Abstract

An algorithm for computing correlation filters based on synthetic discriminant functions that can be displayed on current spatial light modulators is presented. The procedure is nondivergent, computationally feasible, and capable of producing multiple solutions, thus overcoming some of the pitfalls of previous methods.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  2. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  3. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  4. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  5. Ph. Réfrégier, “Méthodes de reconnaissance des formes pour la corrélation optique,” Rev. Tech. Thomson-CSF 22, 649–734 (1990).
  6. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  7. V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulators filters,” Appl. Opt. 33, 4465–4471 (1994).
    [CrossRef] [PubMed]
  8. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  9. D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (1986).
    [CrossRef] [PubMed]
  10. 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]
  11. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discriminant-function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  12. 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 (1989).
  13. D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementation on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 10–20 (1992).
  14. U. Mahlab, J. Shamir, “Phase-only entropy-optimized filter generated by simulated annealing,” Opt. Lett. 14, 1168–1170 (1989).
    [CrossRef] [PubMed]
  15. J. Rosen, J. Shamir, “Application of the projection-onto-constraint-sets algorithm for optical pattern recognition,” Opt. Lett. 16, 752–754 (1991).
    [CrossRef] [PubMed]
  16. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1-theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
    [CrossRef] [PubMed]
  17. Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
    [CrossRef]
  18. B. V. K. Vijaya Kumar, Z. Bahri, A. Mahalanobis, “Constrained phase optimization in minimum variance synthetic discriminant functions,” Appl. Opt. 27, 409–413 (1988).
    [CrossRef]
  19. Ph. Réfrégier, J. P. Huignard, “Phase selection of synthetic discriminant functions filters,” Appl. Opt. 29, 4772–4778 (1990).
    [CrossRef] [PubMed]
  20. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]

1994 (1)

1993 (1)

1992 (1)

1991 (2)

1990 (3)

1989 (2)

1988 (2)

1987 (1)

1986 (3)

1985 (1)

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1-theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef] [PubMed]

Bahri, Z.

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

B. V. K. Vijaya Kumar, Z. Bahri, A. Mahalanobis, “Constrained phase optimization in minimum variance synthetic discriminant functions,” Appl. Opt. 27, 409–413 (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 (1989).

Carlson, D. W.

D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementation on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 10–20 (1992).

Casasent, D.

Ennis, D. J.

Gianino, P. D.

Hassebrook, L.

Horner, J. L.

Huignard, J. P.

Jared, D. A.

Juday, R. D.

Kallman, R. R.

Laude, V.

Mahalanobis, A.

Mahlab, U.

Réfrégier, Ph.

Rosen, J.

Rozzi, W. A.

Shamir, J.

Vijaya Kumar, B. V. K.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1-theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef] [PubMed]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1-theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef] [PubMed]

Appl. Opt. (11)

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulators filters,” Appl. Opt. 33, 4465–4471 (1994).
[CrossRef] [PubMed]

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

D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (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]

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

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

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

B. V. K. Vijaya Kumar, Z. Bahri, A. Mahalanobis, “Constrained phase optimization in minimum variance synthetic discriminant functions,” Appl. Opt. 27, 409–413 (1988).
[CrossRef]

Ph. Réfrégier, J. P. Huignard, “Phase selection of synthetic discriminant functions filters,” Appl. Opt. 29, 4772–4778 (1990).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1-theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef] [PubMed]

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

Opt. Lett. (3)

Rev. Tech. Thomson-CSF (1)

Ph. Réfrégier, “Méthodes de reconnaissance des formes pour la corrélation optique,” Rev. Tech. Thomson-CSF 22, 649–734 (1990).

Other (2)

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 (1989).

D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementation on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 10–20 (1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Structure of the algorithm. CONS, constrained filter.

Fig. 2
Fig. 2

Projection of the filter values onto the allowed domain for (a) a phase-only filter, (b) a binary-phase-only filter, (c) an arbitrary domain.

Fig. 3
Fig. 3

Sketch of the two successive projections that form an iteration of the algorithm.

Fig. 4
Fig. 4

Block diagram of the proposed algorithm.

Fig. 5
Fig. 5

Training set: (a) true-class images, (b) false-class images.

Fig. 6
Fig. 6

Different coding domains used to test the method: (a) phase-only domain, (b) binary-phase-only domain, (c) spiral coupling between amplitude and phase, (d) arbitrary domain.

Fig. 7
Fig. 7

Plot showing the convergence of the algorithm for a phase-only SDF.

Fig. 8
Fig. 8

Impulse response of three phase-only SDF’s designed with the same training set but with different starting points: (a) the phase of a MACE filter, (b) a phase-only random vector, (c) a constant plane in Fourier space.

Fig. 9
Fig. 9

Central correlations obtained with two different phase-only SDF’s: (a) with a starting point formed by the sum of the 10 tanks of the training set [Fig. 5(a)], (b) with a starting point formed by the sum of the whole set of 20 tanks. Open circles represent the correlations with the trucks. Filled circles represent those of the tanks.

Fig. 10
Fig. 10

Central correlations between a binary phase-only SDF and the images of the training set. Open circles represent the correlations with the trucks; filled circles represent those of the tanks.

Fig. 11
Fig. 11

Central correlations between a spirally constrained SDF filter and the images of the training set. Open circles represent the correlations with the trucks; filled circles represent those of the tanks.

Fig. 12
Fig. 12

Central correlations between an SDF with values on the domain of Fig. 6(d) and the images of the training set. Open circles represent the correlations with the trucks; filled circles represent those of the tanks.

Fig. 13
Fig. 13

Test input scene. Both images belong to the training set.

Fig. 14
Fig. 14

Correlation between the phase-only SDF and the input scene of Fig. 13.

Fig. 15
Fig. 15

Correlation between the binary-phase-only SDF and the input scene of Fig. 13.

Fig. 16
Fig. 16

Correlation between the spirally constrained SDF and the input scene of Fig. 13.

Fig. 17
Fig. 17

Correlation between the arbitrarily constrained SDF [Fig. 6(d)] and the input scene of Fig. 13.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

i = 1 N | h k + 1 i a k + 1 i | 2 < i = 1 N | h k i a k i | 2 ,
i = 1 N | h k + 1 i a k + 1 i | 2 i = 1 N | h k i a k i | 2 ,
i = 1 N | h k + 1 i a k + 1 i | 2 i = 1 N | h k + 1 i a k i | 2 i = 1 N | h k i a k i | 2
i = 1 N | h k + 1 i a k + 1 i | 2 i = 1 N | h k + 1 i a k i | 2 ,
E ( a ) = i = 1 N | h k + 1 i a i | 2 ;
i = 1 N | h k + 1 i a k i | 2 i = 1 N | h k i a k i | 2 ,
E ( h ) = i = 1 N | h i a k i | 2 ,
h + X = c T ,
E ( h ) = i = 1 N | h i a i | 2
E ( h ) = i = 1 N [ ( h i a i ) * ( h i a i ) ] = i = 1 N [ ( h i ) * h i ( h i ) * a i ( a i ) * h i + ( a i ) * a i ] = h + h h + a a + h + a + a .
h R T X R + h I T X I = c R T , h R T X I h I T X R = c I T ,
E ( h R , h I ) = h R T h R + h I T h I 2 h R T a R 2 h I T a I + a R T a R + a I T a I .
L ( h R , h I ) = h R T h R + h I T h I 2 h R T a R 2 h I T a I + a R T a R + a I T a I 2 ( h R T X R + h I T X I c R T ) u 2 ( h R T X I + h I T X R c I T ) v ,
h = a + X w * ,
( a + + w T X + ) X = c T ,
w * = ( X + X ) 1 ( c * X + a ) .
h = X ( X + X ) 1 c * + [ I N X ( X + X ) 1 X + ] a comp + Pa ,
h = a + Δ a = a + X ( X + X ) 1 ( c * X + a ) ,
h k = α comp + P a k 1
a k i = arg min s D ( | h k i s | 2 )
E k = i = 1 N | h k i a k i | 2 < θ f ,
| E k E k 1 | < θ m ,
[ I X ( X + X ) 1 X + ] a = a X ( X + X ) 1 X + a = a S ( X + a ) .
a k = comp + P v ,
h k + 1 = comp + P ( comp + P v ) = comp + P ( comp ) + P 2 v = comp + 0 + P v = a k ,
a k = a k 1 + v ,
h k + 1 = comp + P a k = comp + P ( a k 1 + v ) = comp + P a k 1 = h k ,
a k = a k + 1 = a k + 2 = , h k = h k + 1 = h k + 2 = ,
h k h k + 1 , a k = a k + 1 ,
a k = a k + 1 = a k + 2 = , h k + 1 = h k + 2 = ,
a 0 = i = 1 M c i X i ,
E NORM ( h ) = ( i = 1 N | h k i a k i | 2 ) / ( i = 1 N | h k i | 2 ) ,
E = h + h a + h h + a + a + a ,
h = α comp + P a ,
E ( α ) = [ α ( comp ) + + a + P ] ( α comp + P a ) a + ( α comp + P a ) [ α ( comp ) + + a + P ] a + a + a = α 2 ( comp ) + comp + α ( comp ) + P a + α a + P ( comp ) + a + P 2 a α a + comp a + P a α ( comp ) + a a + P a + a + a .
( comp ) + P a = 0 , a + P ( comp ) = 0 , a + P 2 a = a + P a a + P 2 a a + P a = 0 ,
E ( α ) = α 2 ( comp ) + comp 2 α a + comp a + P a + a + a ,
E α = 0 2 α ( comp ) + comp 2 a + comp = 0 ,
α = a + comp ( comp ) + comp .

Metrics