Abstract

A mathematical analysis of the distortion tolerance in correlation filters is presented. A good measure for distortion performance is shown to be a generalization of the minimum average correlation energy criterion. To optimize the filter’s performance, we remove the usual hard constraints on the outputs in the synthetic discriminant function formulation. The resulting filters exhibit superior distortion tolerance while retaining the attractive features of their predecessors such as the minimum average correlation energy filter and the minimum variance synthetic discriminant function filter. The proposed theory also unifies several existing approaches and examines the relationship between different formulations. The proposed filter design algorithm requires only simple statistical parameters and the inversion of diagonal matrices, which makes it attractive from a computational standpoint. Several properties of these filters are discussed with illustrative examples.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. B. VanderLugt, “Signal detection by complex matched spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. C.F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  3. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  4. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 2324–2335 (1984).
    [CrossRef]
  5. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  6. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  7. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  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 filter,” Appl. Opt. 26, 3633–3640 (1986).
    [CrossRef]
  10. A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
    [CrossRef] [PubMed]
  11. S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
    [CrossRef]
  12. Ph. Réfrédgier, J. Figue, “Optimal trade-off filter for pattern recognition and their comparison with Weiner approach,” Opt. Computer Process. 1, 3–10 (1991).
  13. G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
    [CrossRef] [PubMed]
  14. D. Casasent, W. T. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
    [CrossRef] [PubMed]
  15. D. Casasent, G. Ravichandran, S. Bollapragada, “Gaussian minimum average correlation energy filters,” Appl. Opt. 30, 5176–5181 (1991).
    [CrossRef] [PubMed]
  16. A. Mahalanobis, D. Casasent, B. V. K. Vijaya Kumar, “Spatial-temporal correlation filter for in-plane distortion invariance,” Appl. Opt. 25, 4466–4472 (1986).
    [CrossRef] [PubMed]
  17. B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
    [CrossRef]
  18. G. W. Stewart, An Introduction to Matrix Computations (Academic, Orlando, Fla., 1973).

1992 (3)

1991 (3)

1990 (1)

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[CrossRef]

1988 (1)

1986 (4)

1984 (2)

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

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 2324–2335 (1984).
[CrossRef]

1982 (1)

1980 (1)

1964 (1)

A. B. VanderLugt, “Signal detection by complex matched spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Arsenault, H. H.

Bahri, Z.

Bollapragada, S.

Casasent, D.

Chang, W. T.

Epperson, J. F.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Figue, J.

Ph. Réfrédgier, J. Figue, “Optimal trade-off filter for pattern recognition and their comparison with Weiner approach,” Opt. Computer Process. 1, 3–10 (1991).

Gianino, P. D.

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 2324–2335 (1984).
[CrossRef]

Hester, C.F.

Horner, J. L.

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 2324–2335 (1984).
[CrossRef]

Hsu, Y. N.

Mahalanobis, A.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
[CrossRef] [PubMed]

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1986).
[CrossRef]

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

Ravichandran, G.

Réfrédgier, Ph.

Ph. Réfrédgier, J. Figue, “Optimal trade-off filter for pattern recognition and their comparison with Weiner approach,” Opt. Computer Process. 1, 3–10 (1991).

Sims, S. R. F.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Song, S.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Stewart, G. W.

G. W. Stewart, An Introduction to Matrix Computations (Academic, Orlando, Fla., 1973).

Sudharsanan, S. I.

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[CrossRef]

Sundareshan, M. K.

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[CrossRef]

VanderLugt, A. B.

A. B. VanderLugt, “Signal detection by complex matched spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

Appl. Opt. (11)

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

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

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 2324–2335 (1984).
[CrossRef]

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

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

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1986).
[CrossRef]

A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
[CrossRef] [PubMed]

G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
[CrossRef] [PubMed]

D. Casasent, W. T. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
[CrossRef] [PubMed]

D. Casasent, G. Ravichandran, S. Bollapragada, “Gaussian minimum average correlation energy filters,” Appl. Opt. 30, 5176–5181 (1991).
[CrossRef] [PubMed]

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

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, “Signal detection by complex matched spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

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

Opt. Computer Process. (1)

Ph. Réfrédgier, J. Figue, “Optimal trade-off filter for pattern recognition and their comparison with Weiner approach,” Opt. Computer Process. 1, 3–10 (1991).

Opt. Eng. (2)

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “A unified framework for the synthesis of synthetic discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[CrossRef]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Other (1)

G. W. Stewart, An Introduction to Matrix Computations (Academic, Orlando, Fla., 1973).

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 (1)

Fig. 1
Fig. 1

Target in background at (a) broadside view and (b) end view.

Tables (1)

Tables Icon

Table 1 Performancea of Unconstrained Correlation Filters and Improvements Compared with the Minimum Average Correlation Energy Filter

Equations (43)

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

g i = X i h ,
ASE = 1 N i = 1 N ( g i - f ) + ( g i - f ) .
f ( ASE ) = 2 N i = 1 N ( g i - f ) = 0
f opt = 1 N i = 1 N g i = g ¯ ,
g ¯ = 1 N i = 1 N g i = 1 N i = 1 N X i h = X ¯ h
ASM = 1 N i = 1 N ( g i - g ¯ ) + ( g i - g ¯ ) = 1 N i = 1 N ( X i h - X ¯ h ) + ( X i h - X ¯ h ) = h + [ 1 N i = 1 N ( X i - X ¯ ) * ( X i - X ¯ ) ] h = h + S x h ,
S x = 1 N i = 1 N ( X i - X ¯ ) * ( X i - X ¯ )
g ¯ ( 0 , 0 ) 2 = ( h + x ¯ ) 2 = h + xx ¯ + h .
J ( h ) = h + xx ¯ + h h + S x h
ACE = 1 N i = 1 N h + Y i Y i * h = h + ( 1 N i = 1 N Y i Y i * ) h = h + D y h ,
J ( h ) = h + xx ¯ + h h + S x h + h + D y h = h + xx ¯ + h h + ( S x + D y ) h .
h [ J ( h ) ] = 2 xx ¯ + h h + ( S x + D y ) h - 2 ( h + xx ¯ + h ) ( S x + D y ) h [ h + ( S x + D y ) h ] 2 = 0 .
1 h + ( D y + S x ) h [ xx ¯ + h - λ ( D y + S x ) h ] = 0 ,
λ = h + xx ¯ + ¯ h h + ( S x + D y ) h
xx ¯ + h - λ ( D y + S x ) h = 0 ,
( D y + S x ) - 1 xx ¯ + h = λ h .
α ( D y + S x ) - 1 x ¯ = λ h ,
h = c ( D y + S x ) - 1 x ¯ ,
ASM = h + [ 1 N i = 1 N ( X i - X ¯ ) * ( X i - X ¯ ) ] h = h + ( 1 N i = 1 N X i X i * ) h - h + XX ¯ * h = h + D x h - h + XX ¯ * h = ACE - h + XX ¯ * h .
h mach = S x - 1 x ¯ .
h mse = S x - 1 X ( X + S x - 1 X ) - 1 u = S x - 1 Xa = S x - 1 x ^ ,
h mach = D x - 1 x ¯ .
h mace = D x - 1 X ( X + D x - 1 X ) - 1 u = D x - 1 Xb = D x - 1 x ^ ,
c i = h + x i ,             i = 1 , 2 , , N .
c ¯ = 1 N i = 1 N c i = h + x ¯ ,
σ 2 = 1 N i = 1 N c i - c 2 = 1 N i = 1 N h + ( x i - x ¯ ) ( x i - x ¯ ) + h = h + Σ x h ,
Σ x = 1 N i = 1 N ( x i - x ¯ ) ( x i - x ¯ ) +
J ( h ) = c ¯ 2 σ 2 + ACE = h + xx ¯ + h h + Σ x h + h + D y h .
xx ¯ + h = ( Σ x + D y ) h .
h = c ( Σ x + D y ) - 1 x ¯ ,
Maximize h h + xx ¯ + h
ϕ ( h ) = h + xx ¯ + h - α ( h + S x h - δ ) - β ( h + D y h - E 0 ) .
( xx ¯ + - α S x - β D y ) h = 0 ,
( α S x + β D y ) h = x ¯ ( x ¯ + h ) ,
h = k ( α S x + β D y ) - 1 x ¯ .
h = c ( S x + γ D y ) - 1 x ¯ .
X i ( k , l ) = A ( k , l ) exp [ j ϕ ( k , l ) ] ,
X ^ i ( k , l ) = exp [ j ϕ ( k , l ) ] .
ACE = h + Ih = h + h .
J = h + xx ¯ + h h + ( S x + δ I ) h .
h = ( S x + δ I ) - 1 x ¯ .
h mace = X ( X + X ) - 1 u .
h umace = x ¯

Metrics