Abstract

A technique in which the filter modulation is included in the synthesis of a synthetic-discriminant-function matched spatial filter (SDF MSF) is presented. In the filter synthesis, a system of simultaneous nonlinear equations is solved with an iteration procedure. A computer simulation of the new method using thresholded images of the Space Shuttle over a range of aspect angles was performed for phase-only filters (POFs) and binary-phase-only filters (BPOFs). The filters constructed are capable of obtaining the specified peak-correlation response to within 1% with a high signal-to-clutter-ratio for the one-class problem, the two-class problem, and the multilevel problem. In contrast, conventional projection SDF POFs and BPOFs are unable to produce the desired peak-correlation response.

© 1989 Optical Society of America

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References

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  1. D. P. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. G. F. Schils, D. W. Sweeney, “Rotational Invariant Correlation Filtering,” J. Opt. Soc. Am. A 2, 1411 (1985).
    [CrossRef]
  4. G. F. Schils, D. W. Sweeney, “Iterative Technique for the Synthesis of Optical-Correlation Filters,” J. Opt. Soc. Am. A 3, 1433 (1986).
    [CrossRef]
  5. T. Szoplik, H. H. Arsenault, “Shift and Scale-Invariant Anamorphic Fourier Correlator Using Multiple Circular Harmonic Filters,” Appl. Opt. 24, 3179 (1985).
    [CrossRef] [PubMed]
  6. J. L. Horner, P. D. Gianino, “Applying the Phase-Only Filter Concept to Synthetic Discriminant Function Correlation Filters,” Appl. Opt. 24, 851 (1985).
    [CrossRef] [PubMed]
  7. J. Riggins, S. Butler, “Simulation of Synthetic Discriminant Function Optical Implementation,” Opt. Eng. 23, 721 (1984).
    [CrossRef]
  8. R. R. Kallman, “Optimal Low Noise Phase-Only and Binary Phase-Only Optical Correlation Filters for Threshold Detectors,” Appl. Opt. 25, 4216 (1986).
    [CrossRef] [PubMed]
  9. D. P. Casasent, W. A. Rozzi, “Computer Generated Phase-Only Synthetic Discriminant Function Filters,” Appl. Opt. 25, 3767 (1986).
    [CrossRef] [PubMed]
  10. R. R. Kallman, “Direct Construction of Phase-Only Filters,” Appl. Opt. 26, 5200 (1987).
    [CrossRef] [PubMed]
  11. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
    [CrossRef] [PubMed]
  12. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filter,” Appl. Opt. 24, 609 (1985).
    [CrossRef] [PubMed]
  13. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
    [CrossRef]
  14. J. P. Allebach, J. J. Keegan, “Computer Synthesis of Binary Fourier Transform Holograms Using Ordered Dither,” J. Opt. Soc. Am. 68, 1440 (1978).
  15. A. W. Lohmann, D. P. Paris, “Computer Generated Spatial Filters for Coherent Optical Data Processing,” Appl. Opt. 7, 651 (1968).
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  16. D. A. Jared, D. J. Ennis, “Learned Distortion Invariant Pattern Recognition Using Synthetic-Discriminant-Functions,” Proc. Soc. Photo-Opt. Instrum. Eng. 638, 91 (1986).

1987 (1)

1986 (5)

1985 (4)

1984 (4)

J. Riggins, S. Butler, “Simulation of Synthetic Discriminant Function Optical Implementation,” Opt. Eng. 23, 721 (1984).
[CrossRef]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
[CrossRef] [PubMed]

D. P. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
[CrossRef] [PubMed]

1978 (1)

J. P. Allebach, J. J. Keegan, “Computer Synthesis of Binary Fourier Transform Holograms Using Ordered Dither,” J. Opt. Soc. Am. 68, 1440 (1978).

1968 (1)

Allebach, J. P.

J. P. Allebach, J. J. Keegan, “Computer Synthesis of Binary Fourier Transform Holograms Using Ordered Dither,” J. Opt. Soc. Am. 68, 1440 (1978).

Arsenault, H. H.

Butler, S.

J. Riggins, S. Butler, “Simulation of Synthetic Discriminant Function Optical Implementation,” Opt. Eng. 23, 721 (1984).
[CrossRef]

Casasent, D. P.

Chang, Wen-Thong

Ennis, D. J.

D. A. Jared, D. J. Ennis, “Learned Distortion Invariant Pattern Recognition Using Synthetic-Discriminant-Functions,” Proc. Soc. Photo-Opt. Instrum. Eng. 638, 91 (1986).

Gianino, P. D.

Horner, J. L.

Jared, D. A.

D. A. Jared, D. J. Ennis, “Learned Distortion Invariant Pattern Recognition Using Synthetic-Discriminant-Functions,” Proc. Soc. Photo-Opt. Instrum. Eng. 638, 91 (1986).

Kallman, R. R.

Keegan, J. J.

J. P. Allebach, J. J. Keegan, “Computer Synthesis of Binary Fourier Transform Holograms Using Ordered Dither,” J. Opt. Soc. Am. 68, 1440 (1978).

Leger, J. R.

Lohmann, A. W.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Paris, D. P.

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Riggins, J.

J. Riggins, S. Butler, “Simulation of Synthetic Discriminant Function Optical Implementation,” Opt. Eng. 23, 721 (1984).
[CrossRef]

Rozzi, W. A.

Schils, G. F.

Sweeney, D. W.

Szoplik, T.

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Appl. Opt. (10)

J. Opt. Soc. Am. (1)

J. P. Allebach, J. J. Keegan, “Computer Synthesis of Binary Fourier Transform Holograms Using Ordered Dither,” J. Opt. Soc. Am. 68, 1440 (1978).

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

Opt. Eng. (2)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

J. Riggins, S. Butler, “Simulation of Synthetic Discriminant Function Optical Implementation,” Opt. Eng. 23, 721 (1984).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. A. Jared, D. J. Ennis, “Learned Distortion Invariant Pattern Recognition Using Synthetic-Discriminant-Functions,” Proc. Soc. Photo-Opt. Instrum. Eng. 638, 91 (1986).

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Figures (6)

Fig. 1
Fig. 1

Mapping of a complex-valued MSF S(u,v) to a filter S′(u,v) with allowed modulation values of −1 and 1 (i.e., BPOF).

Fig. 2
Fig. 2

Examples of the threshold out-of-plane-rotated training images of the Space Shuttle at (a) 0°, (b) 80°, and (c) 160° aspect angles.

Fig. 3
Fig. 3

Example of the convergence of iterating Eq. (7) to the desired filter response (case 1) for a BPOF.

Fig. 4
Fig. 4

Peak-correlation response (open squares) peak-clutter response (solid diamonds) using the fSDF method for POFs and BPOFs for three cases of desired filter response.

Fig. 5
Fig. 5

Peak-correlation response (open squares) and peak-clutter response (solid diamonds) using the conventional pSDF method for POFs and BPOFs for three cases of desired filter response.

Fig. 6
Fig. 6

Typical (a) true- and (b) false-class correlation planes for the fSDF-BPOF for Case 3.

Tables (1)

Tables Icon

Table I Definitions for the Filter Modulation Operator

Equations (20)

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R t n ( x , y ) s * ( x , y ) d x d y = t n ( x , y ) s ( x , y ) = c n ,
s ( x , y ) = n = 0 l a n t n ( x , y ) .
a = R - 1 c ,
S ( u , v ) = F [ s ( x , y ) ] * ,
S ( u , v ) = M [ S ( u , v ) ] ,
t n ( x , y ) s ( x , y ) = c n ,
t n ( x , y ) F - 1 M F m = 0 l a m t m ( x , y ) = c n .
m = 0 l a m * t n ( x , y ) F - 1 T m ( u , v ) | p = 0 l a p T p ( u , v ) | = c n ,
S ( u , v ) = M [ S ( u , v ) ] = χ ( θ ) ,
θ = S ( u , v ) S ( u , v ) .
S ( u , v ) = χ ( θ ) = n f n exp ( i n θ ) .
S ( u , v ) = ` n f n exp [ i n m l a m T m ( u , v ) | p l a p T p ( u , v ) | ] .
S ( u , v ) = n f n exp [ i n d m l a m T m ( u , v ) d | p l a p T p ( u , v ) | ] .
a n i + 1 = a n i + β [ c n - c 0 ( m n i m 0 i ) ] ,
m n i = t n ( x , y ) s ( x , y , a i ) .
P = c ( 0 , 0 ) 2 .
C = max ( x , y ) B c ( x , y ) 2 .
Case 1 c n = 1 ;
Case 2 c n = exp ( - n 2 20 ) + exp [ - ( n - 9 ) 2 20 ] ,
Case 3 c n = { 1 true class : for training images from 0 to 80 ° ; 0 false class : for training images from 100 to 180 ° .

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