Abstract

Multiplex coding of optical images is discussed for photon-counting operation in both the signal- and dark-count-dominated domains. Both intensity measurement and signal detection are discussed. The advantage of multiplex operation in both situations is demonstrated and comparison made with alternative imaging techniques. The effects of code noise, detector area, response time, and dynamic range are discussed. It is shown that in dark-count-dominated operation, e.g., ir imaging applications, the limiting factor on multiplex gain is the light collection following image coding. Higher bandwidth detection is also required: for instance, a typical ir detector transmitting a 100 × 100 element image with individual element resolution of 25 will need a total dynamic range of 212 and an exponential response time a factor of 2.4 faster than for a single detector.

© 1976 Optical Society of America

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References

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  2. H. J. Hecker, M. P. Nordseth, H. M. Joseph, U.S. Patent3, 616 (1967).
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  20. J. A. Decker, Appl. Opt. 10, 510 (1971).
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    [Crossref]
  24. J. I. Marcum, IRE Trans. Inf. Theory IT-6, 59 (1960).
    [Crossref]
  25. P. Swerling, IRE Trans. Inf. Theory IT-6, 269 (1960).
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  26. J. L. Walsh, Am. J. Math. 45, 5 (1923).
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    [Crossref] [PubMed]

1974 (1)

1973 (1)

1972 (4)

J. G. Berny, J. P. Bourgoin, Opt. Commun. 4, 341 (1972).
[Crossref]

D. L. Fried, Appl. Opt. 11, 1268 (1972).
[Crossref] [PubMed]

A. Boksenberg, D. E. Burgess, Adv. Electron. Electron Phys. 33B, 835 (1972).
[Crossref]

L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 84, 161 (1972).
[Crossref]

1971 (3)

1970 (3)

1969 (2)

1968 (4)

1967 (1)

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. (Paris) Suppl. 28, 3–4, C2 (1967).

1966 (1)

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

1963 (1)

1960 (2)

J. I. Marcum, IRE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

P. Swerling, IRE Trans. Inf. Theory IT-6, 269 (1960).
[Crossref]

1959 (1)

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[Crossref]

1958 (1)

P. B. Fellgett, J. Phys. Radium 19, 187 (1958).
[Crossref]

1951 (1)

1949 (1)

1923 (1)

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[Crossref]

Andrews, H. C.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[Crossref]

Aspinall, D.

Berny, J. G.

J. G. Berny, J. P. Bourgoin, Opt. Commun. 4, 341 (1972).
[Crossref]

Boksenberg, A.

A. Boksenberg, D. E. Burgess, Adv. Electron. Electron Phys. 33B, 835 (1972).
[Crossref]

A. Boksenberg, Astronomical Use of Television-Type Sensors, Princeton Symposium, 1970, NASA SP-256 (1971), p. 77.

Bourgoin, J. P.

J. G. Berny, J. P. Bourgoin, Opt. Commun. 4, 341 (1972).
[Crossref]

Burgess, D. E.

A. Boksenberg, D. E. Burgess, Adv. Electron. Electron Phys. 33B, 835 (1972).
[Crossref]

Decker, J. A.

Eberhardt, E. H.

Edgar, R. F.

R. F. Edgar, Infrared Phys. 8, 183 (1968).
[Crossref]

Fellgett, P. B.

P. B. Fellgett, J. Phys. Radium 19, 187 (1958).
[Crossref]

P. B. Fellgett, Cambridge Univ. Ph.D. thesis (1951)quoted by L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Fine, T.

Fried, D. L.

Girard, A.

Golay, M. J. E.

Gottlieb, P.

P. Gottlieb, IEEE Trans. Inf. Theory IT-14, 428 (1968).
[Crossref]

Grainger, J. F.

R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
[Crossref] [PubMed]

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. (Paris) Suppl. 28, 3–4, C2 (1967).

Harwit, M.

Hecker, H. J.

H. J. Hecker, M. P. Nordseth, H. M. Joseph, U.S. Patent3, 616 (1967).

Hertel, R. J.

Ibbett, R. N.

Joseph, H. M.

H. J. Hecker, M. P. Nordseth, H. M. Joseph, U.S. Patent3, 616 (1967).

Kahn, F. D.

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[Crossref]

Kane, J.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[Crossref]

Marcum, J. I.

J. I. Marcum, IRE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

McNall, J.

J. McNall, L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 82, 837 (1970).
[Crossref]

Nordseth, M. P.

H. J. Hecker, M. P. Nordseth, H. M. Joseph, U.S. Patent3, 616 (1967).

Oliver, C. J.

Phillips, P. G.

Pike, E. R.

Pratt, W. K.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[Crossref]

Ring, J.

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. (Paris) Suppl. 28, 3–4, C2 (1967).

Robinson, L.

L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 84, 161 (1972).
[Crossref]

J. McNall, L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 82, 837 (1970).
[Crossref]

Siegman, A. E.

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

Sloane, N. J. A.

Stell, J. H.

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. (Paris) Suppl. 28, 3–4, C2 (1967).

Swerling, P.

P. Swerling, IRE Trans. Inf. Theory IT-6, 269 (1960).
[Crossref]

Walsh, J. L.

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[Crossref]

Wampler, E. J.

L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 84, 161 (1972).
[Crossref]

J. McNall, L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 82, 837 (1970).
[Crossref]

Adv. Electron. Electron Phys. (1)

A. Boksenberg, D. E. Burgess, Adv. Electron. Electron Phys. 33B, 835 (1972).
[Crossref]

Am. J. Math. (1)

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[Crossref]

Appl. Opt. (12)

Astrophys. J. (1)

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[Crossref]

IEEE Trans. Inf. Theory (1)

P. Gottlieb, IEEE Trans. Inf. Theory IT-14, 428 (1968).
[Crossref]

Infrared Phys. (1)

R. F. Edgar, Infrared Phys. 8, 183 (1968).
[Crossref]

IRE Trans. Inf. Theory (2)

J. I. Marcum, IRE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

P. Swerling, IRE Trans. Inf. Theory IT-6, 269 (1960).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. (Paris) Suppl. (1)

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. (Paris) Suppl. 28, 3–4, C2 (1967).

J. Phys. Radium (1)

P. B. Fellgett, J. Phys. Radium 19, 187 (1958).
[Crossref]

Opt. Commun. (1)

J. G. Berny, J. P. Bourgoin, Opt. Commun. 4, 341 (1972).
[Crossref]

Proc. IEEE (2)

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[Crossref]

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

Publ. Astron. Soc. Pac. (2)

J. McNall, L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 82, 837 (1970).
[Crossref]

L. Robinson, E. J. Wampler, Publ. Astron. Soc. Pac. 84, 161 (1972).
[Crossref]

Other (3)

H. J. Hecker, M. P. Nordseth, H. M. Joseph, U.S. Patent3, 616 (1967).

A. Boksenberg, Astronomical Use of Television-Type Sensors, Princeton Symposium, 1970, NASA SP-256 (1971), p. 77.

P. B. Fellgett, Cambridge Univ. Ph.D. thesis (1951)quoted by L. Mertz, Transformations in Optics (Wiley, New York, 1965).

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Figures (9)

Fig. 1
Fig. 1

Block schematic of the coding and decoding scheme using a pseudorandom generator, electrooptic shutters, and cross-correlation decoding.

Fig. 2
Fig. 2

A comparison of multiplex and independent detection in dark-count-dominated operation. Experimental and theoretical values of the rms SNR for multiplex operation (M = 48, solid line on left) with different values of R are compared with independent detectors (C = 1, 12, and 48; dashed lines).

Fig. 3
Fig. 3

The multiplex advantage in signal-dominated operation. The basic sample time was 5 msec. The image consisted of (a) M = 100 elements, (b) M = 5000 elements giving a total experimental duration of (a) τ = 0.5 sec, (b) τ = 25 sec. In each case multiplex operation was compared with a single scanned detector. The solid line represents the simulated image feature. Where no counts were recorded, no data point is shown.

Fig. 4
Fig. 4

The multiplex advantage in dark-count-dominated operation. The conditions were as for Fig. 3 with the addition of a strong dark-count component and changing the basic sample time to T = 10 msec.

Fig. 5
Fig. 5

Experimental and theoretical (solid line) probability distributions of the signal estimator Ŝ for signal-dominated multiplex operation (M = 48).

Fig. 6
Fig. 6

Experimental and theoretical (solid line) probability distributions of the signal estimator Ŝ for dark-count-dominated multiplex operation (M = 48). Experimental data for R = 0.0208 is shown.

Fig. 7
Fig. 7

The consequence diagram of Pm against Pf for signal-dominated operation. Independent-detector performance (C = 1, 3, 6, 12; dashed lines) is compared with multiplex operation (M = 48; solid line) for various values of R.

Fig. 8
Fig. 8

The consequence diagram of Pm against Pf for dark-count-dominated operation. Independent-detector performance (C = 1, 12, and 48; dashed lines) is compared with multiplex operation (M = 48; full line) for various values of R.

Fig. 9
Fig. 9

The effect of code noise for LM in various combinations of total dark counts Nd, total signal Ns, and R. (A) Nd = 1.6, Ns = 23, R = 0.0208. (B) Nd = 1.6, Ns = 23, R = 0.148. (C) Nd = 234, Ns = 24.7, R = 0.0208. (D) Nd = 234, Ns = 247, R = 0.148.

Equations (63)

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Δ n i 2 = Var n i = n ¯ i
Δ I i / I i = Δ n i / n ¯ i = 1 / ( n ¯ i ) 1 / 2 = 1 ( η I i A c τ ) 1 / 2 .
Δ I i / I i = ( M / η I i A c τ ) 1 / 2 ,
m = 1 M f i ( m ) = M / 2 ,
g i ( r ) = m = 1 M f i ( m ) f i ( m + r ) = M 4 δ r , 0 + M 4 ,
f i ( m ) = f i ( m + j i + M ) ,
f i ( M + m ) = f i ( m )
ρ m ( j ) = i = 1 M f i ( m ) f i + j ( m ) = M 4 δ j , 0 + M 4 ,
ρ m ( j , k ) = i = 1 M f i ( m ) f i + j ( m ) f i + k ( m ) , = M 8 δ j , 0 δ k , 0 + M 4 ( δ j , 0 + δ k , 0 ) + M 8 ,
n T ( m ) = n d ( m ) + i = 1 M n i ( m ) f i ( m ) ,
n s ( m ) = f s ( m ) n d ( m ) + f s ( m ) i = 1 M n i ( m ) f i ( m )
n b ( m ) = f b ( m ) n d ( m ) + f b ( m ) i = 1 M n i ( m ) f i ( m ) ,
Ŝ s b = N s N b = m = 1 M [ n s ( m ) n b ( m ) ] = m = 1 M { n d ( m ) [ f s ( m ) f b ( m ) ] + i = 1 M n i ( m ) f i ( m ) [ f s ( m ) f b ( m ) ] } .
Ŝ s b = ( M / 4 ) ( n ¯ s n ¯ b ) ,
Var Ŝ s b = S s b 2 S s b 2 = m = 1 M n = 1 M n d ( m ) n d ( n ) [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] + 2 m = 1 M n = 1 M j = 1 M n d ( m ) n j ( n ) [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] f j ( n ) + m = 1 M n = 1 M [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] × j = 1 M i = 1 M n i ( m ) n j ( n ) f i ( m ) f j ( n ) M 2 16 ( n ¯ s n ¯ b ) 2 .
n d ( m ) n j ( n ) = n d n j ,
n i ( m ) n j ( n ) = ( n j 2 n j 2 ) δ i j δ m n + n j n i = n j δ i j δ m n + n j n i .
Var Ŝ s b = ( M / 2 ) n ¯ d + ( M / 4 ) S ,
S = i = 1 M n ¯ i .
SNR s b = Ŝ s b / ( Var Ŝ s b ) 1 / 2 ,
SNR s b = ( n ¯ s n ¯ b ) ( M ) 1 / 2 / 2 ( 2 n ¯ d + S ) 1 / 2 ,
SNR = n ¯ s / ( n ¯ s + 2 n ¯ d ) 1 / 2 .
R = S / M n ¯ s .
E = 1 if Ŝ s b > U , E = 0 if Ŝ s b U .
P m = Ŝ s b = U p ( Ŝ s b ; n ¯ s > 0 )
P f = 1 Ŝ s b = 0 U p ( Ŝ s b ; n ¯ s = 0 ) .
Ŝ s b = N s N b .
p ( Ŝ s b ) = exp [ ( N ¯ s + N ¯ b ) ] ( N ¯ s ) Ŝ s b r = 0 ( N ¯ s N ¯ b ) r ( r + Ŝ s b ) ! r ! ,
N ¯ s = n ¯ s + n ¯ d
N ¯ b = n ¯ d
N ¯ s , b ¯ = N ¯ s N ¯ s , b = ( M / 8 ) ( n ¯ s + S + 2 n ¯ d )
N ¯ b , s ¯ = N ¯ b N ¯ b , s = ( M / 8 ) ( S n ¯ s + 2 n ¯ d ) .
Ŝ s b = N s N b = N s , b ¯ N b , s ¯
p _ ( Ŝ s , b ) = exp [ ( N ¯ s , b ¯ + N b , s ¯ ) ] ( N s , b ¯ ) Ŝ s b r = 0 ( N ¯ s . b ¯ N b . s ¯ ) r ( r + Ŝ s b ) ! r ! .
N s , b ¯ = N s ½ N s
N b , s ¯ = N b ½ N s .
P T = P m + P f .
Ŝ s b = m = 1 L [ f s ( m ) f b ( m ) ] [ I d + i = 1 M f i ( m ) I i ] = m = 1 L [ f s ( m ) f b ( m ) ] I d + i = 1 M m = 1 L I i [ f s ( m ) f i ( m ) f b ( m ) f i ( m ) ] ,
Ŝ s b 2 = I d 2 m = 1 M m = 1 L [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] + 2 I d i = 1 M m = 1 L n = 1 L I i f i ( m ) [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] + i = 1 M j = 1 M m = 1 L n = 1 L I i I j f i ( m ) f j ( n ) [ f s ( m ) f b ( m ) ] [ f s ( n ) f b ( n ) ] .
m = 1 L f i ( m ) = L / 2
g i ( r ) = m = 1 L f i ( m ) f i ( m + r ) = ( L / 4 ) δ r , 0 + ( L / 4 ) ,
ρ m ( j ) = i = 1 L f i ( m ) f i + j ( m ) = i = 1 L f i ( m ) f i + j ( m ) = L / 4 .
Ŝ s b = ( L / 4 ) I s = N ¯ s / 4 ,
Var Ŝ s b = 1 L ( N ¯ d 2 2 + N ¯ d 2 i = 1 M N ¯ i N ¯ s 2 16 + 1 8 i = 1 M N ¯ i 2 + 1 8 [ i = 1 M N ¯ i ] 2 )
Var Ŝ s b = N ¯ d 2 + 1 4 i = 1 M N ¯ i .
N ¯ d / L > 1
N ¯ d i = 1 M N ¯ i ,
1 L i = 1 M N ¯ i > 1
i = 1 M N ¯ i N ¯ d .
F = area of image in coding plane 4 × detector area .
n T ( m ) = n d ( m ) + n i ( m ) .
n ¯ T = n ¯ d + n ¯ i
σ 2 = ( n ¯ d + n ¯ i ) .
n T n ¯ d = n d n ¯ d + n i = σ ( g d d + g i ) ,
V i = σ g i .
n T = σ g d + σ i = 1 M g i .
V T = σ i = 1 M g i .
V ¯ T = ( M / 2 ) V i
Var V T = ( M / 2 ) Var V i .
V ¯ T = ( g i max / 2 ) ( g i max + 1 ) σ ( M / 2 )
Var V T = [ ( g i max + 1 ) / 12 ] ( 2 g i max 1 ) σ 2 ( M / 2 ) .
τ RC = T / 3.5 .
τ RC = T / 8.5 .

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