Abstract

In this paper, we address the problem of correlation peak phase selection for synthetic discriminant function filters. We show that the minimization of the output variance and the optimization of the correlation peak form are problems of the same complexity (analogous to the determination of the ground state of a magnetic disordered system). We propose a general framework and, with examples, we show that, although the variance reduction by a proper selection of the correlation peak phases can be interesting in some situations, the optimization of the sharpness of the correlation peak is often more fruitful.

© 1990 Optical Society of America

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References

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  1. D. Casasent, D. Psaltis, “Position, Rotation and Scale Invariant Optical Correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  2. Y. N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  3. Y. N. Hsu, H. H. Arsenault, “Pattern Discrimination by Multiple Circular Harmonic Components,” Appl. Opt. 23, 841–844 (1984).
    [CrossRef] [PubMed]
  4. R. R. Kallman, “Construction of Low Noise Optical Correlation Filters,” Appl. Opt. 25, 1032–1033 (1986).
    [CrossRef] [PubMed]
  5. B. V. K. V. Kumar, Z. Bahri, A. Mahalanobis, “Constraint Phase Optimization in Minimum Variance Synthetic Discriminant Functions,” Appl. Opt. 27, 409–413 (1988).
    [CrossRef] [PubMed]
  6. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum Average Correlation Energy Filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  7. B. V. K. V. Kumar, “Minimum-Variance Synthetic Discriminant Functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  8. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis, (Wiley, New York, 1973).
  9. K. Binder, A. P. Young, “Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions,” Rev. Mod. Phys. 58, 801–976 (1986).
    [CrossRef]
  10. G. Toulouse, “Theory of the Frustration Effect in Spin Glasses,” Commun. Phys. 2, 115–120 (1977).
  11. S. Kirkpatrick, S. D. Gelatt, M. P. Vecchi, ”Optimization by Simulated Annealing,” IBM Research Report, RC(9955), (1982).
  12. P. J. M. Van Laarhoven, E. H. Aarts, Simulated Annealing: Theory and Applications (D. Reidel, Dordrecht, Holland, 1987).

1988

1987

1986

1984

1982

1977

G. Toulouse, “Theory of the Frustration Effect in Spin Glasses,” Commun. Phys. 2, 115–120 (1977).

1976

Aarts, E. H.

P. J. M. Van Laarhoven, E. H. Aarts, Simulated Annealing: Theory and Applications (D. Reidel, Dordrecht, Holland, 1987).

Arsenault, H. H.

Bahri, Z.

Binder, K.

K. Binder, A. P. Young, “Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Casasent, D.

Duda, R. O.

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

Gelatt, S. D.

S. Kirkpatrick, S. D. Gelatt, M. P. Vecchi, ”Optimization by Simulated Annealing,” IBM Research Report, RC(9955), (1982).

Hart, P. E.

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

Hsu, Y. N.

Kallman, R. R.

Kirkpatrick, S.

S. Kirkpatrick, S. D. Gelatt, M. P. Vecchi, ”Optimization by Simulated Annealing,” IBM Research Report, RC(9955), (1982).

Kumar, B. V. K. V.

Mahalanobis, A.

Psaltis, D.

Toulouse, G.

G. Toulouse, “Theory of the Frustration Effect in Spin Glasses,” Commun. Phys. 2, 115–120 (1977).

Van Laarhoven, P. J. M.

P. J. M. Van Laarhoven, E. H. Aarts, Simulated Annealing: Theory and Applications (D. Reidel, Dordrecht, Holland, 1987).

Vecchi, M. P.

S. Kirkpatrick, S. D. Gelatt, M. P. Vecchi, ”Optimization by Simulated Annealing,” IBM Research Report, RC(9955), (1982).

Young, A. P.

K. Binder, A. P. Young, “Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Appl. Opt.

Commun. Phys.

G. Toulouse, “Theory of the Frustration Effect in Spin Glasses,” Commun. Phys. 2, 115–120 (1977).

J. Opt. Soc. Am. A

Rev. Mod. Phys.

K. Binder, A. P. Young, “Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Other

S. Kirkpatrick, S. D. Gelatt, M. P. Vecchi, ”Optimization by Simulated Annealing,” IBM Research Report, RC(9955), (1982).

P. J. M. Van Laarhoven, E. H. Aarts, Simulated Annealing: Theory and Applications (D. Reidel, Dordrecht, Holland, 1987).

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

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Figures (3)

Fig. 1
Fig. 1

Example of a magnetic Ising system with frustration; many equivalent configurations exist and many local minima are present.

Fig. 2
Fig. 2

Maximum values of the correlation functions on the training set for Problem 2. These values are different from central values which are equal to 0 or 1, i.e., the desired output, and corresponds to pixels around the center. The images with numbers 1–10 are the trucks, i.e., desired output = 1; and the ones with numbers 11–20 are the tanks, i.e., desired output = 0.

Fig. 3
Fig. 3

Example of tank and truck images used as training data. We show only the following values of the view angles: θ = 0 and θ = 60° and φ = 0 and φ = 45°.

Tables (4)

Tables Icon

Table I Different Solutions of the Matrix of the Generic Form to Optimize for the Different Analyzed Cases In the Paper

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Table II Values of the Criteria MSE and ACE for Problem 1 and for Different Configurations of Phases

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Table III Idem of Table II but for Problem 2

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Table IV Evolution from the Optimized Value for Each Criterion When the Optimization is Performed on the Other Criterion

Equations (26)

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h · x l = d l ,             l = 1 , , M ,
X h = d ,
d = ( d 1 , d 2 , , d M ) .
h = A - 1 X ( X A - 1 X ) - 1 d ,
v ˜ = F · v ,             v = F · v ˜ ,
h ˜ = A ˜ - 1 X ˜ ( X ˜ A ˜ - 1 X ˜ ) - 1 d ,
A ˜ = F A F ,
MSE = l = 1 M d l - ( x l + n ) · h 2 = h · n 2 = h · C · h ,
h = C - 1 X ( X C - 1 X ) - 1 d ,
h = X pi d = X ( X X ) - 1 d ,
D ˜ ( k , k ) = l = 1 M x ˜ k 1 2 ,
ACE = i = 1 M k = 1 N x ˜ k l · h ˜ k 2 = h ˜ · D ˜ · h ˜ .
h ˜ = D ˜ - 1 X ˜ ( X ˜ D ˜ - 1 X ˜ ) - 1 d .
S l = F S ˜ l F S ( i , j ) l = S l ( i - j ) = h = 0 N - 1 S ˜ ( k , k ) l exp [ i 2 π k ( i - j ) ] .
h = D - 1 X ( X D - 1 X ) - 1 d ,
h · x l = d l s l .
E = - i = 1 M j = 1 M S i * J ( i j ) s j ,
J ( i , j ) ( 1 , 1 ) = - d i [ ( X C - 1 X ) - 1 ] ( i , j ) d j .
J ( i , j ) ( 1 , 2 ) = - d i [ ( X X ) - 1 X D X ( X X ) - 1 ] ( i , j ) d j .
J ( i , j ) ( 2 , 2 ) = - d i [ ( X D - 1 X ) - 1 ] ( i , j ) d j .
E = - i = 1 M j = 1 M s i * J ( i j ) s j ,
h = A - 1 X ( X A - 1 X ) - 1 d ,
E = h · B · h .
E = d [ ( X A - 1 X ) - 1 X A - 1 B A - 1 X ( X A - 1 X ) - 1 ] ( i , j ) · d .
E = - i = 1 M j = 1 M s i * J ( i j ) s j ,
J ( i , j ) = - d i [ ( X A - 1 X ) - 1 X A - 1 B A - 1 X ( X A - 1 X ) - 1 ] ( i , j ) d j .

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