Abstract

We present a new group of processors, optimal in a maximum-likelihood sense, for target location in images with a discrete number of gray levels. The discrete gray-level distribution can be of any arbitrary form. We compare the performance of the processor derived for five discrete levels with the performance of a processor derived for a continuous Gaussian distribution and show that there are cases when only the processor derived for discrete levels will exhibit satisfactory performance. We give an explanation of this difference based on moment analysis and show how the correlation orders are related to statistical moments of the input scene.

© 2000 Optical Society of America

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References

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  1. P. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  2. P. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  3. A. Mahalanobis, B. V. K. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  4. B. V. K. Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  5. B. Javidi, J. Wang, “Optimum filter for detecting a target in multiplicative noise and additive noise,” J. Opt. Soc. Am. A 14, 836–844 (1997).
    [CrossRef]
  6. B. Javidi, J. Wang, “Optimum distortion-invariant filter for detecting a noisy distorted target in nonoverlapping background noise,” J. Opt. Soc. Am. A 12, 2604–2614 (1995).
    [CrossRef]
  7. B. Javidi, A. H. Fazlollahi, P. Willet, P. Réfrégier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 34, 3858–3868 (1995).
    [CrossRef] [PubMed]
  8. B. Javidi, P. Réfrégier, P. Willet, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
  9. F. Guerault, P. Réfrégier, “Unified statistically independent region processor for deterministic and fluctuating targets in nonoverlapping background,” Opt. Lett. 23, 412–414 (1998).
    [CrossRef]
  10. F. Goudail, P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
    [CrossRef] [PubMed]
  11. F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
    [CrossRef]
  12. F. Guerault, P. Réfrégier, “Optimal χ2 filtering method and application to targets and backgrounds with random correlated gray levels,” Opt. Lett. 22, 630–632 (1997).
    [CrossRef]
  13. P. Réfrégier, O. Germain, T. Gaidon, “Optimal snake segmentation of target and background with independent gamma density probabilities, application to speckled and preprocessed images,” Opt. Commun. 137, 382–388 (1997).
    [CrossRef]
  14. H. Sjöberg, F. Goudail, P. Réfrégier, “Optimal algorithms for target location in nonhomogeneous images,” J. Opt. Soc. Am. A 15, 1–100 (1998).
  15. 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]
  16. K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
    [CrossRef]
  17. A. Mahalanobis, B. V. K. Kumar, “Polynomial filters for higher order correlation and multi-input information fusion,” in Optoelectronic Information Processing, Press Monograph PM54, P. Réfrégier, K. Javidi, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 221–232.

1998

1997

1996

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

F. Goudail, P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

1995

1993

1991

1990

1987

1986

Casasent, D.

Fazlollahi, A. H.

Gaidon, T.

P. Réfrégier, O. Germain, T. Gaidon, “Optimal snake segmentation of target and background with independent gamma density probabilities, application to speckled and preprocessed images,” Opt. Commun. 137, 382–388 (1997).
[CrossRef]

Garvin, C. G.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Germain, O.

P. Réfrégier, O. Germain, T. Gaidon, “Optimal snake segmentation of target and background with independent gamma density probabilities, application to speckled and preprocessed images,” Opt. Commun. 137, 382–388 (1997).
[CrossRef]

Goudail, F.

Guerault, F.

Javidi, B.

Jones, B. K.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Kang, K.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Kirsch, J. C.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Kumar, B. V. K.

A. Mahalanobis, B. V. K. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. V. K. Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
[CrossRef]

A. Mahalanobis, B. V. K. Kumar, “Polynomial filters for higher order correlation and multi-input information fusion,” in Optoelectronic Information Processing, Press Monograph PM54, P. Réfrégier, K. Javidi, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 221–232.

Li, J.

Mahalanobis, A.

A. Mahalanobis, B. V. K. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. Kumar, “Polynomial filters for higher order correlation and multi-input information fusion,” in Optoelectronic Information Processing, Press Monograph PM54, P. Réfrégier, K. Javidi, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 221–232.

Powell, J. S.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Réfrégier, P.

H. Sjöberg, F. Goudail, P. Réfrégier, “Optimal algorithms for target location in nonhomogeneous images,” J. Opt. Soc. Am. A 15, 1–100 (1998).

F. Guerault, P. Réfrégier, “Unified statistically independent region processor for deterministic and fluctuating targets in nonoverlapping background,” Opt. Lett. 23, 412–414 (1998).
[CrossRef]

P. Réfrégier, O. Germain, T. Gaidon, “Optimal snake segmentation of target and background with independent gamma density probabilities, application to speckled and preprocessed images,” Opt. Commun. 137, 382–388 (1997).
[CrossRef]

F. Guerault, P. Réfrégier, “Optimal χ2 filtering method and application to targets and backgrounds with random correlated gray levels,” Opt. Lett. 22, 630–632 (1997).
[CrossRef]

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

F. Goudail, P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

B. Javidi, A. H. Fazlollahi, P. Willet, P. Réfrégier, “Performance of an optimum receiver designed for pattern recognition with nonoverlapping target and scene noise,” Appl. Opt. 34, 3858–3868 (1995).
[CrossRef] [PubMed]

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

P. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

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

Sjöberg, H.

Stack, R. D.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Towghi, N.

Trezza, J. A.

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

Wang, J.

Willet, P.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

P. Réfrégier, O. Germain, T. Gaidon, “Optimal snake segmentation of target and background with independent gamma density probabilities, application to speckled and preprocessed images,” Opt. Commun. 137, 382–388 (1997).
[CrossRef]

Opt. Lett.

Other

K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A. Trezza, J. C. Kirsch, B. K. Jones, “Optical image correlation using high-speed multiple quantum well spatial light modulators,” in Optical Pattern Recognition X, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3715, 97–107 (1999).
[CrossRef]

A. Mahalanobis, B. V. K. Kumar, “Polynomial filters for higher order correlation and multi-input information fusion,” in Optoelectronic Information Processing, Press Monograph PM54, P. Réfrégier, K. Javidi, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 221–232.

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Figures (9)

Fig. 1
Fig. 1

Probability of correct location versus varying values of μ. α=0.3.

Fig. 2
Fig. 2

Probability of correct location versus varying values of μ. α=0.3.

Fig. 3
Fig. 3

(a) Scene with airplane. α=0.3 and μ=0.06. (b) α=0.3 and μ=0.16. (c) α=0.3 and μ=0.26. (d) α=0.3 and μ=0.36. (e) α=0.3 and μ=0.46. (f) Binary reference target w.

Fig. 4
Fig. 4

(a) Scene with airplane. α1=α3=α4=0.2, μ1=0.1, μ2=0.05, μ4=0.4, and μ4=0.2. α2=0.05. (b) Same as in (a) but with α2=0.35.

Fig. 5
Fig. 5

Performance of different versions of the optimal processor on images with discrete arbitrary distribution. The processor labeled MLML is the ML processor with ML estimates of the parameters. The processor labeled 4-factors is the ML processor with known values of the parameters, and 2-factors is the processor when only the lower orders of correlation are included. The parameter values are known. The curve labeled 2-factors est. is the ML processor when the parameters are estimated with only lower orders of correlation. α1=α3=α4=0.2, μ1=0.1, μ2=0.05, μ3=0.4, and μ4=0.2.

Fig. 6
Fig. 6

(a) Scene with airplane. mt=2, st=sb=2, and mb=0.5. (b) mt=2, st=sb=2, and mb=1.5. (c) mt=2, st=sb=2, and mb=3.5. (d) mt=mb=2, st=1, sb=0.5. (e) mt=mb=2, st=1, and sb=0.9. (f) mt=mb=2, st=1, and sb=1.3.

Fig. 7
Fig. 7

Performance of different versions of the optimal processor on images with discrete Gaussian distribution. The processor labeled MLML is the ML processor with ML estimates of the parameters. The processor labeled 4-factors is the ML processor with known values of the parameters, and 2-factors is the processor when only the lower orders of correlation are included. The parameter values are known. The curve labeled 2-factors est. is the ML processor when the parameters are estimated with only lower orders of correlation. mt=2, sb=2, and st=2.

Fig. 8
Fig. 8

Performance of the optimal Gaussian processor (Opt. Gauss) compared with the optimal MLML processor. The gray-level distributions on the images were discrete Gaussian, and the parameters were mt=2 and st=sb=2.

Fig. 9
Fig. 9

Performance of the optimal Gaussian processor (Opt. Gauss) compared with the optimal MLML processor. The gray-level distributions on the images were arbitrary, with the following parameters: α1=α3=α4=0.2, μ1=0.1, μ2=0.05, μ3=0.4, and μ4=0.2.

Tables (2)

Tables Icon

Table 1 Influence of the Number of Levels in the Input Image on the Form of the Optimal Processor a

Tables Icon

Table 2 Four First Moments of the Target and Background Regions for Different Values of the Parameters a

Equations (53)

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si=nitwi-j+nib(1-wi-j),
nit=1withprobabilityα0withprobability1-α,
nib=1withprobabilityμ0withprobability1-μ.
nit=1withprobabilityα12withprobabilityα23withprobabilityα3NwithprobabilityαN0withprobability1-i=1Nαi,
nib=1withprobabilityμ12withprobabilityμ23withprobabilityμ3NwithprobabilityμN0withprobability1-i=1Nμi.
L(j)=iwj[siα1+(1-si)(1-α1)]×iwj[siμ1+(1-si)(1-μ1)],
l(j)=ln L(j)=Nr[ln(1-α1)-ln(1-μ1)]
+N ln(1-μ1)+Ns[ln μ1+ln(1-μ1)]+w*s[ln α1-ln(1-α1)-ln μ1+ln(1-μ1)],
α1j=(w*s)jNr,
μ1j=Ns-(w*s)jN-Nr.
L(j)=iwj(1-si)(2-si)2 (1-α1-α2)+(2-si)siα1+(1-si)si-2 α2iwj(1-si)(2-si)2 (1-μ1-μ2)+(2-si)siμ1+(1-si)si-2 μ2.
l(j)=ln L(j)=p1(w*s)j+p2(w*s2)j+p3Nr+p4Ns+p5Ns2,
p1=2 ln α1-32ln(1-α1-α2)-12ln α2-2 ln μ1+32ln(1-μ1-μ2)+12ln μ2,
p2=12ln(1-α1-α2)-ln α1+12ln α2-12ln(1-μ1-μ2)+ln μ1-12ln μ2,
p3=ln(1-α1-α2)-ln(1-μ1-μ2),
p4=2 ln μ1-32ln(1-μ1-μ2)-12ln μ2,
p5=12ln (1-μ1-μ2)-ln μ1+12ln μ2.
αij=4(w*s)j-2(w*s2)j2Nr,
α2j=-(w*s)j+(w*s2)j2Nr,
μ1j=4(w*s)j-2(w*s2)j-4Ns+2Ns22Nr-2N,
μ2j=-(w*s)j+(w*s2)j+Ns-Ns22Nr-2N.
l(j)=f5[w*s, w*s2, w*s3, w*s4].
yi=exp-(xi-m)2s2,
FjGauss=(st2-sb2)[sc2*w]j-2(mbst2-mtsb2)[sc*w]j,
w*sE[sz],
w*s2E[sz2],
w*s3E[sz3],
w*s4E[sz4],
E[sz]=v1,
E[sz2]=v2+v12,
E[sz3]=v3+3v2v1+v13,
E[sz4]=v4+4v3v1+6v2v12+v14,
l(j)=iwj(1-si)(2-si)(3-si)(4-si)24ln(1-α1-α2-α3-α4)+si(2-si)(3-si)(4-si)6ln α1+si(1-si)(3-si)(4-si)-4ln α2+si(1-si)(2-si)4-si)6ln α3+si(1-si)(2-si)(3-si)-24ln α4+iwj(1-si)(2-si)(3-si)(4-si)24ln(1-μ1-μ2-μ3μ4)+si(2-si)(3-si)(4-si)6ln μ1+si(1-si)(3-si)(4-si)-4ln μ2+si(1-si)(2-si)(4-si)6ln μ3+si(1-si)(2-si)(3-si)-24ln μ4,
l(j)=t1(w*s)j+t2(w*s2)j+t3(w*s3)j+t4(w*s4)j+t5+t6Ns+t7Ns2+t8Ns3+t9Ns4+t10N,
t1=2512ln(1-μ1-μ2-μ3-μ4)-4 ln μ1-14ln α4+4 ln α1+14ln μ4+43ln α3-3 ln α2+3 ln μ2-2512ln(1-α1-α2-α3-α4)-43ln μ3,
t2=-3524ln(1-μ1-μ2-μ3-μ4)-194ln μ2+133ln μ1+3524ln(1-α1-α2-α3-α4)+73ln μ3+194ln α2-133 α1-73ln α3-1124ln μ4+1124ln α4,
t3=512ln(1-μ1-μ2-μ3-μ4)+2 ln μ2-32ln μ1-512ln(1-α1-α2-α3-α4)-76ln μ3-2 ln α2+32ln α1+14ln μ4+76ln α3-14ln(α4),
t4=124ln(1-α1-α2-α3-α4)-14lnμ2+16lnμ1-16ln α3+16ln μ3-16ln α1-124ln(1-μ1-μ2-μ3-μ4)+14lnα2+124ln(α4)-124ln μ4,
t5=[ln(1-α1-α2-α3-α4)-ln(1-μ1-μ2-μ3-μ4)]Nr,
t6=43ln μ3-2512ln(1-μ1-μ2-μ3-μ4)-3 ln μ2+4 ln μ1-14ln μ4,
t7=3524ln(1-μ1-μ2-μ3-μ4)+194ln μ2-133ln μ1-73ln μ3+1124ln μ4,
t8=32ln μ1-2 ln μ2-512ln(1-μ1-μ2-μ3-μ4)+76ln μ3-14ln μ4,
t9=-16ln μ1+124ln(1-μ1-μ2-μ3-μ4)+14ln μ2-16ln μ3+124ln μ4,
t10=ln(1-μ1-μ2-μ3-μ4),
α1j=-96(w*s)j+104(w*s2)j-36(w*s3)j+4(w*s4)j24Nr,
α2j=72(w*s)j-114(w*s2)j-48(w*s3)j-6(w*s4)j24Nr,
α3j=-32(w*s)j+56(w*s2)j-28(w*s3)j+4(w*s4)j24Nr,
α4j=-6(w*s)j+11(w*s2)j-6(w*s3)j+(w*s4)j24Nr,
μ1j=96[Ns-(w*s)j]-104[Ns2-(w*s2)j]+36[Ns3-(w*s3)j]-[Ns4-(w*s4)j]24(N-Nr),
μ2j=72[Ns-(w*s)j]+114[Ns2-(w*s2)j]-48[Ns3-(w*s3)j]+6[Ns4-(w*s4)j]24(N-Nr),
μ3j=32[Ns-(w*s)j]-56[Ns2-(w*s2)j]+28[Ns3-(w*s3)j]-4[Ns4-(w*s4)j]24(N-Nr),
μ4j=-6[Ns-(w*s)j]+112[Ns2-(w*s2)j]-6[Ns3-(w*s3)j]+[Ns4-(w*s4)j]24(N-Nr).
l(j)=t1(w*s)j+t2(w*s2)j+t3(w*s3)j+t4(w*s4)j.

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