Abstract

A generalization of the conventional synthetic discriminant function (SDF) solution in both spatial and frequency domains is provided. It is based on the concept of the generalized inverse. The results of this generalization are shown to specialize to previous work (namely, the minimum-output-variance SDF). A link is established between the quantities characterizing the spatial-domain SDF solutions and the frequency-domain ones through the pseudo-discrete Fourier-transform operation. Several properties of the various relevant matrices are presented to provide a general framework for characterizing SDF solutions. To help to illustrate the advantages of the generalized SDF solution method, two examples of SDF’s (a phase-only and a two-level SDF) are investigated.

© 1988 Optical Society of America

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References

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  1. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I.
  2. A. Vander Lugt, “Signal detection by complex spatial filtering” IEEE Trans. Inf. Theory 10, 139–145 (1964).
    [CrossRef]
  3. D. Casasent, A. Furman, “Sources of correlation degradation,” Appl. Opt. 16, 1652–1661 (1977).
    [CrossRef] [PubMed]
  4. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  5. C. F. Hester, D. Casasent, “Multivariant technique for multi-class pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  6. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  7. Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  8. Z. Gu, S. H. Lee, “Classification of multiclassed stochastic images buried in additive noise,” J. Opt. Soc. Am. A 4, 712–719 (1987).
    [CrossRef]
  9. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am A 3, 1433–1442 (1986).
    [CrossRef]
  10. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of distortion-invariant optical correlation filters,” Opt. Lett. 12, 307–309 (1987).
    [CrossRef] [PubMed]
  11. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Spatial-temporal correlation filter for in-plane distortion invariance,” Appl. Opt. 25, 4466–4472 (1986).
    [CrossRef] [PubMed]
  12. B. V. K. V. Kumar, E. Pochapsky, “Signal to noise ratio considerations in modified matched spatial filters,” J. Opt Soc Am. A 3, 777–786 (1986).
    [CrossRef]
  13. B. V. K. V. Kumar, “Optimally of projection synthetic discriminant functions,” in Intelligent Robots and Computer Vision, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt Instrum. Eng.579, 86–95 (1985).
    [CrossRef]
  14. B. V. K. V. Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  15. R. R. Kallman, “The construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
    [CrossRef] [PubMed]
  16. R. R. Kallman, “Optimal low noise phase-only and binary phase-only optical correlation filters for threshold detectors,” Appl. Opt. 25, 4216–4217 (1986).
    [CrossRef] [PubMed]
  17. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  18. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  19. C. Radhakrishna Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).
  20. D. Casasent, “Computer generated holograms in pattern recognition: a review,” Opt. Eng. 24, 724–730 (1985).
  21. J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
    [CrossRef]
  22. Z. Bahri, “Some generalizations of synthetic discriminant functions,” master’s thesis (Carnegie Mellon University, Pittsburgh, Pa., 1986).
  23. W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
    [CrossRef]
  24. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  25. G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).
  26. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  27. Z. Bahri, B. V. K. Vijaya Kumar, “Computational considerations in the determination of synthetic discriminant functions,” in Intelligent Robots and Computer Vision: Fifth in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.726, 46–55 (1986).
    [CrossRef]
  28. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to synthetic discriminant function correlator filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]

1987 (3)

1986 (6)

1985 (2)

1984 (3)

1983 (1)

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

1982 (1)

1980 (1)

1977 (1)

1969 (1)

1964 (1)

A. Vander Lugt, “Signal detection by complex spatial filtering” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Anderson, R. H.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Arsenault, H. H.

Bahri, Z.

Z. Bahri, “Some generalizations of synthetic discriminant functions,” master’s thesis (Carnegie Mellon University, Pittsburgh, Pa., 1986).

Z. Bahri, B. V. K. Vijaya Kumar, “Computational considerations in the determination of synthetic discriminant functions,” in Intelligent Robots and Computer Vision: Fifth in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.726, 46–55 (1986).
[CrossRef]

Butler, S.

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Casasent, D.

Caulfield, H. J.

Duda, R. O.

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

Furman, A.

Gianino, P. D.

Gu, Z.

Hart, P. E.

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

Hester, C. F.

Horner, J. L.

Hsu, Y. N.

Kallman, R. R.

Kumar, B. V. K. V.

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

B. V. K. V. Kumar, E. Pochapsky, “Signal to noise ratio considerations in modified matched spatial filters,” J. Opt Soc Am. A 3, 777–786 (1986).
[CrossRef]

A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Spatial-temporal correlation filter for in-plane distortion invariance,” Appl. Opt. 25, 4466–4472 (1986).
[CrossRef] [PubMed]

B. V. K. V. Kumar, “Optimally of projection synthetic discriminant functions,” in Intelligent Robots and Computer Vision, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt Instrum. Eng.579, 86–95 (1985).
[CrossRef]

Lee, S. H.

Mahalanobis, A.

Maloney, W. T.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Pochapsky, E.

B. V. K. V. Kumar, E. Pochapsky, “Signal to noise ratio considerations in modified matched spatial filters,” J. Opt Soc Am. A 3, 777–786 (1986).
[CrossRef]

Psaltis, D.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Radhakrishna Rao, C.

C. Radhakrishna Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).

Riggins, J.

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Ross, W. E.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schils, G. F.

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of distortion-invariant optical correlation filters,” Opt. Lett. 12, 307–309 (1987).
[CrossRef] [PubMed]

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am A 3, 1433–1442 (1986).
[CrossRef]

Stewart, G. W.

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).

Sweeney, D. W.

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of distortion-invariant optical correlation filters,” Opt. Lett. 12, 307–309 (1987).
[CrossRef] [PubMed]

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am A 3, 1433–1442 (1986).
[CrossRef]

Van Trees, H. L.

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

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex spatial filtering” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

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, “Computational considerations in the determination of synthetic discriminant functions,” in Intelligent Robots and Computer Vision: Fifth in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.726, 46–55 (1986).
[CrossRef]

Appl. Opt. (11)

D. Casasent, A. Furman, “Sources of correlation degradation,” Appl. Opt. 16, 1652–1661 (1977).
[CrossRef] [PubMed]

H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

C. F. Hester, D. Casasent, “Multivariant technique for multi-class pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

R. R. Kallman, “The construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
[CrossRef] [PubMed]

R. R. Kallman, “Optimal low noise phase-only and binary phase-only optical correlation filters for threshold detectors,” Appl. Opt. 25, 4216–4217 (1986).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

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

A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Spatial-temporal correlation filter for in-plane distortion invariance,” Appl. Opt. 25, 4466–4472 (1986).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to synthetic discriminant function correlator filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, “Signal detection by complex spatial filtering” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

J. Opt Soc Am. A (1)

B. V. K. V. Kumar, E. Pochapsky, “Signal to noise ratio considerations in modified matched spatial filters,” J. Opt Soc Am. A 3, 777–786 (1986).
[CrossRef]

J. Opt. Soc. Am A (1)

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am A 3, 1433–1442 (1986).
[CrossRef]

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

Opt. Eng. (3)

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-dimensional magneto-optic spatial light modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

D. Casasent, “Computer generated holograms in pattern recognition: a review,” Opt. Eng. 24, 724–730 (1985).

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Opt. Lett. (1)

Other (8)

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

Z. Bahri, “Some generalizations of synthetic discriminant functions,” master’s thesis (Carnegie Mellon University, Pittsburgh, Pa., 1986).

B. V. K. V. Kumar, “Optimally of projection synthetic discriminant functions,” in Intelligent Robots and Computer Vision, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt Instrum. Eng.579, 86–95 (1985).
[CrossRef]

C. Radhakrishna Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).

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

Z. Bahri, B. V. K. Vijaya Kumar, “Computational considerations in the determination of synthetic discriminant functions,” in Intelligent Robots and Computer Vision: Fifth in a Series, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.726, 46–55 (1986).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Frequency-plane optical correlator.

Fig. 2
Fig. 2

3×3 images used for synthesizing two-valued SDF’s.

Fig. 3
Fig. 3

8 × 8 tank images used for synthesizing approximate phase-only SDF’s.

Fig. 4
Fig. 4

Normalized magnitude of the entries of the proposed SDF.

Fig. 5
Fig. 5

Normalized magnitude of the entries of the conventional SDF.

Equations (92)

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h T x i = c , i = 1 , , N t ,
h = a 1 x 1 + + a N t x N t = j = 1 N t a j x j ,
[ j = 1 N t a j x j ] T x i = 1 , i = 1 , , N t j = 1 N t a j x i T x j = 1 , i = 1 , , N t .
r i j = x i T x j , i = 1 , , N t , j = 1 , , N t ,
R υ a = u ,
a = R υ 1 u .
Ax = y ,
y = x 1 A 1 + + x d A d ,
A A A = A ,
x p = A y
y span { A 1 , , A d } ( t 1 , , t d ) R d / y = t 1 A 1 + + t d A d y = A t , t = [ t 1 t d ] T .
x p = A At .
A x p = A A At = At = y .
x h 0 = ( I d A A ) z ,
x = x p + x h 0 ,
X = [ x 1 x N t ] T .
Xh = u ,
h 0 = a 1 x 1 + + a N t x N t = X T a .
X X T a = u .
a = ( X X T ) 1 u .
h 0 = X T ( X X T ) 1 u .
E = X T ( X X T ) 1 X
F = I d E .
h = h 0 + Fz ,
XE = X , E h 0 = h 0 .
XF = O , F h 0 = 0 .
ker ( P F ) = { h R d / P F [ h ] Fh = 0 }
Im ( P F ) = { h R d / z R d ; h = Fz } ,
R d = ker ( P F ) Im ( P F ) ,
ker ( P F ) = [ Im ( P F ) ] ,
EF = FE = O .
Xh = 0 z R d / h = Fz .
h 0 Fz ; z R d .
Xh = 0 h T h 0 = 0 ; h R d ( the converse not always true ) .
Xh = u Eh = h 0 ; h R d .
Xh = 0 Fh = 0 ; h R d .
S = F E ,
S = S 1 = S T .
c ( 0 , 0 ) = n = 0 N 1 m = 0 M 1 f ( n , m ) g ( n , m ) .
f ( k ) = f ( n , m ) , g ( k ) = g ( n , m ) ,
c ( 0 , 0 ) = k = 0 M N 1 f ( k ) g ( k ) = f T g ,
f = [ f ( 0 ) f ( M N 1 ) ] T , g = [ g ( 0 ) g ( M N 1 ) ] T .
c ( 0 , 0 ) = m = 0 N 1 m = 0 M 1 f ( n , m ) g ( n , m ) = 1 M N p = 0 N 1 q = 0 M 1 f ̂ * ( p , q ) ĝ ( p , q ) = 1 M N p = 0 N 1 q = 0 M 1 f ̂ ( p , q ) ĝ * ( p , q ) .
c ( 0 , 0 ) = f T g = 1 d f ̂ T * ĝ = 1 d ĝ T * f ̂ .
x ̂ = Qx
Q Q T * = Q T * Q = d I d .
Q = P N P M ,
Q T = [ P N P M ] T = P N T P M T = P N P M = Q .
QQ * = Q * Q = d I d .
Q r = P N r P M r P N i P M i
Q i = P N r P M i + P N i P M r ,
Q r 2 + Q i 2 = d I d
Q r Q i = Q i Q r .
[ x ̂ is the pseudo DFT of x ] [ x ̂ = Qx ] [ x = 1 d Q * x ] .
ĥ T * x ̂ i = x ̂ i T * ĥ = d ; i = 1 , , N t .
X ̂ = [ x ̂ 1 x ̂ N t ] T * ,
X ̂ ĥ = d ,
ĥ 0 = X ̂ T * ( X X T * ) 1 d .
Ê = X ̂ T * ( X X T * ) 1 X ̂
F ̂ = I d Ê .
ĥ = ĥ 0 + F ̂ ,
X ̂ T * = Q X T X ̂ = XQ * X = 1 d X ̂ Q .
X ̂ X ̂ T * = X Q T * Q X T = d X X T .
ĥ 0 = Q h 0 .
Ê = 1 d QEQ * .
F ̂ = 1 d QFQ * .
ĥ = Q h 0 + 1 d [ QFQ * ] ,
ĥ = ( Q r + j Q i ) h 0 + 1 d [ ( Q r + j Q i ) F ( Q r i Q i ) ] ( r + j i ) = [ Q r h 0 + 1 d ( Q r F Q r + Q i F Q i ) r + 1 d ( Q r F Q i Q i F Q r ) i ] + j [ Q i h 0 + 1 d ( Q i F Q r Q r F Q i ) r + 1 d ( Q r F Q r + Q i F Q i ) i ] .
x in = x + w ,
y = h T x + h T w .
σ 2 = h T C w h .
v = C w 1 / 2 h h = C w 1 / 2 v ,
v T v = ( C w 1 / 2 h ) T ( C w 1 / 2 h ) = h T ( C w 1 / 2 ) T C w 1 / 2 h = h T C w h = σ 2 .
( X C w 1 / 2 ) v = u .
X = X C w 1 / 2 .
X v = u .
rank ( X ) = N t .
X X T = ( X C w 1 / 2 ) ( X C w 1 / 2 ) T = X C w 1 X T .
v = v 0 + F z ,
v 0 = X T ( X X T ) 1 u = C w 1 / 2 X T ( X C w 1 X T ) 1 u ,
F = I d X T ( X X T ) 1 X .
F v 0 = 0 .
R d = ker ( P F ) Im ( P F )
ker ( P F ) = [ Im ( P F ) ] .
σ 2 = v T v = ( v 0 + F z ) T ( v 0 + F z ) = v 0 T v 0 + 2 z T F v 0 + z T F 2 z = v 0 T v 0 + z T F z .
[ z T F z = 0 ] [ F z = 0 ] ; z R d .
z = z + z
F z = F ( z + z ) = F z + 0 = z .
z T F z = 0 ( z T + z T ) z = 0 z T z + z T z = 0 z 2 + 0 = 0 z = 0 z = z F z = F z = 0 .
σ 2 ( = v T v ) is minimized z T F z = 0 F z = 0 v opt = v 0 ,
h opt = C w 1 / 2 v opt = C w 1 X T ( X C w 1 X T ) 1 u .
h opt = σ n 2 X T ( σ n 2 X X T ) 1 u = X T ( X X T ) 1 u = h 0 ,

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