Abstract

Minimizing a Euclidean distance in the complex plane optimizes a wide class of correlation metrics for filters implemented on realistic devices. The algorithm searches over no more than two real scalars (gain and phase). It unifies a variety of previous solutions for special cases (e.g., a maximum signal-to-noise ratio with colored noise and a real filter and a maximum correlation intensity with no noise and a coupled filter). It extends optimal partial information filter theory to arbitrary spatial light modulators (fully complex, coupled, discrete, finite contrast ratio, and so forth), additive input noise (white or colored), spatially nonuniform filter modulators, and additive correlation detection noise (including signal-dependent noise). An appendix summarizes the algorithm as it is implemented in available computer code.

© 1993 Optical Society of America

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References

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  1. B. V. K. Vijaya Kumar, “A tutorial review of partial-information filter designs for optical correlators,” Asia Pacific Eng. J. Part A 2, 203–215 (1992).
  2. R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988) pp. 238–241.
  3. R. D. Juday, “Correlation with a spatial light modulator having phase and amplitude cross coupling,” Appl. Opt. 28, 4865–4869 (1989).
    [CrossRef] [PubMed]
  4. M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
    [CrossRef]
  5. B. V. K. Vijaya Kumar, R. D. Juday, P. K. Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
    [CrossRef]
  6. R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 499–504 (1991).
  7. J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 487–498 (1991).
  8. E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.
  9. D. A. Gregory, J. C. Kirsch, “Full complex modulation using liquid crystal televisions,” Appl. Opt. 31, 163–165 (1992).
    [CrossRef] [PubMed]
  10. See, for example, R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991), and references therein.
    [CrossRef] [PubMed]
  11. G.-G. Mu, X.-M. Wang, Z.-Q. Wang, “Amplitude-compensated matched filtering,” Appl. Opt. 27, 3461–3463 (1988).
    [CrossRef] [PubMed]
  12. L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
    [CrossRef] [PubMed]
  13. R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimum real correlation filters,” Appl. Opt. 30, 520–522 (1991).
    [CrossRef] [PubMed]
  14. B. V. K. Vijaya Kumar, W. Shi, C. Hendrix, “Phase-only filters with maximally sharp correlation peaks,” Opt. Lett. 15, 807–809 (1990).
    [CrossRef]
  15. Ph. Réfrégier, “Optimal trade-off filter for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  16. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  17. B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complex ternary matched filters with increased signal-to-noise ratios for colored noise,” Opt. Lett. 16, 1025–1027 (1991).
    [CrossRef]
  18. F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
    [CrossRef]
  19. K. Lu, B. E. A. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” Appl. Opt. 30, 2354–2362 (1991).
    [CrossRef] [PubMed]
  20. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  21. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  22. D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
    [CrossRef] [PubMed]
  23. F. M. Dickey, B. D. Hansche, “Quad-phase-only filters for pattern recognition,” Appl. Opt. 28, 1611–1613 (1989).
    [CrossRef] [PubMed]
  24. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992), and references therein.
    [CrossRef] [PubMed]
  25. J. D. Downie, B. P. Hine, M. B. Reid, “Effects and correction of magneto-optic spatial light modulator phase errors in an optical correlator,” Appl. Opt. 31, 636–643 (1992).
    [CrossRef] [PubMed]
  26. “medof: minimum Euclidean distance optimal filter,” computer code available under NT control number MSC-22380 from COSMIC (Computer Software Management Information Center), University of Georgia, 382 East Broad Street, Athens, Ga. 30602-4272. Telephone (706) 542–3265.

1992 (6)

1991 (5)

1990 (2)

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
[CrossRef]

B. V. K. Vijaya Kumar, W. Shi, C. Hendrix, “Phase-only filters with maximally sharp correlation peaks,” Opt. Lett. 15, 807–809 (1990).
[CrossRef]

1989 (4)

1988 (3)

1984 (1)

Bahri, Z.

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,” Opt. Eng. 29, 994–1001 (1990).
[CrossRef]

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,” Opt. Eng. 29, 994–1001 (1990).
[CrossRef]

F. M. Dickey, B. D. Hansche, “Quad-phase-only filters for pattern recognition,” Appl. Opt. 28, 1611–1613 (1989).
[CrossRef] [PubMed]

Downie, J. D.

Eschbach, R.

Farn, M. W.

Flannery, D. L.

Florence, J. M.

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 499–504 (1991).

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 487–498 (1991).

Gianino, P. D.

Goodman, J. W.

Gregory, D. A.

D. A. Gregory, J. C. Kirsch, “Full complex modulation using liquid crystal televisions,” Appl. Opt. 31, 163–165 (1992).
[CrossRef] [PubMed]

E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.

Hansche, B. D.

Hendrix, C.

Hine, B. P.

Horner, J. L.

Juday, R. D.

B. V. K. Vijaya Kumar, R. D. Juday, P. K. Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complex ternary matched filters with increased signal-to-noise ratios for colored noise,” Opt. Lett. 16, 1025–1027 (1991).
[CrossRef]

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimum real correlation filters,” Appl. Opt. 30, 520–522 (1991).
[CrossRef] [PubMed]

R. D. Juday, “Correlation with a spatial light modulator having phase and amplitude cross coupling,” Appl. Opt. 28, 4865–4869 (1989).
[CrossRef] [PubMed]

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 499–504 (1991).

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988) pp. 238–241.

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 487–498 (1991).

Kirsch, J. C.

Kirsch, J. M.

E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.

Loomis, J. S.

Lu, K.

Milkovich, M. E.

Mu, G.-G.

Rajan, P. K.

Réfrégier, Ph.

Reid, M. B.

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,” Opt. Eng. 29, 994–1001 (1990).
[CrossRef]

Saleh, B. E. A.

Shi, W.

Tam, E. C.

E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.

Vijaya Kumar, B. V. K.

Wang, X.-M.

Wang, Z.-Q.

Yaroslavsky, L. P.

Zhou, S.

E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.

Appl. Opt. (15)

R. D. Juday, “Correlation with a spatial light modulator having phase and amplitude cross coupling,” Appl. Opt. 28, 4865–4869 (1989).
[CrossRef] [PubMed]

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

D. A. Gregory, J. C. Kirsch, “Full complex modulation using liquid crystal televisions,” Appl. Opt. 31, 163–165 (1992).
[CrossRef] [PubMed]

See, for example, R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991), and references therein.
[CrossRef] [PubMed]

G.-G. Mu, X.-M. Wang, Z.-Q. Wang, “Amplitude-compensated matched filtering,” Appl. Opt. 27, 3461–3463 (1988).
[CrossRef] [PubMed]

L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
[CrossRef] [PubMed]

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimum real correlation filters,” Appl. Opt. 30, 520–522 (1991).
[CrossRef] [PubMed]

K. Lu, B. E. A. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” Appl. Opt. 30, 2354–2362 (1991).
[CrossRef] [PubMed]

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

D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
[CrossRef] [PubMed]

F. M. Dickey, B. D. Hansche, “Quad-phase-only filters for pattern recognition,” Appl. Opt. 28, 1611–1613 (1989).
[CrossRef] [PubMed]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992), and references therein.
[CrossRef] [PubMed]

J. D. Downie, B. P. Hine, M. B. Reid, “Effects and correction of magneto-optic spatial light modulator phase errors in an optical correlator,” Appl. Opt. 31, 636–643 (1992).
[CrossRef] [PubMed]

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

Asia Pacific Eng. J. Part A (1)

B. V. K. Vijaya Kumar, “A tutorial review of partial-information filter designs for optical correlators,” Asia Pacific Eng. J. Part A 2, 203–215 (1992).

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

Opt. 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,” Opt. Eng. 29, 994–1001 (1990).
[CrossRef]

Opt. Lett. (3)

Other (5)

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 499–504 (1991).

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” in Wave Propagation and Scattering in Varied Media, V. K. Varadan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1558, 487–498 (1991).

E. C. Tam, S. Zhou, D. A. Gregory, J. M. Kirsch, “Phase/amplitude spatial light modulators using 90° twisted nematic liquid crystal devices,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8 November 1991.

“medof: minimum Euclidean distance optimal filter,” computer code available under NT control number MSC-22380 from COSMIC (Computer Software Management Information Center), University of Georgia, 382 East Broad Street, Athens, Ga. 30602-4272. Telephone (706) 542–3265.

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988) pp. 238–241.

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Figures (8)

Fig. 1
Fig. 1

Region of realizable complex filter values, regarded as the union of its boundary 1 and its interior 2. Different constraints apply to filter values in each.

Fig. 2
Fig. 2

Relationships among coupled filter’s parameters. Drive V produces filter value H with amplitude μ and phase ν. As V increases differentially, the filter change has magnitude Γ and phase γ.

Fig. 3
Fig. 3

Optimal realizable value H0 compared with the target H+, calculated as given in text. Points 1 and 2 (projections onto the direction of tangent) must actually be the same, so H0 is the closest realizable point to H+.

Fig. 4
Fig. 4

SNR as a function of H m if all other filter values were fixed. H+ would yield peak SNR, if realizable (and consistent). H m is chosen at the indicated H0, since the SNR is highest there of all realizable values. This is a useful but slightly specious drawing; H+ would not be fixed (nor would the iso-SNR curves) as H m is moved about, except in the limit of small spectral SNR. In that limit the iso-SNR curves are circles, supporting the claim that the Euclidean-closest value is selected.

Fig. 5
Fig. 5

Determination of H+ for peak SNR. Individual samples of S(f) are plotted in the complex signal space on the left, and their corresponding complex filter values are plotted on the right. The signal values are scaled (G), whitened (P n ), conjugated (−ϕ), and rotated (β) to give H+. Since they lie outside the realizable region, frequencies represented by the open symbols are saturated and must be mapped to realizable filter values.

Fig. 6
Fig. 6

Discrete SLM’s produce straight-line decision boundaries for the determination of the closest realizable filter value. Dots are realizable complex values H0, and they lie in their regions of associated target filter values H+, as indicated by shading: (a) general binary filter; (b) binary phase-only filter; (c) binary filter plus a region of support; interestingly, calculated filter values of arbitrarily large magnitude in the clear segment are mapped to zero; (d) quad-phase filter plus a region of support, which is the same as for a complex ternary filter.

Fig. 7
Fig. 7

Some continuous SLM’s and filter mappings. Hollow dots are target filter values H+, and solid dots are realized values H0: (a) a passive real positive SLM, such as photographic film; (b) the detour phase computer-generated hologram, which has a realizable interior, indicated by solid square dots at which H+ = H0;(c) a phase-only SLM with a region of support; (d) an eccentric phase SLM, such as one version of a deformable mirror device. The constant-phase dashed lines show that the optimal filter on such a device is not a matched-phase filter; note the different phases from mapping of the two indicated points.

Fig. 8
Fig. 8

Mappings for a phase-only filter SLM having less than 2π modulation range. Hollow dots are target filter values H+, and solid dots are realized values H0: (a) zero is not possible; there is no sensitivity to gain G; (b) zero is possible; G might put a value of H+ inside or outside the dashed boundary.

Tables (1)

Tables Icon

Table 1 Elements of the Metric Function for Various Metrics on the Filtered Signal

Equations (48)

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H ( V ) = d d V { μ ( V ) exp [ j ν ( V ) ] } = : Γ ( V ) exp [ j γ ( V ) ] ,
μ μ Γ = μ cos ( γ - ν ) = : τ ,
c ( 0 ) = B exp ( j β ) : = k H k S k .
B 2 θ m = 2 B A m M m sin ( β - ϕ m - θ m ) ,
B 2 M m = 2 B A m cos ( β - ϕ m - θ m ) ,
B 2 V m = 2 B Γ m A m cos ( β - γ m - ϕ m ) .
Intensity = B 2 ,
SNR = B 2 σ d 2 + k P n k M k 2 ,
PCE = B 2 k A k 2 M k 2 ,
PTE = B 2 σ d 2 + k M k 2 ( P n k + A k 2 ) .
t correlation plane c ( t ) 2 = k A k 2 M k 2
T = g [ B 2 ( { H k } ) ] p [ B 2 ( { H k } ) ] + h [ { M k 2 } ] .
G : = B [ g ( p + h ) - g p ] g .
θ m = β - ϕ m + n π ,
H + = G A m h M m 2 exp [ j ( β - ϕ m ) ] .
G A m h 2 M m 2 cos [ β - γ ( V m ) - ϕ m ] = μ m μ m Γ m = τ ( V m ) .
P 1 = G A m h M m 2 cos ( β - γ m - ϕ m ) ,
2 H + - H 0 H cos = { M + exp [ - j ( β - ϕ ) ] - μ exp [ - j ν ] } Γ exp [ + j γ ] + { M + exp [ + j ( β - ϕ ) ] - μ exp [ + j ν ] } Γ exp [ - j γ ] = M + cos ( β - ϕ - γ ) - μ cos ( ν - γ ) = M + cos ( β - ϕ - γ ) - μ μ Γ = 0 ,
H SNR + = G SNR A m P n , m exp [ j ( β - ϕ m ) ] ,
H PCE + = G PCE 1 A m exp [ j ( β - ϕ m ) ] ,
H PTE + = G PTE A m A m 2 + P n , m exp [ j ( β - ϕ m ) ] ,
G SNR : = σ d 2 + k P n k M k 2 B ,
G PCE : = k A k 2 M k 2 B ,
G PTE : = σ d 2 + k ( P n k + A k 2 ) M k 2 B .
( B 2 V m = 0 ) [ 2 A m B Γ m cos ( β - ϕ m - γ m ) = 0 ] .
γ m = β - ϕ m + 2 n + 1 2 π ,
p ( B 2 ) = σ d 2 + c B 2 ,
G = p + h - B 2 p B 2 = σ d 2 + c B 2 - B 2 c + h B 2 = σ d 2 + h B 2 ,
- π ( 1 - M ) θ + π ( 1 - M ) .
μ = ν / π ,
D - = σ d 2 + k m P n k M k 2 ,
B - exp ( j β - ) = k m H k S k ,
SNR - = ( B - ) 2 D - .
SNR = [ B - exp ( j β - ) + H m S m ] [ B - exp ( - j β - ) + H m * S m * ] D - + P n m M m 2 = ( B - ) 2 + A m 2 M m 2 + 2 A m M m B - cos ( β - - ϕ m - θ m ) D - + P n m M m 2 .
M m = D - B - A m P n m .
H + = D - B - A m P n m exp [ j ( β - ϕ m ) ] .
H m = H + + H + d exp j δ = H + ( 1 + d exp j δ ) ,
SNR m = B - exp j β - + A m exp j ϕ m H + ( 1 + d exp j δ ) 2 D - + P n m H + ( 1 + d exp j δ ) 2 ,
K : = A m 2 / P n m ( B - ) 2 / D - .
SNR m = 1 + 2 K ( 1 + d cos δ ) + K 2 ( 1 + 2 d cos δ + d 2 ) 1 + K ( 1 + 2 d cos δ + d 2 ) ,
SNR 1 + K ( 1 - d 2 ) + K 2 d 2 ( 1 + 2 d cos δ + d 2 ) .
SNR = B 2 σ native 2 + k P n k M k 2 ,
SNR = R B 2 σ native 2 + R k P n k M k 2 ,
s ( x ) = cos ( k x )
I ( x ) = W cos ( k x ) 2 = W 2 cos 2 ( k x ) = W 2 2 + W 2 2 cos ( 2 k x ) ,
D 2 = R W 2 2 ,
R = 2 D 2 W 2 .
SNR = B 2 σ native 2 R + k P n k M k 2 ,

Metrics