Abstract

Very efficient suboptimal algorithms for the design of phase-only filters and binary phase-only filters are presented. A reduction of 2- to 3-orders of magnitude in computer time is obtained over previous algorithms. The loss in signal-to-noise ratio is negligible (<0.001 dB).

© 1990 Optical Society of America

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References

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  1. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  2. R. R. Kallman, “Direct Construction of Phase-Only Filters,” Appl. Opt. 26, 5200–5201 (1987).
    [CrossRef] [PubMed]
  3. J. L. Horner, H. O. Bartelt, “Two-Bit Correlation,” Appl. Opt. 24, 2889–2893 (1985).
    [CrossRef] [PubMed]
  4. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
    [CrossRef]
  5. D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Design Elements of Binary Phase-Only Correlation Filters,” Appl. Opt. 27, 4231–4235 (1988).
    [CrossRef] [PubMed]
  6. M. W. Farn, J. W. Goodman, “Optimal Binary Phase-Only Matched Filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  7. F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth Considerations for Binary Phase-Only Filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  8. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters With Improved Signal-to-Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  9. B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing a Ternary Valued Filter Yielding Maximum Signal to Noise Ratio,” Appl. Opt. 28, 1919–25 (1989).
    [CrossRef]
  10. Z. Bahri, “Phase-Only and Binary Phase-Only Filters for Optical Correlators,” Ph.D. Dissertation, Carnegie Mellon University, 1989.
  11. F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Optical Eng. 29, (1989).
  12. R. Conway, D. Gries, An Introduction to Programming: A Structured Approach Using PL/I and PL/C (Winthrop Publishers, Cambridge, MA, 1979).

1989 (3)

1988 (3)

1987 (1)

1985 (1)

1984 (2)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

Bahri, Z.

Bartelt, H. O.

Connelly, J. M.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Optical Eng. 29, (1989).

Conway, R.

R. Conway, D. Gries, An Introduction to Programming: A Structured Approach Using PL/I and PL/C (Winthrop Publishers, Cambridge, MA, 1979).

Dickey, F. M.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Optical Eng. 29, (1989).

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth Considerations for Binary Phase-Only Filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Farn, M. W.

Flannery, D. L.

Gianino, P. D.

Goodman, J. W.

Gries, D.

R. Conway, D. Gries, An Introduction to Programming: A Structured Approach Using PL/I and PL/C (Winthrop Publishers, Cambridge, MA, 1979).

Horner, J. L.

Kallman, R. R.

Loomis, J. S.

Mason, J. J.

Milkovich, M. E.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Romero, L. A.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Optical Eng. 29, (1989).

Stalker, K. T.

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Vijaya Kumar, B. V. K.

Appl. Opt. (8)

Opt. Eng. (1)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Optical Eng. (1)

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Optical Eng. 29, (1989).

Other (2)

R. Conway, D. Gries, An Introduction to Programming: A Structured Approach Using PL/I and PL/C (Winthrop Publishers, Cambridge, MA, 1979).

Z. Bahri, “Phase-Only and Binary Phase-Only Filters for Optical Correlators,” Ph.D. Dissertation, Carnegie Mellon University, 1989.

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Figures (2)

Fig. 1
Fig. 1

The 32 × 32 tank image used in the simulations.

Fig. 2
Fig. 2

The 32 × 32 pliers image used in the simulations.

Equations (20)

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SNR OPOF = [ R S ( f ) d f ] 2 R P n ( f ) d f .
S ( 1 ) S ( 2 ) S ( d ) ,
SNR ( K ) = α [ i = 1 K S ( i ) ] 2 K ,
H ( f ) = I R ( f ) Sgn [ S r ( f ) cos ( θ * ) + S i ( f ) sin ( θ * ) ] ,
Sgn ( x ) = { + 1 if x 0 - 1 otherwise .
R 1 = { f R : S r ( f ) cos ( θ ) > S i ( f ) sin ( θ ) }
R 2 = { f R : S r ( f ) cos ( θ ) S i ( f ) sin ( θ ) } .
SNR = [ R 1 S r ( f ) d f ] 2 + [ R 2 S i ( f ) d f ] 2 R 1 R 2 P n ( f ) d f ,
P 1 = { n : S r ( n ) cos ( θ ) > S i ( n ) sin ( θ ) } ,
P 2 = { n : S r ( n ) cos ( θ ) S i ( n ) sin ( θ ) } .
SNR = α [ n = 1 K 1 S r n ] 2 + [ n = 1 K 2 S i n ] 2 K 1 + K 2 ,
SNR ( K 0 ) SNR ( K 0 - 1 ) .
β = i = 1 K 0 S ( i ) .
SNR ( K 0 + 1 ) - SNR ( K 0 ) = ( β + S ( K 0 ) ) 2 K 0 + 1 - β 2 K 0 = 1 K 0 ( K 0 + 1 ) [ K 0 ( S ( K 0 ) 2 + 2 β S ( K 0 ) ) - β 2 ] .
SNR ( K 0 ) - SNR ( K 0 - 1 ) = β 2 K 0 - ( β - S ( K 0 ) ) 2 ( K 0 - 1 ) = [ - β 2 + K 0 ( 2 β S ( K 0 ) - S ( K 0 ) 2 ) ] / [ K 0 ( K 0 - 1 ) ] 0.
K 0 ( 2 β S ( K 0 ) + S ( K 0 ) 2 ) - β 2 0.
SNR ( K 0 ) < SNR ( K 0 - 1 ) .
K opt < K 0 or K opt K 0 + 1 ,
SNR ( K ) = [ i = 1 K S ( i ) ] 2 + a K + b ,
M 1 = Max { S r ( n ) , n P 1 } , M 2 = Max { S i ( n ) , n P 2 } , M = Max [ M 1 , M 2 ] , S ( n ) S ( n ) / M , n = 1 , , d .

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