Abstract

Information theory is used to formulate a single figure of merit for assessing the performance of line-scan imaging systems as a function of their spatial response (PSF or MTF), sensitivity, and sampling and quantization intervals and of the statistical properties of a random radiance field. Information density and efficiency (i.e., the ratio of information density to data density) tend to be optimum when the MTF and sampling passband of the imaging system are matched to the Wiener spectrum of the radiance field. Computational results for the statistical properties of natural radiance fields and the responses of common line-scan imaging mechanisms indicate that information density and efficiency are not strongly sensitive to variations in typical statistical properties of the radiance field and that the best practically realizable performance is approached when the sampling intervals are ∼0.5–0.7 times the equivalent diameter of the PSF.

© 1981 Optical Society of America

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References

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  1. P. Mertz, F. Grey, Bell Syst. Tech. J. 13, 464 (1934).
  2. M. W. Baldwin, Bell Syst. Tech. J. 19, 563 (1940).
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  4. O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 131 (1951);J. Soc. Motion Pict. Telev. Eng.58, 181 (1952);J. Soc. Motion Pict. Telev. Eng.61, 97 (1953);J. Soc. Motion Pict. Telev. Eng.64, 593 (1955);J. Soc. Motion Pict. Telev. Eng.73, 81 (1964).
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  12. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
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    [CrossRef] [PubMed]
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  22. N. Halyo, S. T. Stallman, A Parametric Study of Aliasing Error for a Narrow Field-of- View Scanning Radiometer, NASA CR-3294 (U.S. GPO, Washington, D.C., 1980).
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  25. N. Halyo, G. A. McAlpine, IEEE Trans. Audio Electroacoust. AU-19, 3 (1971).
  26. A. B. Carlson, Communications Systems (McGraw-Hill, New York, 1968).
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  29. A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
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  30. S. Tutumi, J. Electr. Commun. Jpn. 51-C (1966).
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  32. Y. Itakura et al., Infrared Phys. 14 (1974).
    [CrossRef]

1980

1976

R. A. Gonsalves, P. S. Considine, Opt. Eng. 15, 64 (1976).
[CrossRef]

1975

1974

Y. Itakura et al., Infrared Phys. 14 (1974).
[CrossRef]

1973

1971

N. Halyo, G. A. McAlpine, IEEE Trans. Audio Electroacoust. AU-19, 3 (1971).

1970

1968

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C (1968).

1966

S. Tutumi, J. Electr. Commun. Jpn. 51-C (1966).

1963

1962

R. Shaw, Photogr. Sci. Eng. 6, 281 (1962).

R. C. Jones, J. Opt. Soc. Am. 52, 493 (1962).
[CrossRef]

1961

E. H. Linfoot, J. Photogr. Sci. 9, 188 (1961).

R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
[CrossRef]

1960

1955

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).
[CrossRef]

1951

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 131 (1951);J. Soc. Motion Pict. Telev. Eng.58, 181 (1952);J. Soc. Motion Pict. Telev. Eng.61, 97 (1953);J. Soc. Motion Pict. Telev. Eng.64, 593 (1955);J. Soc. Motion Pict. Telev. Eng.73, 81 (1964).

1948

C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); orC. Shannon, W. Weaver, The Mathematical Theory of Communication (U Illinois Press, Urbana, 1964).

W. R. Bennett, Bell Syst. Tech. J. 27, 446 (1948).

1940

M. W. Baldwin, Bell Syst. Tech. J. 19, 563 (1940).

R. D. Kell et al., RCA Rev. 5, 8 (1940).

1934

P. Mertz, F. Grey, Bell Syst. Tech. J. 13, 464 (1934).

Baldwin, M. W.

M. W. Baldwin, Bell Syst. Tech. J. 19, 563 (1940).

Bennett, W. R.

W. R. Bennett, Bell Syst. Tech. J. 27, 446 (1948).

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

Brown, W. M.

Callahan, L. G.

Carlson, A. B.

A. B. Carlson, Communications Systems (McGraw-Hill, New York, 1968).

Considine, P. S.

R. A. Gonsalves, P. S. Considine, Opt. Eng. 15, 64 (1976).
[CrossRef]

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, P. S. Considine, Opt. Eng. 15, 64 (1976).
[CrossRef]

Grey, F.

P. Mertz, F. Grey, Bell Syst. Tech. J. 13, 464 (1934).

Halyo, N.

F. O. Huck, N. Halyo, S. K. Park, Appl. Opt. 19, 2174 (1980).
[CrossRef] [PubMed]

N. Halyo, G. A. McAlpine, IEEE Trans. Audio Electroacoust. AU-19, 3 (1971).

N. Halyo, S. T. Stallman, A Parametric Study of Aliasing Error for a Narrow Field-of- View Scanning Radiometer, NASA CR-3294 (U.S. GPO, Washington, D.C., 1980).

Huck, F. O.

Itakura, Y.

Y. Itakura et al., Infrared Phys. 14 (1974).
[CrossRef]

Jones, R. C.

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Katzberg, S. J.

Kell, R. D.

R. D. Kell et al., RCA Rev. 5, 8 (1940).

Khinchin, A. I.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

Linfoot, E. H.

E. H. Linfoot, J. Photogr. Sci. 9, 188 (1961).

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).
[CrossRef]

Macovski, A.

McAlpine, G. A.

N. Halyo, G. A. McAlpine, IEEE Trans. Audio Electroacoust. AU-19, 3 (1971).

Mertz, P.

P. Mertz, F. Grey, Bell Syst. Tech. J. 13, 464 (1934).

Park, S. K.

Pearson, D. E.

D. E. Pearson, Transmission and Display of Pictorial Information (Wiley, New York, 1975).

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Robinson, A. H.

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Schade, O. H.

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 131 (1951);J. Soc. Motion Pict. Telev. Eng.58, 181 (1952);J. Soc. Motion Pict. Telev. Eng.61, 97 (1953);J. Soc. Motion Pict. Telev. Eng.64, 593 (1955);J. Soc. Motion Pict. Telev. Eng.73, 81 (1964).

Shannon, C.

C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); orC. Shannon, W. Weaver, The Mathematical Theory of Communication (U Illinois Press, Urbana, 1964).

Shaw, R.

R. Shaw, Photogr. Sci. Eng. 6, 281 (1962).

Stallman, S. T.

N. Halyo, S. T. Stallman, A Parametric Study of Aliasing Error for a Narrow Field-of- View Scanning Radiometer, NASA CR-3294 (U.S. GPO, Washington, D.C., 1980).

Takagi, T.

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C (1968).

Tutumi, S.

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C (1968).

S. Tutumi, J. Electr. Commun. Jpn. 51-C (1966).

Appl. Opt.

Bell Syst. Tech. J.

W. R. Bennett, Bell Syst. Tech. J. 27, 446 (1948).

C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); orC. Shannon, W. Weaver, The Mathematical Theory of Communication (U Illinois Press, Urbana, 1964).

P. Mertz, F. Grey, Bell Syst. Tech. J. 13, 464 (1934).

M. W. Baldwin, Bell Syst. Tech. J. 19, 563 (1940).

IEEE Trans. Audio Electroacoust.

N. Halyo, G. A. McAlpine, IEEE Trans. Audio Electroacoust. AU-19, 3 (1971).

Infrared Phys.

Y. Itakura et al., Infrared Phys. 14 (1974).
[CrossRef]

J. Electr. Commun. Jpn.

S. Tutumi, J. Electr. Commun. Jpn. 51-C (1966).

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C (1968).

J. Opt. Soc. Am.

J. Photogr. Sci.

E. H. Linfoot, J. Photogr. Sci. 9, 188 (1961).

J. Soc. Motion Pict. Telev. Eng.

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 131 (1951);J. Soc. Motion Pict. Telev. Eng.58, 181 (1952);J. Soc. Motion Pict. Telev. Eng.61, 97 (1953);J. Soc. Motion Pict. Telev. Eng.64, 593 (1955);J. Soc. Motion Pict. Telev. Eng.73, 81 (1964).

Opt. Eng.

R. A. Gonsalves, P. S. Considine, Opt. Eng. 15, 64 (1976).
[CrossRef]

Philos. Trans. R. Soc. London

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

Photogr. Sci. Eng.

R. Shaw, Photogr. Sci. Eng. 6, 281 (1962).

RCA Rev.

R. D. Kell et al., RCA Rev. 5, 8 (1940).

Other

L. M. Biberman, Ed., Perception of Displayed Information (Plenum, New York, 1973).
[CrossRef]

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

D. E. Pearson, Transmission and Display of Pictorial Information (Wiley, New York, 1975).

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

N. Halyo, S. T. Stallman, A Parametric Study of Aliasing Error for a Narrow Field-of- View Scanning Radiometer, NASA CR-3294 (U.S. GPO, Washington, D.C., 1980).

R. M. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

A. B. Carlson, Communications Systems (McGraw-Hill, New York, 1968).

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

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Figures (7)

Fig. 1
Fig. 1

Model of line-scan imaging process.

Fig. 2
Fig. 2

Wiener spectrum of radiance field.

Fig. 3
Fig. 3

Properties of photon-detection mechanisms.

Fig. 4
Fig. 4

Variation of blurring σb, aliasing σa, information density hi, sampling density hs, and information per sample hi/hs vs sampling intervals, X = Y, for several mean spatial widths μr of the radiance field; hi and hi/hs include curves for a (white) Wiener spectrum that remains constant with frequency. Variables are normalized with respect to the equivalent diameter γ of the PSF. Electronic noise and quantization are excluded.

Fig. 5
Fig. 5

Variation of blurring σb, aliasing σa, and information per sample hi/hs vs sampling intervals, X = Y, for a circular photosensor aperture with and without electronic filter. Mean spatial width of the radiance field is equal to the diameter of the aperture (i.e., μr = γ). Electronic noise and quantization are excluded.

Fig. 6
Fig. 6

Variation of information density hi, data density hd, and information efficiency hi/hd vs sampling intervals, X = Y, for several mean spatial widths μr of the radiance field. Variables are normalized with respect to the equivalent diameter γ of the PSF.

Fig. 7
Fig. 7

Variation of information density hi, data density hd, and information efficiency hi/hd, vs sampling intervals, X = Y, for several encoding levels η. Mean width of the radiance field is equal to the equivalent diameter of the PSF (i.e., μr = γ), and variables are normalized with respect to the equivalent diameter γ.

Tables (1)

Tables Icon

Table I Design Parameters Favorable to High Information Efficiency

Equations (55)

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S ( x , y ; X , Y ) = { KL ( x , y ) * τ ( x , y ) + N ( y ) * [ δ ( x ) τ e ( y ) ] } ( x X , y Y ) .
τ ( x , y ) = τ l ( x , y ) * τ p ( x , y ) * [ δ ( x ) τ e ( y ) ] ,
( x X , y Y ) = m = n = δ ( x X m , y Y n ) = X Y m = n = δ ( x X m , y Y n ) .
g ̂ ( υ , ω ) = g ( x , y ) exp [ i 2 π ( υ x + ω y ) ] dxdy , g ( x , y ) = g ̂ ( υ , ω ) exp [ i 2 π ( x υ + y ω ) ] d υ d ω .
s ̂ ( υ , ω ; X , Y ) = [ K L ̂ ( υ , ω ) τ ̂ ( υ , ω ) + N ̂ ( ω ) τ ̂ e ( ω ) ] * XY ( X υ , Y ω ) ,
τ ̂ ( υ , ω ) = τ ̂ l ( υ , ω ) τ ̂ p ( υ , ω ) τ ̂ e ( υ , ω ) , XY ( X υ , Y ω ) = XX m = n = δ ( X υ m , Y ω n ) = m = n = δ ( υ m X , ω n Y ) .
s ̂ ( υ , ω ; X , Y ) = s ̂ ( υ , ω ) + s ̂ a ( υ , ω ; X , Y ) ,
s ̂ ( υ , ω ) = K L ̂ ( υ , ω ) τ ̂ ( υ , ω ) + N ̂ ( ω ) τ ̂ e ( ω ) ,
s ̂ a ( υ , ω ; X , Y ) = m = n = s ̂ ( υ m X , ω n X ) . ( m , n ) ( 0 , 0 )
Π ( X υ , Y ω ) = { 1 , | ν | < 1 2 X , | ω | < 1 2 Y , 0 , elsewhere .
r ̂ ( υ , ω ) = s ̂ ( υ , ω ; X , Y ) Π ( X υ , Y ω )
r ( x , y ) = m = n = s ( mX , nY ) sinc ( x mX X ) sinc ( y nY Y ) ,
Φ ̂ s ( υ , ω ; X , Y ; κ ) = Φ ̂ s ( υ , ω ) + Φ ̂ a ( υ , ω ; X , Y ) + Φ ̂ n ( ω ; Y ) + Φ ̂ q ( κ ) .
Φ ̂ s ( υ , ω ) = K 2 Φ ̂ L ( υ , ω ) | τ ̂ ( υ , ω ) | 2 .
Φ ̂ a ( υ , ω ; X , Y ) = m = n = Φ ̂ s ( υ m X , ω n X ) . ( m , n ) ( 0 , 0 )
Φ ̂ n ( ω ; Y ) = n = Φ ̂ N ( ω n Y ) | τ ̂ e ( ω n Y ) | 2 .
σ s 2 = K 2 Φ ̂ L ( υ , ω ) d υ d ω = K 2 σ L 2
p ( n κ ) = { κ / 2 c σ s , c σ s / κ n κ c σ s / κ , 0 , elsewhere .
σ q 2 = c σ s / κ c σ s / κ n κ 2 p ( n κ ) dn κ = 1 3 ( c σ s κ ) 2 .
Φ ̂ q ( υ , ω ; κ ) = Φ ̂ q ( κ ) = σ q 2 = 1 3 ( c σ s κ ) 2 .
H g = ½ | A | 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 4 π Φ ̂ g ( υ , ω ) ] d υ d ω .
H i = ½ | A | 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 4 π Φ ̂ s ( υ , ω ; X , Y ; κ ) ] d υ d ω ½ | A | 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 { 4 π [ Φ ̂ a ( υ , ω ; X , Y ) + Φ ̂ n ( ω ; Y ) + Φ ̂ q ( κ ) ] } d υ d ω ,
h i = H i | A | = ½ 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 1 + Φ ̂ s ( υ , ω ) Φ ̂ a ( υ , ω ; X , Y ) + Φ ̂ n ( ω ; Y ) + Φ ̂ q ( κ ) ] d υ d ω ,
H d = MN log 2 κ = | A | XY log 2 κ .
h d = H d | A | = 1 XY log 2 κ = η XY .
Φ ̂ L ( υ , ω ) = { σ L 2 , | υ | < 1 2 X , | ω | < 1 2 Y , 0 , elsewhere .
τ ̂ ( υ , ω ) = { 1 , | υ | < 1 2 X , | ω | < 1 2 Y , 0 , elsewhere .
h i = 1 2 XY log 2 ( 1 + σ s 2 σ n 2 + σ q 2 ) ,
h i = 1 2 XY log 2 ( 1 + σ s 2 σ q 2 ) = 1 2 XY log 2 ( 1 + 3 κ 2 c 2 ) 1 XY log 2 3 κ c .
h i = 1 2 XY log 2 ( 1 + σ s 2 σ n 2 ) .
Φ L ( r ) = σ L 2 exp ( r / μ r ) ,
Φ ̂ L ( υ , ω ) = Φ ̂ L ( ρ ) = 2 π μ r 2 σ L 2 [ 1 + ( 2 π μ r ρ ) 2 ] 3 / 2 ,
σ L 2 = Φ L ( 0 ) = Φ ̂ L ( υ , ω ) d υ d ω = 2 π 0 ρ Φ ̂ L ( ρ ) d ρ ;
τ ̂ ( υ , ω ) = τ ̂ p ( υ , ω ) τ ̂ e ( ω ) ,
τ p ( x , y ) dxdy = τ ̂ p ( 0 , 0 ) = 1 .
τ p ( x , y ) = { 4 / π γ 2 , ( x , y ) ϵ aperture area , 0 , elsewhere ,
τ ̂ p ( υ , ω ) = τ ̂ p ( ρ ) = exp [ ( π γ ρ / 2 ) 2 ] .
τ ̂ p ( υ , ω ) = τ ̂ p ( ρ ) = J 1 ( π γ ρ ) π γ ρ / 2 .
τ ̂ p ( υ , ω ) = sinc ½ ( π γ υ / 2 + γ ω ) sinc ½ ( π γ υ / 2 γ ω ) .
τ ̂ e ( ω ) = { 1 ( 2 Y ω ) 4 , | ω | < 1 / 2 Y , 0 , | ω | > 1 / 2 Y ,
σ b 2 = 1 / 2 Y 1 / 2 Y 1 / 2 X 1 / 2 X Φ ̂ L ( υ , ω ) | 1 τ ̂ ( υ , ω ) | 2 d υ d ω .
σ a 2 = 1 / 2 Y 1 / 2 Y 1 / 2 X 1 / 2 X m = n = Φ ̂ L ( υ m X , ω n Y ) ( m , n ) ( 0 , 0 ) × | τ ̂ ( υ m X , ω n Y ) | 2 d υ d ω .
h i = ½ 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 1 + Φ ̂ L ( υ , ω ) | τ ̂ ( υ , ω ) | 2 m = n = Φ ̂ L ( υ m X , ω n Y ) | τ ̂ ( υ m X , ω n Y ) | 2 ( m , n ) ( 0 , 0 ) ] d υ d ω .
h s = 1 / ( XY ) .
h i = ½ 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 1 + Φ ̂ L ( υ , ω ) | τ ̂ ( υ , ω ) | 2 m = n = Φ ̂ L ( υ m X , ω n Y ) | τ ̂ ( υ m X , ω n Y ) | 2 + ( σ s σ n ) 2 + κ 2 ( m , n ) ( 0 , 0 ) ] d υ d ω .
σ n 2 = 1 / 2 Y 1 / 2 Y Φ ̂ N ( ω ; Y ) | τ ̂ e ( ω ) | 2 d ω .
h d = 1 X Y log 2 κ = η XY ,
1 2 υ r ̂ = 0.05 X / γ 1 2 υ r ̂ = 0.4 ,
K = ( π 4 ) 2 D 2 γ 2 0 L ( λ ) τ ( λ ) d λ ,
Φ ̂ s ( υ , ω ) = K 2 σ L 2 Φ ̂ L ( υ , ω ) | τ ̂ ( υ , ω ) | 2 ,
Φ ̂ a ( υ , ω ; X , Y ) = K 2 σ L 2 m = n = Φ ̂ L ( υ m X , ω n Y ) × | τ ̂ ( υ m X , ω n Y ) | 2 ( m , n ) ( 0 , 0 ) ,
Φ ̂ n ( ω ; Y ) = σ N 2 n = Φ ̂ N ( ω n Y ) | τ ̂ e ( ω n Y ) | 2 .
Φ ̂ n ( ω ; Y ) = σ N 2 Φ ̂ N ( ω ) | τ ̂ e ( ω ) | 2 .
h i = ½ 1 / 2 X 1 / 2 X 1 / 2 Y 1 / 2 Y log 2 [ 1 + Φ ̂ L ( υ , ω ) | τ ̂ ( υ , ω ) | 2 m = n = Φ ̂ L ( υ m X , ω n Y ) | τ ̂ ( υ m X , ω n Y ) | 2 + ( σ N K σ L ) 2 Φ ̂ N ( ω ) | τ ̂ e ( ω ) | 2 + c 2 κ 2 3 ( m , n ) ( 0 , 0 ) ] d υ d ω .
σ n 2 = σ N 2 1 / 2 Y 1 / 2 Y Φ ̂ N ( ω ) | τ ̂ e ( ω ) | 2 d ω .

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