Abstract

This paper is a 1-D analysis of the degradation caused by image sampling and interpolative reconstruction. The analysis includes the sample-scene phase as an explicit random parameter and provides a complete characterization of this image degradation as the sum of two terms: one term accounts for the mean effect of undersampling (aliasing) and nonideal reconstruction averaged over all sample-scene phases; the other term accounts for variations about this mean. The results of this paper have application to the design and performance analysis of image scanning, sampling, and reconstruction systems.

© 1982 Optical Society of America

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References

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  1. P. Mertz, F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
  2. O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (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).
  3. M. W. Baldwin, Bell Syst. Tech. J. 19, 563 (1940).
  4. R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).
  5. O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).
  6. L. G. Callahan, W. M. Brown, Appl. Opt. 2, 401 (1963).
    [CrossRef]
  7. A. Macovski, Appl. Opt. 9, 1906 (1970).
    [PubMed]
  8. A. H. Robinson, Appl. Opt. 12, 2344 (1973).
    [CrossRef] [PubMed]
  9. F. O. Huck, S. K. Park, Appl. Opt. 14, 2508 (1975).
    [CrossRef] [PubMed]
  10. R. A. Gonsalves, P. S. Considine, Opt. Eng. 15, 64 (1976).
    [CrossRef]
  11. L. M. Biberman, Ed., Perception of Displayed Information (Plenum, New York, 1973).
    [CrossRef]
  12. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  13. S. J. Katzberg, F. O. Huck, S. D. Wall, Appl. Opt. 12, 1054 (1973).
    [CrossRef] [PubMed]
  14. F. O. Huck, N. Halyo, S. K. Park, Appl. Opt. 19, 2174 (1980).
    [CrossRef] [PubMed]
  15. R. J. Arguello, Proc. Soc. Photo-Opt. Instrum. Eng. 271, 86 (1981).
  16. W. W. Marshall, E. Behane, Mead Tech. Lab. Final Report 75-3 (1975).
  17. F. C. Billingsley, Photogram. Eng. Rem. Sens. 48, 3 (1982).
  18. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  19. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  20. S. Shlien, Can. J. Remote Sensing 5, 74 (1979).
  21. D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
    [CrossRef]
  22. R. Bernstein, IBM J. Res. Dev. 40 (1976).
  23. S. K. Park, R. A. Schowengerdt, Computer Graphics and Image Processing, (to be published, 1983).

1982 (1)

F. C. Billingsley, Photogram. Eng. Rem. Sens. 48, 3 (1982).

1981 (1)

R. J. Arguello, Proc. Soc. Photo-Opt. Instrum. Eng. 271, 86 (1981).

1980 (1)

1979 (1)

S. Shlien, Can. J. Remote Sensing 5, 74 (1979).

1976 (2)

R. Bernstein, IBM J. Res. Dev. 40 (1976).

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

1975 (1)

1973 (2)

1970 (1)

1964 (1)

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).

1963 (1)

1962 (1)

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

1951 (1)

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (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).

1940 (2)

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

R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).

1934 (1)

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

Arguello, R. J.

R. J. Arguello, Proc. Soc. Photo-Opt. Instrum. Eng. 271, 86 (1981).

Baldwin, M. W.

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

Bedford, A. V.

R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).

Behane, E.

W. W. Marshall, E. Behane, Mead Tech. Lab. Final Report 75-3 (1975).

Bernstein, R.

R. Bernstein, IBM J. Res. Dev. 40 (1976).

Billingsley, F. C.

F. C. Billingsley, Photogram. Eng. Rem. Sens. 48, 3 (1982).

Bracewell, R.

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

Brown, W. M.

Callahan, L. G.

Considine, P. S.

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

Fredendall, G. L.

R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gonsalves, R. A.

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

Gray, F.

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

Halyo, N.

Huck, F. O.

Katzberg, S. J.

Kell, R. D.

R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).

Macovski, A.

Marshall, W. W.

W. W. Marshall, E. Behane, Mead Tech. Lab. Final Report 75-3 (1975).

Mertz, P.

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

Middleton, D.

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Park, S. K.

Peterson, D. P.

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Pratt, W. K.

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

Robinson, A. H.

Schade, O. H.

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (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).

Schowengerdt, R. A.

S. K. Park, R. A. Schowengerdt, Computer Graphics and Image Processing, (to be published, 1983).

Shlien, S.

S. Shlien, Can. J. Remote Sensing 5, 74 (1979).

Wall, S. D.

Appl. Opt. (6)

Bell Syst. Tech. J. (2)

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

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

Can. J. Remote Sensing (1)

S. Shlien, Can. J. Remote Sensing 5, 74 (1979).

IBM J. Res. Dev. (1)

R. Bernstein, IBM J. Res. Dev. 40 (1976).

Inf. Control (1)

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

J. Soc. Motion Pict. Telev. Eng. (2)

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (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).

O. H. Schade, J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).

Opt. Eng. (1)

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

Photogram. Eng. Rem. Sens. (1)

F. C. Billingsley, Photogram. Eng. Rem. Sens. 48, 3 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. J. Arguello, Proc. Soc. Photo-Opt. Instrum. Eng. 271, 86 (1981).

RCA Rev. (1)

R. D. Kell, A. V. Bedford, G. L. Fredendall, RCA Rev. 5, 8 (1940).

Other (6)

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

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

W. W. Marshall, E. Behane, Mead Tech. Lab. Final Report 75-3 (1975).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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

S. K. Park, R. A. Schowengerdt, Computer Graphics and Image Processing, (to be published, 1983).

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Figures (9)

Fig. 1
Fig. 1

Image scanning, sampling, and reconstruction process.

Fig. 2
Fig. 2

(a) Blurring introduced by scanning; (b) additional blurring introduced by sampling and reconstruction.

Fig. 3
Fig. 3

Ramp image sample dependence upon the sample-scene phase parameter.

Fig. 4
Fig. 4

SR blur dependence upon the sample-scene phase parameter. Three distinct situations are depicted corresponding to (a) 0 ≤ u < 1 − 0.5s, (b) 1 − 0.5su ≤ 0.5s, and (c) 0.5 < u ≤ 1, respectively.

Fig. 5
Fig. 5

Dependence of SR blur upon the sample-scene phase parameter. Curves correspond to s = 1.0, 1.25, 1.50, 1.75, and 2.00, respectively. For s = 1.50, SR 2 is nearly constant.

Fig. 6
Fig. 6

Probability density (vertical axis) of SR 2 for the case s = 1. Extreme values of SR 2 are significantly more likely than the average value E ( SR 2 ).

Fig. 7
Fig. 7

SR blur variance vs sampling rate (samples/IFOV) for edge reconstruction.

Fig. 8
Fig. 8

Function e2(ν) for three common interpolators, nearest neighbor, linear, cubic and the ideal interpolator, sinc. The inset illustrates the small hump at low frequencies for cubic.

Fig. 9
Fig. 9

Average SR blur as a function of sampling rate (samples/IFOV) for the interpolators, nearest neighbor, linear, cubic, and sinc. The image is an edge scanned with an ideal aperture.

Tables (2)

Tables Icon

Table I Characteristics of Three Common Digital Reconstruction Functions, Nearest Neighbor, Linear, Cubic, and the Ideal Interpolator, Sinc

Tables Icon

Table II Calculation of SR Blur: the Image Spectrum ĝ(ν) is Assumed to be Zero for all |ν| > νc

Equations (72)

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g ( x - u ) = h ( x ) * f ( x - u )
g ( x - u ) comb ( x ) ,
comb ( x ) = n = - δ ( x - n ) .
g r ( x ; u ) = [ g ( x - u ) comb ( x ) ] * r ( x ) ,
g r ( x ; u ) = n = - g ( n - u ) r ( x - n ) ,
image blur = I 2 = - [ f ( x - u ) - g ( x - u ) ] 2 d x ,
sampling and reconstruction blur = SR 2 ( u ) = - [ g ( x - u ) - g r ( x ; u ) ] 2 d x .
f ( x - u ) = step ( x - u ) = { 1 x u , 0 otherwise .
h ( x ) = 1 s rect ( x s ) = { 1 s x s 2 , 0 otherwise ,
g ( x - u ) = h ( x ) * f ( x - u ) = { 1             x > u + s 2 , x - u s + ½             x - u s 2 , 0             x < u - s 2 .
I 2 = 1 12 s
r ( x ) = tri ( x ) = { 1 - x x 1 , 0 oherwise .
SR ( u ) 2 = { 1 48 s 2 [ ( s - 2 u ) 2 ( 2 + 2 u - s ) 2 + ( s + 2 u ) 2 ( 2 - 2 u - s ) 2 ]             0 u 1 - 0.5 s , 1 48 s 2 [ ( s - 2 u ) 2 ( 2 + 2 u - s ) 2 + ( s + 2 u - 2 ) 2 ( 4 - 2 u - s ) 2 ]             1 - 0.5 s < u < 0.5 s , 1 48 s 2 [ ( s - 2 u ) 2 ( 2 - 2 u + s ) 2 + ( s + 2 u - 2 ) 2 ( 4 - 2 u - s ) 2 ]             0.5 s u 1.
E ( SR 2 ) = 0 1 SR 2 d u ,
var ( SR 2 ) = 0 1 [ SR 2 - E ( SR 2 ) ] 2 d u .
E ( SR 2 ) = 1 45 s 2             1 s 2 ,
E ( SR 2 ) = 1 6 - 5 18 s + 1 6 s 2 - 1 30 s 3 ,             s < 1.
F ( x ) F F ^ ( ν )
F ^ ( ν ) = - F ( x ) exp ( - 2 π x ν i ) d x ,
F ( x ) = - F ^ ( ν ) exp ( 2 π x ν i ) d ν .
f ( x - u ) F exp ( - 2 π u ν i ) f ^ ( ν ) ,
g ( x - u ) F exp ( - 2 π u ν i ) h ^ ( ν ) f ^ ( ν ) ,
g r ( x ; u ) F r ^ ( ν ) n = - exp [ - 2 π u ( ν - n ) i ] h ^ ( ν - n ) f ^ ( ν - n ) ,
g r ( x ; u ) F { [ exp ( - 2 π u ν i ) h ^ ( ν ) f ^ ( ν ) ] * comb ( ν ) } r ^ ( ν ) ,
h ^ ( 0 ) = - h ( x ) d x = 1 ,
r ^ ( 0 ) = - r ( x ) d x = 1.
I 2 = - 1 - h ^ ( ν ) 2 f ^ ( ν ) 2 d ν ,
SR 2 ( u ) = m = - a m exp ( 2 π u m i ) ,
a m = 0 1 SR 2 exp ( - 2 π u m i ) d u .
E ( SR 2 ) = a 0 .
var ( SR 2 ) = 2 m = 1 a m 2 .
g ( x - u ) - g r ( x ; u ) F exp ( - 2 π u ν i ) n = - [ δ n - r ^ ( ν ) ] × h ^ ( ν - n ) f ^ ( ν - n ) exp ( 2 π u n i ) ,
a m = - n = - n ¯ = - [ δ n - r ^ ( ν ) ] [ δ n ¯ - r ^ * ( ν ) ] h ^ ( ν - n ) × h ^ * ( ν = n ¯ ) f ^ ( ν - n ) f ^ * ( ν - n ¯ ) × [ 0 1 exp [ - 2 π u ( m - n + n ¯ ) i ] d u ] d ν .
a m = - [ δ m - r ^ ( ν ) - r ^ * ( ν - m ) + n = - r ^ ( ν - n ) 2 ] × h ^ * ( ν ) h ^ ( ν - m ) f ^ * ( ν ) f ^ ( ν - m ) d ν .
f ^ ( ν ) = ½ δ ( ν ) + 1 i 2 π ν ,
h ^ ( ν ) = sinc ( s ν ) ,
r ^ ( ν ) = sinc 2 ( ν ) .
I 2 = - [ 1 - sinc ( s ν ) ] 2 4 π 2 ν 2 d ν ,
SR 2 ( u ) = E ( SR 2 ) + 2 m = 1 a m cos ( 2 π m u ) ,
E ( SR 2 ) = - [ 1 - 2 sinc 2 ( ν ) + n = - sinc 4 ( ν - n ) ] sinc 2 ( s ν ) 4 π 2 ν 2 d ν ,
a m = - [ - sinc 2 ( ν ) - sinc 2 ( ν - m ) + n = - sinc 4 ( ν - n ) ] × sinc ( s ν ) sinc [ s ( ν - m ) ] 4 π 2 ν ( ν - m ) d ν ;             m = 1 , 2 , .
E ( SR 2 ) = - e 2 ( ν ) h ^ ( ν ) f ^ ( ν ) 2 d ν ,
e 2 ( ν ) = 1 - 2 R [ r ^ ( ν ) ] + n = - r ^ ( ν - n ) 2
= 1 - r ^ ( ν ) 2 + n 0 r ^ ( ν - n ) 2 ,
e ( ν ) h ^ ( ν ) f ^ ( ν ) = 0
h ^ ( ν ) f ^ ( ν ) = 0
e 2 ( ν ) = 0 ,             ν ν c
r ^ ( ν ) = { 1 ν ν c , arbitrary ν c     < ν 0.5 , 0 ν > 0.5.
SR 2 ( u ) = E ( SR 2 ) + m 0 a m exp ( 2 π u m i ) .
E ( SR 2 ) = R 2 + S 2 ,
R 2 = - 1 - r ^ ( ν ) 2 h ^ ( ν ) f ^ ( ν ) 2 d ν ,
S 2 = - [ n 0 r ^ ( ν - n ) 2 ] h ^ ( ν ) f ^ ( ν ) 2 d ν .
n 0 r ^ ( ν - n ) 2 ,
S 2 = - [ n 0 r ^ ( ν - n ) 2 ] h ^ ( ν ) f ^ ( ν ) 2 d ν = - r ^ ( ν ) 2 [ n 0 h ^ ( ν - n ) f ^ ( ν - n ) 2 ] d ν .
SR 2 ( u ) = S 2 + R 2 + ϕ ( u ) ,
ϕ ( u ) = m 0 a m exp ( 2 π u m i ) = 2 m = 1 R [ a m exp ( 2 π u m i ) ] .
2 m = 1 a m 2 .
R 2 = - [ 1 - sinc 2 ( ν ) ] 2 [ sinc ( s ν ) 2 π ν ] 2 d ν ,
S 2 = - [ n 0 sinc 4 ( ν - n ) ] [ sinc ( s ν ) 2 π ν ] 2 d ν
n 0 r ^ ( ν - n ) 2
E ( SR 2 ) = - e 2 ( ν ) sinc 2 ( s ν ) 4 π 2 ν 2 d ν ,
n = - r ^ ( ν - n ) 2
n = - r ^ ( ν - n ) 2 .
n = - r ^ ( ν - n ) 2 = ( r * r ) ( 0 ) + 2 n = 1 ( r * r ) ( n ) cos ( 2 π n ν ) .
n = - r ^ ( ν - n ) 2 = r ^ 2 ( ν ) * comb ( ν ) = n = - ( r * r ) ( n ) exp ( - 2 π n ν i ) .
n r ^ ( ν - n ) 2 ,
g ^ ( ν ) = h ^ ( ν ) f ^ ( ν ) ,
v ( ν ; m ) = n = - r ^ ( ν - n ) 2 - r ^ ( ν ) - r ^ * ( ν - m )
s ν c = constant .
2 ( u ) = - [ f ( x - u ) - g r ( x ; u ) ] 2 d x .
2 ( u ) I 2 + SR 2 ( u ) ,
E ( 2 ) I 2 + E ( SR 2 ) .

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