard deviation is σd=0.2. (b) std(qc|H0)/E(qc|H0) versus the background noise bandwidth B.

Tables (1)

Tables Icon

Table 1 Results of the Example of Subsection 7.B for the Images Shown in Figs. 3(a) and 3(c), Where the Background Noise is Nonzero Mean and Colored

Equations (49)

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q(tj)=-i=1mr2(ti)w(ti-tj)-i=1mr(ti)s(ti-tj)2s,
q(tj)=-i=jj+W-1r2(ti)-(1/s)i=jj+W-1r(ti)s(ti-tj)2,
q=q(tj)=-RTR-1s(RTS)2=RTQR,
Q=1sSST-IW×W
H0:R=NB+Nd,
H1:R=aS+Nd,
E(q|H1)=-σd2(W-1),
var(q|H1)=2σd4(W-1),
E(q|H0)=-(σB2+σd2)(W-1),
var(q|H0)=2(σB2+σd2)2(W-1).
E(qc|H0)=E(RcTQRc|H0)=E[trace(RcTQRc)|H0],
E(qc|H0)=trace[E(RcRcTQ)|H0]=trace[(M0+R¯cR¯cT)Q].
var(qc|H0)=2 trace[(QM0)2]+4R¯cTQM0R¯c.
PF=pr(q>τ|H0),
PM=pr{q<τ|H1},
Φq(ω)=E[exp(iωq)],ω,
Φq(ω)=E[exp(iωq)]=exp12[R¯TM-1(IW×W-2iωMQ)-1R¯-R¯TM-1R¯][det(IW×W-2iωMQ)]1/2,
PF=12πτ0- [exp(-iωq)][Φq(ω)|H0]dωdq,
PM=12π-τ- exp[-iωq][Φq(ω)|H1]dωdq.
Pe=π0PF+π1PM.
PFexp[-sτ+μq,0(s)],
PMexp[-tτ+μq,1(t)]
μq,0(s)=log(E[exp(sq)|H0]),
μq,1(t)=log(E[exp(tq)|H1]).
PMexp{-tτ-0.5 log[det(IW×W-2tM1Q)]},
PMexp-tτ-0. 5 logk=1W(1-2tλ1k).
k=1W λ1k1-2t0λ1k=τ.
t0=τ-(W-1)λ12τλ1,λ1=-σd2.
E(q|H1)=-σd2(W-1)=-12.25,
E(q|H0)=-(σB2+σd2)(W-1)=-189.2,
std(q|H1)=[2σd4(W-1)]1/2=2.475,
std(q|H0)=[2(σB2+σd2)2(W-1)]1/2=38.22.
PFMF=12erfcτMFσj2s,
PMMF=12erfcs-τMFσd2s,
erfc(x)=2πxexp(-t2)dt.
PeMF=14erfcs-τMFσd2s+14erfcτMFσj2s
τMFoptimum=sσj2-{s2σj2-σB2[s2σj2-sσj2σd2 log(σj2/σd2)]}1/2σB2.
E(qc|H0)=trace(M0Q)=-172.55,
std(qc|H0)={2 trace[(QM0)2]}1/2=72.86.
H(α, β)=exp-ln 22α2Bα2+β2Bβ2,
RnB(j)=A exp-Bα24 ln 22πj2562,
J=aSTHaS,
H=M-1(I-2tMQ)-1-M-1=(M-M2tQM)-1-M-1.
2tQ=DDT.
H=[M-(MD)I(MD)T]-1-M-1.
(A+BCBT)-1=A-1-A-1B(BTA-1B+C-1)-1BTA-1,
H=M-1-M-1MD(DTMTM-1MD-I)-1×DTMTM-1-M-1=D(I-DTMD)-1DT.
H=D[I-DT(DDT+M-1)-1D]DT=2tQ-(4t2)Q(2tQ+M-1)-1Q.
J=aSTHaS=2ta2STQS-(4a2t2)STQ(2tQ+M-1)-1QS=0.

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