Abstract

The classical matched-filter definition of the signal-to-noise ratio (SNR) is used as the performance measure to present bounds for the performances of phase-only filters (POF’s), binary phase-only filters (BPOF’s), and N-ary phase-only filters (NPOF’s). Specifically, signal-dependent bounds on the SNR of the POF relative to the ideal matched spatial filter for additive white noise are developed. In addition, the tightest possible bounds on the SNR of the BPOF relative to the POF for any type of additive noise are developed. These bounds are then extended to develop the tightest possible bounds on the SNR of the NPOF relative to the POF. The extension to the generalized BPOF and the generalized NPOF is also discussed.

© 1990 Optical Society of America

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References

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  1. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  2. B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of phase-only filters,” in Computer-Generated Holography, S. H. Lee, ed. Proc. Soc. Photo-Opt. Instrum. Eng.884, 146–152 (1988).
    [CrossRef]
  3. F. M. Dickey, L. A. Romero, “Dual optimality of the phase-only filter,” Opt. Lett. 14, 4–5 (1989).
    [CrossRef] [PubMed]
  4. R. R. Kallman, “Direct construction of phase-only filters,” Appl. Opt. 26, 5200–5201 (1987).
    [CrossRef] [PubMed]
  5. R. R. Kallman, “Optimal low-noise phase-only and binary phase-only optical correlation filters for threshold detectors,” Appl. Opt. 25, 4216–4217 (1986).
    [CrossRef] [PubMed]
  6. J. Rosen, J. Shamir, “Distortion invariant pattern recognition with phase filters,” Appl. Opt. 26, 2315–2319 (1987).
    [CrossRef] [PubMed]
  7. J. L. Horner, H. O. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).
    [CrossRef] [PubMed]
  8. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  9. J. L. Horner, P. D. Gianino, “Additional properties of the phase-only correlation filter,” Opt. Eng. 23, 695–697 (1984).
  10. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  11. F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  12. F. M. Dickey, J. J. Mason, K. T. Stalker, “Analysis of binarized Hartley phase-only filter performance with respect to stochastic noise,” Opt. Eng. 28, 8–13 (1989).
    [CrossRef]
  13. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  14. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  15. B. V. K. Vijaya Kumar, Z. Bahir, “Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio,” Appl. Opt. 28, 1919–1925 (1989).
    [CrossRef]
  16. 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, 3755–3761 (1987).
    [CrossRef] [PubMed]
  17. S. E. Monroe, J. Knopp, R. D. Juday, “Laboratory comparison of continuous versus binary phase-mostly filters,” in Optical Pattern Recognition II, H.-K. Liu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1053, 244–252 (1989).
  18. F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).
  19. M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
    [CrossRef]
  20. H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

1989 (5)

1988 (2)

1987 (3)

1986 (1)

1985 (3)

1984 (2)

J. L. Horner, P. D. Gianino, “Additional properties of the phase-only correlation filter,” Opt. Eng. 23, 695–697 (1984).

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

1979 (1)

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

Bahir, Z.

Bahri, Z.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
[CrossRef]

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of phase-only filters,” in Computer-Generated Holography, S. H. Lee, ed. Proc. Soc. Photo-Opt. Instrum. Eng.884, 146–152 (1988).
[CrossRef]

Bartelt, H. O.

Cottrell, D. M.

Dammann, H.

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

Davis, J. A.

Day, T.

Dickey, F. M.

Farn, M. W.

Gianino, P. D.

Goodman, J. W.

Horner, J. L.

Juday, R. D.

S. E. Monroe, J. Knopp, R. D. Juday, “Laboratory comparison of continuous versus binary phase-mostly filters,” in Optical Pattern Recognition II, H.-K. Liu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1053, 244–252 (1989).

Kallman, R. R.

Knopp, J.

S. E. Monroe, J. Knopp, R. D. Juday, “Laboratory comparison of continuous versus binary phase-mostly filters,” in Optical Pattern Recognition II, H.-K. Liu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1053, 244–252 (1989).

Leger, J. R.

Lilly, R. A.

Mason, J. J.

F. M. Dickey, J. J. Mason, K. T. Stalker, “Analysis of binarized Hartley phase-only filter performance with respect to stochastic noise,” Opt. Eng. 28, 8–13 (1989).
[CrossRef]

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Monroe, S. E.

S. E. Monroe, J. Knopp, R. D. Juday, “Laboratory comparison of continuous versus binary phase-mostly filters,” in Optical Pattern Recognition II, H.-K. Liu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1053, 244–252 (1989).

Romero, L. A.

Rosen, J.

Shamir, J.

Stalker, K. T.

F. M. Dickey, J. J. Mason, K. T. Stalker, “Analysis of binarized Hartley phase-only filter performance with respect to stochastic noise,” Opt. Eng. 28, 8–13 (1989).
[CrossRef]

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

Vijaya Kumar, B. V. K.

Yu, F. T. S.

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

Appl. Opt. (13)

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

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

R. R. Kallman, “Optimal low-noise phase-only and binary phase-only optical correlation filters for threshold detectors,” Appl. Opt. 25, 4216–4217 (1986).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Distortion invariant pattern recognition with phase filters,” Appl. Opt. 26, 2315–2319 (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, 3755–3761 (1987).
[CrossRef] [PubMed]

R. R. Kallman, “Direct construction of phase-only filters,” Appl. Opt. 26, 5200–5201 (1987).
[CrossRef] [PubMed]

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
[CrossRef]

B. V. K. Vijaya Kumar, Z. Bahir, “Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio,” Appl. Opt. 28, 1919–1925 (1989).
[CrossRef]

M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
[CrossRef]

J. L. Horner, H. O. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).
[CrossRef] [PubMed]

J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

Opt. Eng. (2)

J. L. Horner, P. D. Gianino, “Additional properties of the phase-only correlation filter,” Opt. Eng. 23, 695–697 (1984).

F. M. Dickey, J. J. Mason, K. T. Stalker, “Analysis of binarized Hartley phase-only filter performance with respect to stochastic noise,” Opt. Eng. 28, 8–13 (1989).
[CrossRef]

Opt. Lett. (1)

Optik (1)

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

Other (3)

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of phase-only filters,” in Computer-Generated Holography, S. H. Lee, ed. Proc. Soc. Photo-Opt. Instrum. Eng.884, 146–152 (1988).
[CrossRef]

S. E. Monroe, J. Knopp, R. D. Juday, “Laboratory comparison of continuous versus binary phase-mostly filters,” in Optical Pattern Recognition II, H.-K. Liu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1053, 244–252 (1989).

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

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Figures (2)

Fig. 1
Fig. 1

BPOF design algorithm: (a) spectrum S(p, q) in spatial-frequency domain; (b) transform to θ domain, θ being arg[S(p, q)]; (c) designed BPOF in θ domain; (d) reverse mapping to obtain BPOF in spatial-frequency domain.

Fig. 2
Fig. 2

R(θ) transformation: (a) division of B, (b) division of B1, (c) division of B2.

Equations (58)

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SNR = | s 0 ( 0 , 0 ) | 2 σ 2 = | H ( p , q ) S ( p , q ) d p d q | 2 | H ( p , q ) | 2 Φ n ( p , q ) d p d q ,
SNR = | B H ( p , q ) S ( p , q ) d p d q | 2 B | H ( p , q ) | 2 Φ n ( p , q ) d p d q .
S ( p , q ) = | S ( p , q ) | exp [ j θ s ( p , q ) ] .
H MSF ( p , q ) = S * ( p , q ) Φ n ( p , q ) ,
SNR MSF = B | S ( p , q ) | 2 Φ n ( p , q ) d p d q .
H POF ( p , q ) = exp [ j θ s ( p , q ) ] ,
SNR POF = ( B | S ( p , q ) | d p d q ) 2 B Φ n ( p , q ) d p d q .
| B H NPOF ( p , q ) S ( p , q ) d p d q | = max H ( p , q ) NPOF | B H ( p , q ) S ( p , q ) d p d q |
SNR NPOF = [ max H ( p , q ) NPOF | B H ( p , q ) S ( p , q ) d p d q | ] 2 B Φ n ( p , q ) d p d q .
SNR BPOF SNR NPOF SNR POF SNR MSF ,
SNR NPOF SNR GNPOF SNR POF .
M 1 = SNR POF SNR MSF .
M 1 = [ B | S ( p , q ) | d p d q ] 2 B 0 B | S ( p , q ) | 2 d p d q .
| S ( p , q ) | = S ¯ + Δ S ( p , q ) ,
S ¯ = B | S ( p , q ) | d p d q B 0 ,
B Δ S ( p , q ) d p d q = 0 .
M 1 = B 0 S ¯ 2 B 0 S ¯ 2 + B [ Δ S ( p , q ) ] 2 d p d q .
S ¯ S max SNR POF SNR MSF
M 2 = SNR BPOF SNR POF .
m 2 = M 2 = ( max H ( p , q ) BPOF | B H ( p , q ) S ( p , q ) d p d q | ) [ B | S ( p , q ) | d p d q ] .
θ L θ U R ( θ ) d θ = θ L < θ S ( p , q ) < θ U | S ( p , q ) | d p d q ,
0 θ L θ U < 2 π .
m 2 = max H ( θ ) BPOF | 0 2 π H ( θ ) R ( θ ) exp ( j θ ) d θ | 0 2 π R ( θ ) d θ .
H ( θ ) = { + 1 if β + 2 π m θ < β + π + 2 π m for some integer m 1 if β + π + 2 π m θ < β + 2 π + 2 π m for some integer m ,
arg [ 0 2 π H ( θ ) R ( θ ) exp ( j θ ) d θ ] = β + π / 2 .
numerator of m 2 = max H ( θ ) BPOF | 0 2 π H ( θ ) R ( θ ) exp ( j θ ) d θ | = max π / 2 < β π / 2 | β β + π R ( θ ) exp ( j θ ) d θ + β + π β + 2 π R ( θ ) exp [ j ( θ π ) d θ | .
numerator of m 2 = max β [ Re ( { β β + π R ( θ ) exp ( j θ ) d θ + β + π β + 2 π R ( θ ) exp [ j ( θ π ) d θ } exp [ j ( β + π / 2 ) ] ) ] = max β [ β β + π R ( θ ) cos ( θ β π / 2 ) d θ + β β + 2 π R ( θ ) cos ( θ β 3 π / 2 ) d θ ] .
numerator of m 2 = max β [ π / 2 π / 2 R ( θ + β + π / 2 ) cos θ d θ + π / 2 π / 2 R ( θ + β + 3 π / 2 ) cos θ d θ ] = max β [ π / 2 π / 2 T ( θ + β + π / 2 ) cos θ d θ ] = max β [ π / 2 π / 2 T ( θ + β ) cos θ d θ ] .
m 2 = max β [ π / 2 π / 2 T ( θ + β ) cos θ d θ ] π / 2 π / 2 T ( θ ) d θ .
T ( θ ) = T ¯ + Δ T ( θ ) ,
T ¯ = 0 π T ( θ ) d θ π ,
0 π Δ T ( θ ) d θ = 0 .
m 2 = max β [ T ¯ π / 2 π / 2 cos θ d θ + π / 2 π / 2 Δ T ( θ + β ) cos θ d θ T ¯ 0 π d θ = 2 T ¯ + max β [ π / 2 π / 2 Δ T ( θ + β ) cos θ d θ ] π T ¯ .
( 2 / π ) 2 SNR BPOF SNR POF .
M 3 = SNR NPOF SNR POF .
m 3 = M 3 = max H ( p , q ) NPOF | B H ( p , q ) S ( p , q ) d p d q | B | S ( p , q ) | d p d q .
m 3 = max H ( p , q ) NPOF | 0 2 π H ( θ ) R ( θ ) exp ( j θ ) d θ | 0 2 π R ( θ ) d θ .
H ( θ ) = exp ( j k Δ β )
arg [ 0 2 π H ( θ ) R ( θ ) exp ( j θ ) d θ ] = β + Δ β / 2 .
numerator of m 3 = max β { k = 0 N 1 β + k Δ β β + ( k + 1 ) Δ β R ( θ ) cos [ θ β ( k + 1 / 2 ) Δ β ] d θ } .
T ( θ ) = k = 0 N 1 R ( θ + k Δ β ) .
m 3 = max β [ Δ β / 2 Δ β / 2 T ( θ + β ) cos θ d θ ] Δ β / 2 Δ β / 2 T ( θ ) d θ .
m 3 = 2 sin ( Δ β / 2 ) T ¯ + max β [ Δ β / 2 Δ β / 2 Δ T ( θ + β ) cos θ d θ ] Δ β T ¯ .
[ sin ( π / N ) π / N ] 2 SNR NPOF SNR POF
S ¯ S max SNR POF SNR MSF 1 for white noise ,
[ sin ( π / N ) π / N ] 2 SNR NPOF SNR POF 1 for any noise ,
[ sin ( π / N ) π / N ] 2 SNR GNPOF SNR POF 1 for any noise .
max B f 2 ( x ) d x = L 2 f ( x ) = L d x + U 2 f ( x ) = U d x = L 2 B 0 U L U + U 2 B 0 L L U = B 0 L U .
I = θ L θ U R ( θ ) d θ = θ L < θ S ( p ) < θ U | S ( p ) | d p .
I 1 = B 2 ( θ L < θ S ( p ) < θ U ) | S ( p ) | d p .
I 2 = B 2 ( θ L < θ S ( p ) < θ U ) | S ( p ) | d p .
I 1 = p 2 p 3 | S ( p ) | d p + p 3 p 4 | S ( p ) | d p = θ 2 θ 3 | S [ θ S 1 ( θ ) ] | / θ S d θ + θ 3 θ 4 | S [ θ S 1 ( θ ) ] | θ S d θ = θ L θ U R 1 ( θ ) d θ ,
R 1 ( θ ) = all regions in B 1 | S [ θ S 1 ( θ ) ] | / | θ S | .
I 2 = p 1 p 2 | S ( p ) | d p + p 4 p 5 | S ( p ) | d p = θ L θ U R 2 ( θ ) d θ ,
R 2 ( θ ) = all regions in B 2 { θ s ( p ) = θ | S [ θ S 1 ( θ ) ] | d p } δ ( θ θ S ) .
R n = 2 π ( n 1 / 2 ) / N < θ S ( p ) < 2 π ( n + 1 / 2 ) / N | S ( p ) | d p ;
c ( y ) = X f ( x + y ) g ( x ) d x .
X c ( y ) d y = X X f ( x + y ) g ( x ) d x d y = X [ X f ( x + y ) d y ] g ( x ) d x = 0 .

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