Abstract

Many spatial light modulators and computer-generated hologram techniques can very efficiently implement binary phase-only filters. At present, almost all binary phase-only filters are designed by first designing a matched spatial filter and then binarizing it. There is no theoretical basis to expect that such a filter will be optimal. This paper first develops a theory for optimal binary phase-only filters and then presents a numerical algorithm which designs the optimal binary phase-only matched filter for a given image. The filter is optimal in the matched filter sense of maximizing SNR at the output origin. Characteristics of the optimal filter compared to conventional binary phase-only filters are discussed.

© 1988 Optical Society of America

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References

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  1. A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [CrossRef]
  2. G. G. Lendaris, G. L. Stanley, “Diffraction-Pattern Sampling for Automatic Pattern Recognition,” Proc. IEEE 58, 198 (1979).
    [CrossRef]
  3. K. Johnson, A Review of Spatial Light Modulators and Their Applications, Short Course at O-E/Lase ’88 SPIE Conference, Los Angeles (1988).
  4. H. Farhoosh, M. R. Feldman, S. H. Lee, C. C. Guest, Y. Fainman, R. Eschbach, “Comparison of Binary Encoding Schemes for Electron-Beam Fabrication of Computer Generated Holograms,” Appl. Opt. 26, 4361 (1987).
    [CrossRef] [PubMed]
  5. S. H. Lee, “Computer Generated Holography: An Introduction,” Appl. Opt. 26, 4350 (1987).
    [CrossRef] [PubMed]
  6. S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803 (1985).
    [CrossRef]
  7. H. Dammann, “Synthetic Digital-Phase Gratings—Design, Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 72 (1983).
  8. A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529 (1981).
    [CrossRef]
  9. R. R. Kallman, “Direct Construction of Phase-Only Filters,” Appl. Opt. 26, 5200 (1987).
    [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 (1986).
    [CrossRef] [PubMed]
  11. D. Casasent, W. A. Rozzi, “Computer-Generated and Phase-Only Synthetic Discriminant Function Filters,” Appl. Opt. 25, 3767 (1986).
    [CrossRef] [PubMed]
  12. X. Y. Su, G. S. Zhang, L. R. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
    [CrossRef]
  13. J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315 (1987).
    [CrossRef] [PubMed]
  14. H. O. Bartelt, “Applications of the Tandem Component: an Element with Optimum Light Efficiency,” Appl. Opt. 24, 3811 (1985).
    [CrossRef] [PubMed]
  15. J. L. Horner, H. O. Bartelt, “Two-Bit Correlation,” Appl. Opt. 24, 2889 (1985).
    [CrossRef] [PubMed]
  16. J. L. Horner, P. D. Gianino, “Applying the Phase-Only Filter Concept to the Synthetic Discriminant Function Correlation Filter,” Appl. Opt. 24, 851 (1985).
    [CrossRef] [PubMed]
  17. J. L. Horner, P. D. Gianino, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).
  18. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
    [CrossRef] [PubMed]
  19. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609 (1985).
    [CrossRef] [PubMed]
  20. D. M. Cottrell, J. A. Davis, M. P. Schamschula, R. A. Lilly, “Multiplexing Capabilities of the Binary Phase-Only Filter,” Appl. Opt. 26, 934 (1987).
    [CrossRef] [PubMed]
  21. D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755 (1987).
    [CrossRef] [PubMed]
  22. J. A. Davis, G. M. Heissenberger, R. A. Lilly, D. M. Cottrell, M. F. Brownell, “High Efficiency Optical Reconstruction of Binary Phase-Only Filters using the Hughes Liquid Crystal Light Valve,” Appl. Opt. 26, 929 (1987).
    [CrossRef] [PubMed]
  23. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
    [CrossRef]
  24. F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).
  25. B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-Only Filters,” Proc. Soc. Photo-Opt. Instrum Eng. 884, (1988).
  26. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

1988 (1)

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-Only Filters,” Proc. Soc. Photo-Opt. Instrum Eng. 884, (1988).

1987 (8)

1986 (2)

1985 (5)

1984 (3)

J. L. Horner, P. D. Gianino, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).

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]

1983 (1)

H. Dammann, “Synthetic Digital-Phase Gratings—Design, Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 72 (1983).

1981 (1)

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529 (1981).
[CrossRef]

1979 (1)

G. G. Lendaris, G. L. Stanley, “Diffraction-Pattern Sampling for Automatic Pattern Recognition,” Proc. IEEE 58, 198 (1979).
[CrossRef]

1964 (1)

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

Arnold, S. M.

S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803 (1985).
[CrossRef]

Bahri, Z.

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-Only Filters,” Proc. Soc. Photo-Opt. Instrum Eng. 884, (1988).

Bartelt, H. O.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

Brownell, M. F.

Casasent, D.

Cottrell, D. M.

Dammann, H.

H. Dammann, “Synthetic Digital-Phase Gratings—Design, Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 72 (1983).

Davis, J. A.

Day, T.

Eschbach, R.

Fainman, Y.

Farhoosh, H.

Feldman, M. R.

Gianino, P. D.

Guest, C. C.

Guo, L. R.

X. Y. Su, G. S. Zhang, L. R. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[CrossRef]

Heissenberger, G. M.

Horner, J. L.

Johnson, K.

K. Johnson, A Review of Spatial Light Modulators and Their Applications, Short Course at O-E/Lase ’88 SPIE Conference, Los Angeles (1988).

Kallman, R. R.

Lee, S. H.

Leger, J. R.

Lendaris, G. G.

G. G. Lendaris, G. L. Stanley, “Diffraction-Pattern Sampling for Automatic Pattern Recognition,” Proc. IEEE 58, 198 (1979).
[CrossRef]

Lilly, R. A.

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529 (1981).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529 (1981).
[CrossRef]

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]

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]

Rosen, J.

Rozzi, W. A.

Schamschula, M. P.

Shamir, J.

Stanley, G. L.

G. G. Lendaris, G. L. Stanley, “Diffraction-Pattern Sampling for Automatic Pattern Recognition,” Proc. IEEE 58, 198 (1979).
[CrossRef]

Su, X. Y.

X. Y. Su, G. S. Zhang, L. R. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[CrossRef]

VanderLugt, A.

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

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]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-Only Filters,” Proc. Soc. Photo-Opt. Instrum Eng. 884, (1988).

Yu, F. T. S.

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).

Zhang, G. S.

X. Y. Su, G. S. Zhang, L. R. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[CrossRef]

Appl. Opt. (14)

H. Farhoosh, M. R. Feldman, S. H. Lee, C. C. Guest, Y. Fainman, R. Eschbach, “Comparison of Binary Encoding Schemes for Electron-Beam Fabrication of Computer Generated Holograms,” Appl. Opt. 26, 4361 (1987).
[CrossRef] [PubMed]

S. H. Lee, “Computer Generated Holography: An Introduction,” Appl. Opt. 26, 4350 (1987).
[CrossRef] [PubMed]

R. R. Kallman, “Direct Construction of Phase-Only Filters,” Appl. Opt. 26, 5200 (1987).
[CrossRef] [PubMed]

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]

D. Casasent, W. A. Rozzi, “Computer-Generated and Phase-Only Synthetic Discriminant Function Filters,” Appl. Opt. 25, 3767 (1986).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315 (1987).
[CrossRef] [PubMed]

H. O. Bartelt, “Applications of the Tandem Component: an Element with Optimum Light Efficiency,” Appl. Opt. 24, 3811 (1985).
[CrossRef] [PubMed]

J. L. Horner, H. O. Bartelt, “Two-Bit Correlation,” Appl. Opt. 24, 2889 (1985).
[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 (1985).
[CrossRef] [PubMed]

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

J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609 (1985).
[CrossRef] [PubMed]

D. M. Cottrell, J. A. Davis, M. P. Schamschula, R. A. Lilly, “Multiplexing Capabilities of the Binary Phase-Only Filter,” Appl. Opt. 26, 934 (1987).
[CrossRef] [PubMed]

D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755 (1987).
[CrossRef] [PubMed]

J. A. Davis, G. M. Heissenberger, R. A. Lilly, D. M. Cottrell, M. F. Brownell, “High Efficiency Optical Reconstruction of Binary Phase-Only Filters using the Hughes Liquid Crystal Light Valve,” Appl. Opt. 26, 929 (1987).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

Opt. Eng. (4)

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, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).

X. Y. Su, G. S. Zhang, L. R. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[CrossRef]

S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803 (1985).
[CrossRef]

Proc. IEEE (2)

G. G. Lendaris, G. L. Stanley, “Diffraction-Pattern Sampling for Automatic Pattern Recognition,” Proc. IEEE 58, 198 (1979).
[CrossRef]

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529 (1981).
[CrossRef]

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

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-Only Filters,” Proc. Soc. Photo-Opt. Instrum Eng. 884, (1988).

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

H. Dammann, “Synthetic Digital-Phase Gratings—Design, Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 72 (1983).

Other (3)

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

K. Johnson, A Review of Spatial Light Modulators and Their Applications, Short Course at O-E/Lase ’88 SPIE Conference, Los Angeles (1988).

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Figures (9)

Fig. 1
Fig. 1

Example of assignment of optimal H(p).

Fig. 2
Fig. 2

Division of complex S plane.

Fig. 3
Fig. 3

BPOF decision rules: (a) conventional BPOF; (b) optimal BPOF.

Fig. 4
Fig. 4

Spectral elements Ŝ(k) aligned along real axis: (a) Ŝ(k); (b) Ŝ(k)Ĥ(k) for optimal BPOF; (c) Ŝ(k)Ĥ(k) for conventional BPOF.

Fig. 5
Fig. 5

Spectral elements Ŝ(k) aligned along imaginary axis: (a) Ŝ(k); (b) Ŝ(k)Ĥ(k) for optimal BPOF; (c) Ŝ(k)Ĥ(k) for conventional BPOF.

Fig. 6
Fig. 6

Signal with odd dominance.

Fig. 7
Fig. 7

Magnitude squared of correlation of filters with odd-dominated signal: (a) optimal BPOF; (b) conventional BPOF.

Fig. 8
Fig. 8

Signal with even dominance.

Fig. 9
Fig. 9

Magnitude squared of correlation of filters with even-dominated signal: (a) optimal BPOF; (b) conventional BPOF.

Equations (15)

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SNR = s 0 ( 0 ) 2 σ 2 ,
SNR = | - H ( p ) S ( p ) d p | 2 - H ( p ) 2 Φ n ( p ) d p ,
SNR = | B H ( p ) S ( p ) d p | 2 B H ( p ) 2 Φ n ( p ) d p .
H ( p ) = exp ( j ϕ ) or exp ( j γ ) for all p ,
| B H ( p ) S ( p ) d p | 2 B H ( p ) 2 Φ n ( p ) d p | B H ( p ) S ( p ) d p | 2 .
| B H ( p ) S ( p ) d p | 2 | B H ( p ) S ( p ) d p | .
C = B H ( p ) S ( p ) d p = C exp ( j α ) .
C = Re { [ B H ( p ) S ( p ) d p ] exp ( - j α ) } = B Re [ H ( p ) S ( p ) exp ( - j α ) ] d p
max C = max { B Re [ H ( p ) S ( p ) exp ( - j α ) ] d p } = B max { Re [ H ( p ) S ( p ) exp ( - j α ) ] } d p ,
S ϕ = S ( p ) region 1 S ( p ) d p ,
S γ = S ( p ) region 2 S ( p ) d p .
C = exp ( j ϕ ) S ϕ + exp ( j γ ) S γ .
max C = S ϕ + S γ ,
( ϕ - γ ) = arg ( S γ ) - arg ( S ϕ ) ,
H ^ ( k ) = { 1 if Re [ S ^ ( k ) ] > 0 , - 1 otherwise .

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