Abstract

The total system response of a sampling imaging system to a single input spatial frequency is obtained. The first Brillouin zone (familiar to the literature of solid-state physics) is shown to play an important role in discussing the aliasing phenomenon of such systems. The precise quantitative way in which the input energy is distributed among the aliases in the output is determined by the detector array geometry and by the physical characteristics of the detectors and the spots on the output display. A tentative, but plausible, hypothesis is entertained concerning the data processing of the human eye and its relation to the first Brillouin zone. The principal quantitative results of this study show that the usual discussions concerning the modulation transfer function of such systems are oversimplified.

© 1975 Optical Society of America

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References

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  1. D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
    [Crossref]
  2. W. D. Montgomery, J. Electron. Control 17, 437 (1964).
    [Crossref]
  3. W. D. Montgomery, J. Control 1, 7 (1965).
    [Crossref]
  4. D. P. Petersen and D. Middleton, Inf. Control 7, 445 (1964).
    [Crossref]
  5. W. D. Montgomery, IEEE Trans. IT-11, 204 (1965).
  6. A. H. Robinson, Appl. Opt. 12, 2344 (1973).
    [Crossref] [PubMed]
  7. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953).
  8. R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1950).

1973 (1)

1965 (2)

W. D. Montgomery, IEEE Trans. IT-11, 204 (1965).

W. D. Montgomery, J. Control 1, 7 (1965).
[Crossref]

1964 (2)

D. P. Petersen and D. Middleton, Inf. Control 7, 445 (1964).
[Crossref]

W. D. Montgomery, J. Electron. Control 17, 437 (1964).
[Crossref]

1962 (1)

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953).

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1950).

Middleton, D.

D. P. Petersen and D. Middleton, Inf. Control 7, 445 (1964).
[Crossref]

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[Crossref]

Montgomery, W. D.

W. D. Montgomery, J. Control 1, 7 (1965).
[Crossref]

W. D. Montgomery, IEEE Trans. IT-11, 204 (1965).

W. D. Montgomery, J. Electron. Control 17, 437 (1964).
[Crossref]

Petersen, D. P.

D. P. Petersen and D. Middleton, Inf. Control 7, 445 (1964).
[Crossref]

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[Crossref]

Robinson, A. H.

Appl. Opt. (1)

IEEE Trans. (1)

W. D. Montgomery, IEEE Trans. IT-11, 204 (1965).

Inf. Control (2)

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[Crossref]

D. P. Petersen and D. Middleton, Inf. Control 7, 445 (1964).
[Crossref]

J. Control (1)

W. D. Montgomery, J. Control 1, 7 (1965).
[Crossref]

J. Electron. Control (1)

W. D. Montgomery, J. Electron. Control 17, 437 (1964).
[Crossref]

Other (2)

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1950).

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Figures (6)

FIG. 1
FIG. 1

Schematic of some imaging systems.

FIG. 2
FIG. 2

Aliasing in one dimension.

FIG. 3
FIG. 3

Spatial frequencies and their negatives for the sine waves of Fig. 2.

FIG. 4
FIG. 4

Direct and reciprocal lattices.

FIG. 5
FIG. 5

Weighting functions at input and output.

FIG. 6
FIG. 6

Aliasing of a spatial frequency and its negative with respect to the first Brillouin zone (FBZ).

Equations (36)

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t ˆ ( ξ ) = t ( x ) e - 2 π i x · ξ d x ,
t ( x ) = t ˆ ( ξ ) e 2 π i x · ξ d ξ ,
t ˆ ( - ξ ) = t ˆ * ( ξ )
t ( x ) = a 0 e 2 π i x · ξ 0 + a 0 * e - 2 π i x · ξ 0 ,
t ( x ) = 2 a 0 cos [ 2 π x · ξ 0 + a 0 ] ,
a n = n 1 a 1 + n 2 a 2 ,
a i · b j = δ i j ,
b n = n 1 b 1 + n 2 b 2 ,
t ( x ) α ( x ) d x ,
t ( x ) α ( x - a n ) d x = t n .
t n β ( x - a n ) .
U ( x ) = n t n β ( x - a n ) ,
n 1 = - N 1 N 1 n 2 = - N 2 N 2 .
U ( x ) = B n [ α ˆ * ( ξ 0 ) β ˆ ( ξ n ) a 0 e 2 π i x · ξ n + α ˆ ( ξ 0 ) β ˆ * ( ξ n ) a 0 * e - 2 π i x · ξ n ] ,
ξ n = ξ 0 + b n ,
B [ α ˆ * ( ξ 0 ) β ˆ ( ξ 0 ) a 0 e 2 π i x · ξ 0 + α ˆ ( ξ 0 ) β ˆ * ( ξ 0 ) a 0 * e - 2 π i x · ξ 0 ] .
α ( - x ) = α ( x ) ,             β ( - x ) = β ( x ) ,
B α ˆ ( ξ 0 ) β ˆ ( ξ 0 ) [ a 0 e 2 π i x · ξ 0 + a 0 * e - 2 π i x · ξ 0 ] ,
U ( x ) = B α ˆ ( ξ 0 ) n β ˆ ( ξ n ) [ a 0 e 2 π i x · ξ n + a 0 * e - 2 π i x · ξ n ] .
B α ˆ ( ξ 0 ) β ˆ ( ξ 0 ) [ a 0 e 2 π i x · ξ 0 + a 0 * e - 2 π i x · ξ 0 ]
U ˆ ( ξ ) = n t n β ˆ ( ξ ) e - 2 π i a n · ξ ,
β ( x - a n ) e - 2 π i x · ξ d x = β ˆ ( ξ ) e - 2 π i a n · ξ .
t n = a 0 e 2 π i x · ξ 0 α ( x - a n ) d x + a 0 * e - 2 π i x · ξ 0 α ( x - a n ) d x = a 0 α ˆ ( - ξ 0 ) e 2 π i a n · ξ 0 + a 0 * α ˆ ( ξ 0 ) e - 2 π i a n · ξ 0 .
U ˆ ( ξ ) = β ˆ ( ξ ) [ a 0 α ˆ * ( ξ 0 ) n e - 2 π i a n · ( ξ - ξ 0 ) + a 0 * α ˆ ( ξ 0 ) n e - 2 π i a n · ( ξ + ξ 0 ) ] .
n e - 2 π i a n · ( ξ - ξ 0 ) = sin [ π a 1 · ( ξ - ξ 0 ) ( 2 N 1 + 1 ) ] sin [ π a 1 · ( ξ - ξ 0 ) ] × sin [ π a 2 · ( ξ - ξ 0 ) ( 2 N 2 + 1 ) ] sin [ π a 2 · ( ξ - ξ 0 ) ] ,
n e - 2 π i a n · ( ξ + ξ 0 ) = sin [ π a 1 · ( ξ + ξ 0 ) ( 2 N 1 + 1 ) ] sin [ π a 1 · ( ξ + ξ 0 ) ] × sin [ π a 2 · ( ξ + ξ 0 ) ( 2 N 2 + 1 ) ] sin [ π a 2 · ( ξ + ξ 0 ) ] .
a 1 · ( ξ - ξ 0 ) = m 1 , a 2 · ( ξ - ξ 0 ) = m 2 ,
ξ - ξ 0 = δ 1 b 1 + δ 2 b 2 .
δ 1 = m 1 ,             δ 2 = m 2 .
ξ = ξ 0 + b m ,
ξ = - ξ 0 + b m .
β ˆ ( ξ ) n e - 2 π i a n · ( ξ - ξ 0 ) e 2 π i x · ξ d ξ
β ˆ ( ξ ) n e - 2 π i a n · ( ξ + ξ 0 ) e 2 π i x · ξ d ξ .
R n e - 2 π i a n · ( ξ - ξ 0 ) d ξ N ( B / N ) = B
B n β ˆ ( ξ n ) e 2 π i x · ξ n ,             B n β ˆ ( - ξ n ) e - 2 π i x · ξ n ,
U ( x ) = B n [ α ˆ * ( ξ 0 ) β ˆ ( ξ n ) a 0 e 2 π i x · ξ n + α ˆ ( ξ 0 ) β ˆ * ( ξ n ) a 0 * e - 2 π i x · ξ n ] ,