Abstract

A multimodal model for correlation-plane distributions generated by composite filters is presented. From this model a statistical classifier referred to as a composite Bayesian classifier is developed. By exploiting the Gaussian behavior of correlation-plane data, this classifier concisely represents multimodal distributions as composite algebraic functions. These multimodal distributions, each of which is constructed by superposition of many normal distributions, are used to partition a vector signal space into optimum classification regions derived from Bayes’s likelihood ratio test. For the purpose of validating the multimodal model, expected performance for the training images is derived from calibration data and compared with observed performance.

© 1994 Optical Society of America

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  1. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).
  2. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  3. F. M. Dickey, T. K. Stalker, J. M. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  4. D. Flannery, J. Loomis, M. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
    [CrossRef] [PubMed]
  5. J. Downie, B. Hine, M. Reid, “Effects and correction of magneto-optic spatial light modulator phase errors in an optical correlator,” Appl. Opt. 31, 636–643 (1992).
    [CrossRef] [PubMed]
  6. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  7. D. Jared, D. Ennis, “Learned pattern recognition using synthetic discriminant functions,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 91–101 (1986).
  8. Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
    [CrossRef]
  9. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  10. D. Casasent, W. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
    [CrossRef] [PubMed]
  11. T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).
  12. S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Selection of optimum output correlation values in synthetic discriminant function design,” J. Opt. Soc. Am. A 7, 611–616 (1990).
    [CrossRef]
  13. V. B. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  14. Z. Bahri, B. V. K. Vijaya Kumar, “Algorithms for designing phase-only synthetic discriminant functions,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 138–147 (1990).
  15. D. Jared, D. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  16. R. Shiavi, Introduction to Applied Statistical Signal Analysis (Aksen, Boston, Mass., 1991).
  17. C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).
  18. C. Chatfield, Introduction to Multivariate Analysis (Chapman & Hall, London, 1980).
  19. R. Schalkoff, Pattern Recognition (Wiley, New York, 1992).
  20. R. E. Ziemer, W. H. Tranter, Principles of Communications, 3rd ed. (Houghton Mifflin, Boston, Mass., 1990).
  21. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).
  22. D. J. Hand, Discrimination and Classification (Wiley, Chichester, UK, 1981).
  23. K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, Mass., 1990).
  24. P. A. Lachenbruch, Discriminant Analysis (Hafner, New York, 1975).
  25. R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  26. P. Billings, T. Giles, “Multisensor image analysis system demonstration,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 83–91 (1992).
  27. G. W. Carhart, B. F. Draayer, M. K. Giles, “Optical pattern recognition via Bayesian decision theory,” J. Patt. Recog. (to be published).

1992 (2)

1990 (1)

1989 (1)

1988 (4)

1987 (1)

1986 (1)

1984 (1)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).

1980 (1)

Bahri, Z.

Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
[CrossRef]

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithms for designing phase-only synthetic discriminant functions,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 138–147 (1990).

Billings, P.

P. Billings, T. Giles, “Multisensor image analysis system demonstration,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 83–91 (1992).

Carhart, G. W.

G. W. Carhart, B. F. Draayer, M. K. Giles, “Optical pattern recognition via Bayesian decision theory,” J. Patt. Recog. (to be published).

Casasent, D.

Chang, W.

Chatfield, C.

C. Chatfield, Introduction to Multivariate Analysis (Chapman & Hall, London, 1980).

Cravatt, J. E.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).

Dickey, F. M.

Downie, J.

Draayer, B. F.

G. W. Carhart, B. F. Draayer, M. K. Giles, “Optical pattern recognition via Bayesian decision theory,” J. Patt. Recog. (to be published).

Ennis, D.

D. Jared, D. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

D. Jared, D. Ennis, “Learned pattern recognition using synthetic discriminant functions,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 91–101 (1986).

Farn, M. W.

Flannery, D.

Fukunaga, K.

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, Mass., 1990).

Giles, M. K.

G. W. Carhart, B. F. Draayer, M. K. Giles, “Optical pattern recognition via Bayesian decision theory,” J. Patt. Recog. (to be published).

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).

Giles, T.

P. Billings, T. Giles, “Multisensor image analysis system demonstration,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 83–91 (1992).

Goodman, J. W.

Hand, D. J.

D. J. Hand, Discrimination and Classification (Wiley, Chichester, UK, 1981).

Hester, F.

Hine, B.

Jared, D.

D. Jared, D. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

D. Jared, D. Ennis, “Learned pattern recognition using synthetic discriminant functions,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 91–101 (1986).

Johnson, R. A.

R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Kast, B. A.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).

Lachenbruch, P. A.

P. A. Lachenbruch, Discriminant Analysis (Hafner, New York, 1975).

Loomis, J.

Mahalanobis, A.

Mason, J. M.

Milkovich, M.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).

Reid, M.

Schalkoff, R.

R. Schalkoff, Pattern Recognition (Wiley, New York, 1992).

Shiavi, R.

R. Shiavi, Introduction to Applied Statistical Signal Analysis (Aksen, Boston, Mass., 1991).

Stalker, T. K.

Sudharsanan, S. I.

Sundareshan, M. K.

Therrien, C. W.

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

Tranter, W. H.

R. E. Ziemer, W. H. Tranter, Principles of Communications, 3rd ed. (Houghton Mifflin, Boston, Mass., 1990).

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).

Vijaya Kumar, B. V. K.

Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
[CrossRef]

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

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithms for designing phase-only synthetic discriminant functions,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 138–147 (1990).

Vijaya Kumar, V. B. K.

Walsh, T. R.

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).

Wichern, D. W.

R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Ziemer, R. E.

R. E. Ziemer, W. H. Tranter, Principles of Communications, 3rd ed. (Houghton Mifflin, Boston, Mass., 1990).

Appl. Opt. (9)

J. Opt. Soc. Am. A (2)

Opt. Eng (1)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng 23, 698–704 (1984).

Other (15)

D. Jared, D. Ennis, “Learned pattern recognition using synthetic discriminant functions,” in Hybrid Image Processing, D. P. Casasent, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 91–101 (1986).

R. Shiavi, Introduction to Applied Statistical Signal Analysis (Aksen, Boston, Mass., 1991).

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

C. Chatfield, Introduction to Multivariate Analysis (Chapman & Hall, London, 1980).

R. Schalkoff, Pattern Recognition (Wiley, New York, 1992).

R. E. Ziemer, W. H. Tranter, Principles of Communications, 3rd ed. (Houghton Mifflin, Boston, Mass., 1990).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

D. J. Hand, Discrimination and Classification (Wiley, Chichester, UK, 1981).

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, Mass., 1990).

P. A. Lachenbruch, Discriminant Analysis (Hafner, New York, 1975).

R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

P. Billings, T. Giles, “Multisensor image analysis system demonstration,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 83–91 (1992).

G. W. Carhart, B. F. Draayer, M. K. Giles, “Optical pattern recognition via Bayesian decision theory,” J. Patt. Recog. (to be published).

T. R. Walsh, J. E. Cravatt, B. A. Kast, M. K. Giles, “Time-sequenced rotation invariant optical correlator for multiple target recognition,” in Aerospace Pattern Recognition, M. R. Weathersby, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1098, 240–252 (1989).

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithms for designing phase-only synthetic discriminant functions,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 138–147 (1990).

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Figures (12)

Fig. 1
Fig. 1

Statistical correlation assesses the probability, denoted α, that signal x0 is part of the distribution p(x) by integrating the distribution over the hatched region. This region is defined by all signals farther from the mean than the sample signal.

Fig. 2
Fig. 2

Equal-probability contours of a bivariate normal distribution are concentric ellipses albeit arbitrarily oriented. The eigenvectors of the covariance matrix characterizing the distribution define an alternate coordinate system, ( x 1 , x 2 ), that is aligned with the principal axes of the ellipses.

Fig. 3
Fig. 3

Quadratic classification boundaries generated by a statistical correlation classifier: R1R4, classification regions.

Fig. 4
Fig. 4

Statistical correlation versus likelihood. Although sample point x0 lies on the 50% contours of both distributions, the density, δ2, on the pdf-2 contour is greater than the corresponding density, δ1, of pdf 1 because the area enclosed by the pdf-1 contour is larger. The likelihood that x0 is part of either pdf is the ratio of the pdf’s density to the sum of both. Thus by Bayes’s likelihood ratio test, x0 is assigned correctly to pdf 2 because δ2/(δ1 + δ2) > δ1/(δ1 + δ2).

Fig. 5
Fig. 5

Quadratic classifier boundaries versus SC classifier boundaries. Classification according to equal density (quadratic classifier) rather than equal probability (SC classifier) results in a TI assignment to pdf 2 instead of pdf 1 for signals in the hatched region.

Fig. 6
Fig. 6

QU object regions (dark outlines) produced by a union of the TI regions. Given the relative sizes of the density values, δ1, δ2, δ3, and δ4, the QU classifier assigns x0 directly to pdf 2 and consequently indirectly to the object class characterized by pdf’s 1 and 2. In contrast the CB classifier assigns x0 to the object class characterized by pdf’s 3 and 4 because the density of the composite distribution of pdf’s 3 and 4 is greater than the density of the composite distribution for pdf’s 1 and 2. Thus for x0 the proximity of pdf 3 influences the migration of the classification boundary more strongly than the proximity of pdf 1.

Fig. 7
Fig. 7

Integration in the discrete signal space. Overall pdf signal densities (total column height) are formed by renormalization and addition of the densities of the two distributions at each signal. Integration in this space is just summation of the appropriate renormalized density values. For example, the fraction of class 2 signals that can be identified is found by addition of all the column tops that are dark.

Fig. 8
Fig. 8

Computer-controlled hybrid correlator with an Epson LCTV at the input plane, a Semetex MOSLM at the filter plane, and a Burle CCD camera at the correlation plane: SF, spatial filter; L1, L2, lenses; P1, P2, polarizers.

Fig. 9
Fig. 9

Source images used to generate the TI set. The type of plane is given below each image.

Fig. 10
Fig. 10

Bivariate scatter plots obtained from (a) first, (b) second, (c) third, (d) fourth, (e) fifth levels in the tree.

Fig. 11
Fig. 11

(a) Bivariate scatter plot obtained by correlation of two simple filters against a TI associated with one of the filters. (b)–(d) Contours taken near the top, the middle, and the bottom of (a), respectively.

Fig. 12
Fig. 12

(a) Bivariate scatter plot obtained from digital correlations after the input image is subjected to minor pixel variations along the edges. (b)–(d) Contours taken near the top, the middle, and the bottom of (a), respectively.

Tables (2)

Tables Icon

Table 1 Theoretical Versus Observed Span Performance on a Target-by-Target Basis

Tables Icon

Table 2 Expected Versus Observed Performance on a Training-Image by Training-image Basis

Equations (35)

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T = { τ i i = 1 , 2 , , N T } = { t k k = 1 , 2 , , N o } = { τ j k j = 1 , 2 , , N k ; k = 1 , 2 , , N o } .
p ( x T ) = i = 1 N T p ( x τ i ) Pr [ τ i ] .
Pr [ τ i x ] = p ( x τ i ) Pr [ τ i ] p ( x T ) ,
Pr [ τ α x ] Pr [ x ] i α ,             i = 1 , 2 , , N T .
p ( x t k ) = 1 Pr [ t k ] j = 1 N k p ( x τ j k ) Pr [ τ j k ] ,
Pr [ t k ] = j = 1 N k Pr [ τ j k ] .
Pr [ t k x ] = p ( x t k ) Pr [ t k ] p ( x T ) = 1 p ( x T ) j = 1 N k p ( x τ j k ) Pr [ τ j k ] ,
Pr [ t α [ x ] Pr [ t k x ] k α ,             k = 1 , 2 , , N o .
N ( x ) = 1 [ ( 2 π ) n C ] 1 / 2 × exp [ - 1 2 ( x - m ) T C - 1 ( x - m ) ] .
m = E [ x ] ,
C = E [ ( x - m ) ( x - m ) T ] .
m ˜ = 1 N s i = 1 N s x i ,
C ˜ = 1 N s - 1 i = 1 N s ( x i x i T - m ˜ m ˜ T ) .
Pr [ t α x ] Pr [ t k x ] k α ,             k = 1 , 2 , , N o ,
Pr [ t α x ] Pr [ t k x ] 1 k α .
p ( x t α ) Pr [ t α ] p ( x t k ) Pr [ t k ] = j = 1 N α p ( x τ j α ) Pr [ τ j α ] j = 1 N k p ( x τ j k ) Pr [ τ j k ] 1 k α .
j = 1 N α Pr [ τ j α ] [ C j α ] 1 / 2 exp [ - ½ ( x - m j α ) T C j α - 1 ( x - m j α ) ] j = 1 N k Pr [ τ j k ] [ C j k ] 1 / 2 exp [ - ½ ( x - m j k ) T C j k - 1 ( x - m j k ) ] 1 k α .
f α ( x ) = j = 1 N α Pr [ τ j α ] [ C j a ] 1 / 2 exp [ - ½ ( x - m j α ) T C j α - 1 ( x - m j α ) ]
choose class ω α if f α ( x ) = max k f k ( x ) .
f ˜ k ( x ) = j = 1 N k Pr [ τ j k ] [ C ˜ j k ] 1 / 2 exp [ - ½ ( x - m ˜ j k ) T C ˜ j k - 1 ( x - m ˜ j k ) ] ,
choose class ω α if f ˜ α ( x ) = max k f ˜ k ( x ) .
d 2 ( x ) = ( x - m ) T C - 1 ( x - m ) ,
d 2 ( x ) = p = 1 n ( x p - m p σ p ) 2 ,
f k ( x ) = j = 1 N k Pr [ τ j k ] [ C j k ] 1 / 2 exp [ - 1 2 p = 1 n ( x p - m p j k σ p j k ) 2 ] ,
f ˜ k ( x ) = j = 1 N k Pr [ τ j k ] [ C ˜ j k ] 1 / 2 exp [ - 1 2 p = 1 n ( x p - m ˜ p j k σ ˜ p j k ) 2 ] .
Pr [ t k x ] = 1 p ( x T ) j = 1 N k p ( x τ j k ) Pr [ τ j k ] ,
Pr [ x R α τ j α ] = R α p ( x τ j α ) d x .
p ( x T ) = i = 1 N T p ( x τ i ) Pr [ τ i ] = k = 1 N o { j = 1 N k p ( x τ j k ) Pr [ τ j k ] } .
P α = R α p ( x T ) d x = R α k = 1 N o { j = 1 N k p ( x τ j k ) Pr [ τ j k ] } d x .
C α = R α p ( x t α ) Pr [ t α ] d x = R α j = 1 N α p ( x τ j α ) Pr [ τ j α ] d x = j = 1 N α Pr [ x R α τ j α ] Pr [ τ j α ] .
Pr [ ω α x R α ] = C α P α = R α p ( x t α ) Pr [ t α ] d x R α p ( x T ) d x = R α j = 1 N α p ( x τ j α ) Pr [ τ j α ] d x R α k = 1 N o { j = 1 N k p ( x τ j k ) Pr [ τ j k ] } d x .
Pr [ success ] = α = 1 N o Pr [ ω α x R α ] Pr [ x R α ] = α = 1 N o C α .
Pr [ x S α τ α ] x S α p ( x τ j α ) ,
Pr [ ω α x S α ] x S α p ( x t α ) Pr [ t α ] x S α p ( x T ) = x S α { j = 1 N α p ( x t j α ) Pr [ τ j α ] } x S α ( k = 1 N o { j = 1 N k p ( x τ j k ) Pr [ τ j k ] } ) ,
Pr [ success ] α = 1 N o { x S α p ( x t α ) Pr [ t α ] } .

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