Abstract

The synthesis of a new category of spatial filters that produces sharp output correlation peaks with controlled peak values is considered. The sharp nature of the correlation peak is the major feature emphasized, since it facilitates target detection. Since these filters minimize the average correlation plane energy as the first step in filter synthesis, we refer to them as minimum average correlation energy filters. Experimental laboratory results from optical implementation of the filters are also presented and discussed.

© 1987 Optical Society of America

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References

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  1. A. B. VanderLugt, “Signal Detection by Complex Matched Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [Crossref]
  2. D. Casasent, D. Psaltis, “Position, Rotation, and Scale Invariant Optical Correlation,” Appl. Opt. 15, 1795 (1976).
    [Crossref] [PubMed]
  3. Y. N. Hsu, H. H. Arsenault, “Optical Pattern Recognition using the Circular Harmonic Expansion,” Appl. Opt. 21, 4016 (1982).
    [Crossref] [PubMed]
  4. H. J. Caulfield, M. H. Weinberg, “Computer Recognition of 2-D Patterns using Generalized Matched Filters,” Appl. Opt. 21, 1699 (1982).
    [Crossref] [PubMed]
  5. D. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
    [Crossref] [PubMed]
  6. D. Casasent, W. T. Chang, “Correlation Synthetic Discriminant Functions,” Appl. Opt. 25, 2343 (1986).
    [Crossref] [PubMed]
  7. R. R. Kallman, “Construction of Low Noise Optical Correlation Filters,” Appl. Opt. 25, 1032 (1986).
    [Crossref] [PubMed]
  8. B. V. K. Vijaya Kumar, “Minimum Variance Synthetic Discriminant Functions,” J. Opt. Soc. Am. A 3, 1579 (1986).
    [Crossref]
  9. B. V. K. Vijaya Kumar, A. Mahalanobis, “Alternate Interpretation for Minimum Variance Synthetic Discriminant Functions,” Appl. Opt. 25, 2484 (1986).
    [Crossref]
  10. J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).
  11. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
    [Crossref] [PubMed]

1986 (4)

1984 (2)

1982 (2)

1976 (1)

1964 (1)

A. B. VanderLugt, “Signal Detection by Complex Matched Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
[Crossref]

Arsenault, H. H.

Casasent, D.

Caulfield, H. J.

Chang, W. T.

Gianino, P. D.

Golosman, S.

J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).

Hines, D. C.

J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).

Horner, J. L.

Hsu, Y. N.

Kallman, R. R.

Kuglin, C. D.

J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).

Mahalanobis, A.

Pearson, J. J.

J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).

Psaltis, D.

VanderLugt, A. B.

A. B. VanderLugt, “Signal Detection by Complex Matched Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
[Crossref]

Vijaya Kumar, B. V. K.

Weinberg, M. H.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, “Signal Detection by Complex Matched Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
[Crossref]

J. Opt. Soc. Am. A (1)

Other (1)

J. J. Pearson, D. C. Hines, S. Golosman, C. D. Kuglin, “Video-Rate Image Correlation Processor,” Proc. Soc. Photo-Opt. Instrum. Eng.119, 197 (IOCC1977).

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Figures (5)

Fig. 1
Fig. 1

MACE filter as a cascade of the prefilter P and the projection SDF H ¯.

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Fig. 2
Fig. 2

Plot of training image correlation energies before ● and after × iterative scatter reduction.

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Fig. 3
Fig. 3

(A) True class correlation plane for tank; (B) false class correlation plane for APC.

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Fig. 4
Fig. 4

Input plane test scene for optical correlator.

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Fig. 5
Fig. 5

(A) Cross section of class 1 (tanks) optical correlation peaks; (B) cross section of class 1 and 2 (tank and APC) optical correlation peaks.

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Tables (3)

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Table I Iterative Scatter Reduction Algorithm

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Table II Correlation Plane Statistics for Class 1 Image (Tanks)

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Table III Correlation Plane Statistics for Class 2 Image (APC)

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Equations (47)

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x i = [ x i ( 1 ) , x i ( 2 ) … , x i ( d ) ] T .
X = [ X 1 , X 2 , … , X N ]
g i ( n ) = x i ( n ) ⊛ h ( n ) .
E i = ∑ n = 1 d | g i ( n ) | 2 = ( 1 / d ) ∑ k = 1 d | G i ( k ) | 2 = ( 1 / d ) ∑ k = 1 d | H ( k ) | 2 | X i ( k ) | 2 .
E i = H + D i H ,
D i ( k , k , ) = | X i ( k ) | 2 .
g i ( 0 ) = X i + H = u i
E i = H + D i H
X + H = u .
E av = ( 1 / N ) ∑ i = 1 N E i = ( 1 / N ) ∑ i = 1 N H + D i H = ( 1 / N ) H + ( ∑ i = 1 N D i ) H .
D = ∑ i = 1 N α i D i ,
E av = ( 1 / N ) H + D H , α i = 1 , i = 1 , 2 , … , N .
H = D − 1 X ( X + D − 1 X ) − 1 u .
H = AX ( X + AX ) − 1 u
X + H = u .
H = D − 1 X ( X + D − 1 X ) − 1 u ,
H = P ( PX ) ( X + P PX ) − 1 u .
H = P X ¯ ( X ¯ + X ¯ ) − 1 u .
H = P H ¯ .
X ¯ i ( k ) = F ( k ) X i ( k ) .
X ¯ = S − 0.5 X
H ¯ = D ¯ − 1 X ¯ ( X ¯ + D ¯ − 1 X ¯ ) − 1 u .
D ¯ = ∑ i α i D ¯ i
| X ¯ i ( k ) | 2 = | S − 0.5 ( k , k ) | 2 | X i ( k ) | 2 .
S − 1 = S − 0.5 ( S − 0.5 ) + .
D ¯ i = S − 1 D i = S − 0.5 D i ( S − 0.5 ) + ,
D ¯ = S − 1 ∑ i = 1 N α i D i = S − 1 D = DS − 1 ,
D ¯ − 1 = D ¯ − 1 S = S 0.5 D − 1 ( S 0.5 ) + = ( S 0.5 ) + D − 1 S 0.5 .
H ¯ = D − 1 SS − 0.5 X [ X + ( S − 0.5 ) + ( S 0.5 ) + D − 1 S 0.5 S − 0.5 X ] − 1 u = ( S 0.5 ) + D − 1 X ( X + D − 1 X ) − 1 u = ( S 0.5 ) + H .
E ¯ i = H ¯ + D ¯ i H ¯ = [ H + S 0.5 ] [ S − 0.5 D i ( S − 0.5 ) + ] [ ( S 0.5 ) + H ] = H + ( S 0.5 S − 0.5 ) D i ( S − 0.5 S 0.5 ) + H = H + D i H .
D ( k , k ) = | X ( k ) | 2 .
X + D − 1 X = ∑ i = 1 d ∑ j = 1 d X * ( i ) X ( j ) D − 1 ( i , j ) ,
X + D − 1 X = ∑ i = 1 d X * ( i ) X ( i ) | X ( i ) | 2 = d .
H = u D − 1 X ,
H ( k ) = u X ( k ) | X ( k ) | 2 .
X * ( k ) H ( k ) = u X * ( k ) X ( k ) | X ( k ) | 2 = u = constant .
x ( n ) ⊛ h ( n ) = u δ ( n ) ,
σ 2 = 1 N ∑ i = 1 N ( E i − E av ) 2 .
∑ i = 1 N α i E i
X + H = u ,
ϕ = H + A H − 2 λ 1 ( H + X 1 − u 1 ) − … − 2 λ N ( H + X N − u N ) ,
A H = λ 1 X 1 + … + λ N X N ,
H = A − 1 [ ∑ i = 1 N λ i X i ] = ∑ i = 1 N λ i A − 1 X i .
H = A − 1 X L .
X + A − 1 X L = u ,
L = ( X + A − 1 X ) − 1 u .
H = A − 1 X ( X + A − 1 X ) − 1 u .

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