Abstract

Most previous research into the design of correlation filters considered only input noise and filter spatial light modulators (SLM’s) of an implicitly assumed infinite contrast ratio. We introduce a signal-to-noise ratio that also includes correlation-detector noise and finite contrast SLM’s. Filters maximizing this signal-to-noise ratio exhibit saturation at some frequencies and are called saturated filters. We accommodate SLM’s whose amplitude has a finite maximum and a nonzero minimum. We give algorithms for optimum saturated complex-and real-valued filters. Previous results are reproduced as various limiting cases. The phase-only filter and the binary phase-only filter are limiting cases for large detector noise with, respectively, complex and real modulators.

© 1992 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. F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  3. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  4. B. V. K. Vijaya Kumar, Z. Bahri, “Efficient algorithm for designing a ternary-valued filter yielding maximum signal-to-noise ratio,” Appl. Opt. 28, 1919–1925 (1989).
    [CrossRef]
  5. B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complexternary matched filters with increased signal-to-noise ratios for colored noise,” Opt. Lett. 16, 1025–1027 (1991).
    [CrossRef]
  6. B. V. K. Vijaya Kumar, V. Liang, R. D. Juday, “Optimal phase-only correlation filters in colored scene noise,” in Computer and Optically Generated Holographic Optics, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 138–145 (1991).
    [CrossRef]
  7. 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]
  8. A. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  9. R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 30, 520–522 (1991).
    [CrossRef] [PubMed]
  10. 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.
  11. R. D. Juday, “Correlation with a spatial light modulator having phase and amplitude cross coupling,” Appl. Opt. 28, 4865–4869 (1989).
    [CrossRef] [PubMed]

1991 (2)

1990 (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]

1989 (3)

1988 (1)

1984 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

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

Gianino, P. D.

Horner, J. L.

Juday, R. D.

B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complexternary 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, “Optimal 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]

B. V. K. Vijaya Kumar, V. Liang, R. D. Juday, “Optimal phase-only correlation filters in colored scene noise,” in Computer and Optically Generated Holographic Optics, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 138–145 (1991).
[CrossRef]

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.

Liang, V.

B. V. K. Vijaya Kumar, V. Liang, R. D. Juday, “Optimal phase-only correlation filters in colored scene noise,” in Computer and Optically Generated Holographic Optics, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 138–145 (1991).
[CrossRef]

Mason, J. J.

Rajan, P. K.

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]

Stalker, K. T.

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Vijaya Kumar, B. V. K.

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

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. (1)

Other (2)

B. V. K. Vijaya Kumar, V. Liang, R. D. Juday, “Optimal phase-only correlation filters in colored scene noise,” in Computer and Optically Generated Holographic Optics, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 138–145 (1991).
[CrossRef]

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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Equations (69)

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c ( 0 ) = S ( f ) H ( f ) d f ,
E { c ( 0 ) } = μ n H ( 0 ) + S ( f ) H ( f ) d f ,
var { c ( 0 ) } = P n ( f ) H ( f ) 2 d f .
SNR = | S ( f ) H ( f ) d f | 2 P n ( f ) H ( f ) 2 d f .
y = c ( 0 ) + n d .
E { y H 0 } = μ d + E { c ( 0 ) H 0 } = μ d + μ n H ( 0 ) ,
var { y H 0 } = σ d 2 + var { c ( 0 ) H 0 } = σ d 2 + P n ( f ) H ( f ) 2 d f .
E { y H 1 } = μ d + μ n H ( 0 ) + S ( f ) H ( f ) d f ,
var { y H 1 } = σ d 2 + P n ( f ) H ( f ) 2 d f .
SNR E { y H 1 } - E { y H 0 } 2 1 / 2 ( var { y H 1 } + var { y H 0 } ) = | S ( f ) H ( f ) d f | 2 σ d 2 + P n ( f ) H ( f ) 2 d f .
S k = A k exp ( j ϕ k ) ,             0 ϕ k < 2 π ,             0 A k .
H k = M k exp ( j ϕ k ) ,             0 θ k < 2 π ,             0 M k 1.
B exp ( j β ) = k S k H k ,             0 β < 2 π ,             0 B .
SNR = Δ f 2 | k S k H k | 2 σ d 2 + Δ f k P n k M k 2 = B 2 ( σ d 2 / Δ f 2 ) + k ( P n k / Δ f ) M k 2 .
0 D min M k D max 1 ,
ρ M k 1 ,
ρ = D min D max .
θ k + ϕ k = const .
SNR = ( k A k M k ) 2 σ d 2 + k P n k M k 2 .
SNR M m = 0.
SNR M m = ( σ d 2 + k P n k M k 2 ) 2 ( k M k A k ) A m - ( k M k A k ) 2 2 M m P n m ( σ d 2 + k P n k M k 2 ) 2 ,
M m = A m P n m [ σ d 2 + k P n k M k 2 k M k A k ] .
M m = A m P n m G ,
G A m P n m > 1 ,
SNR M m > 0             in             0 M m 1 ,
G A m P n m < ρ ,
SNR M m < 0             in             ρ M m 1 ,
{ { x } } a b sgn ( x ) × max [ a , min ( b , x ) ] .
M m = { { G A m P n m } } ρ 1 ,
A 1 P n 1 A r P n r > A r + 1 P n , r + 1 A i - 1 P n , i - 1 > A i P n i A N P n N .
G a = P n r A r .
ρ A r P n r > A i P n i .
G l = P n l A l .
M l k = { { G l A k P n k } } ρ 1 .
SNR ( G l ) = [ R 1 A k + G l R 2 ( A k 2 / P n k ) + ρ R 3 A k ] 2 σ d 2 + R 1 P n k + G l 2 R 2 ( A k 2 / P n k ) + ρ 2 R 3 P n k ,
G b = σ d 2 + R 1 P n k + G b 2 R 2 ( A k 2 / P n k ) + ρ 2 R 3 P n k R 1 A k + G b R 2 ( A k 2 / P n k ) + ρ R 3 A k ,
G b = σ d 2 + R 1 P n k + ρ 2 R 3 P n k R 1 A k + ρ R 3 A k .
SNR = [ k A k exp ( + j ϕ k ) H k ] [ l A l exp ( - ϕ l ) H l ] σ d 2 + k P n k H k 2 = B 2 σ d 2 + k P n k H k 2 ,
2 H m P n m B 2 = ( σ d 2 + k P n k H k 2 ) B [ A m exp j ( ϕ m - β ) + A m exp j ( β - ϕ m ) ] .
H m = { { G A m P n m cos ( ϕ m - β ) } } ρ 1 ,
G = σ d 2 + k P n k H k 2 | k A k exp ( j ϕ k ) H k | .
G a = P n l A l .
G b = σ d 2 + R 1 P n k + G b 2 R 2 ( A k 2 / P n k ) cos 2 ( ϕ k - β ) + ρ 2 R 3 P n k exp ( - j β ) [ R 1 A k exp ( j ϕ k ) + G b R 2 ( A k 2 / P n k ) exp ( j ϕ k ) cos ( ϕ k - β ) + ρ R 3 A k exp ( j ϕ k ) ] ,
G b = σ d 2 + R 1 P n k + ρ 2 R 3 P n k R 1 A k cos ( ϕ k - β ) + ρ R 3 A k cos ( ϕ k - β ) .
G b = - R 1 A k sin ( ϕ k - β ) + ρ R 3 A k sin ( ϕ k - β ) R 2 ( A k 2 / P n k ) sin ( ϕ k - β ) cos ( ϕ k - β ) .
sin x cos x = exp ( + j x ) - exp ( - j x ) 2 j , sin x cos x = sin 2 x 2 ,
F 2 σ d 2 + R 1 P n k + ρ 2 R 3 P n k ,
E exp ( j ) = R 1 A k exp ( j ϕ k ) + ρ R 3 A k exp ( j ϕ k ) ,
Q 2 exp ( j 2 q ) R 2 [ A k exp ( j ϕ k ) ] 2 P n k .
R 1 A k sin ( ϕ k - β ) + ρ R 3 ( similar ) = R 1 A k 1 2 j { exp [ j ( ϕ k - β ) ] - exp [ - j ( ϕ k - β ) ] } + ρ R 3 ( similar ) = exp ( - j β ) 2 j R 1 A k exp ( + j ϕ k ) - exp ( + j β ) 2 j × R 1 A k exp ( - j ϕ k ) + ρ R 3 ( similar ) = 1 2 j [ exp ( - j β ) E exp ( + j ) - exp ( + j β ) E exp ( - j ) ] = E sin ( - β ) .
R 1 A k cos ( ϕ k - β ) + ρ R 3 A k cos ( ϕ k - β ) = E cos ( - β ) ,
R 2 A 2 P n sin ( ϕ k - β ) cos ( ϕ k - β ) = R 2 A 2 P n 1 2 sin ( 2 ϕ k - 2 β ) = 1 2 [ 1 2 j exp ( - 2 j β ) R 2 A 2 P n exp ( + 2 j ϕ k ) - 1 2 j exp ( + 2 j β ) R 2 A 2 P n exp ( - 2 j ϕ k ) ] = 1 2 Q 2 sin ( 2 q - 2 β ) .
F 2 E cos ( - β ) = E sin ( - β ) 1 2 Q 2 sin ( 2 q - 2 β ) .
sin ( 2 β ) [ E 2 cos ( 2 ) + F 2 Q 2 cos ( 2 q ) ] = cos ( 2 β ) [ E 2 sin ( 2 ) + F 2 Q 2 sin ( 2 q ) ] ,
tan 2 β = E 2 sin 2 + F 2 Q 2 sin 2 q E 2 cos 2 + F 2 Q 2 cos 2 q .
SNR = ( k A k M k ) 2 k P n k M k 2 .
M m = { { A m P n m G } } 0 1 ,
G min k [ P n k A k ] ,
H m = G A m P n m exp ( - j ϕ m ) ,
M m = { { A m P n m G } } ρ 1 ,
G a ( r ) = P n r A r ,
SNR = ( k A k M k ) 2 σ d 2 ,
H k = exp [ - j ( ϕ k + const . ) ] ,
G < min k ( P n k A k ) .
tan 2 β = E 2 sin 2 + F 2 Q 2 sin 2 q E 2 cos 2 + F 2 Q 2 cos 2 q ( F 2 ) tan 2 q .
E exp ( j ) ( σ d 2 ) Ω A k exp ( j ϕ k ) .
tan 2 β ( σ d 2 ) tan 2
β ( σ d 2 ) arg [ Ω A k exp ( j ϕ k ) ] ± n π 2 ,             n = 0 , 1 , 2 , .
H m = sgn [ cos ( ϕ m - β ) ] .

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