Abstract

Sampling generally causes the response of a digital imaging system to be locally shift-variant and not directly amenable to MTF analysis. However, this paper demonstrates that a meaningful system response can be calculated by averaging over an ensemble of point-source system inputs to yield an MTF which accounts for the combined effects of image formation, sampling, and image reconstruction. As an illustration, the MTF of the Landsat MSS system is analyzed to reveal an average effective IFOV which is significantly larger than the commonly accepted value, particularly in the along-track direction where undersampling contributes markedly to an MTF reduction and resultant increase in image blur.

© 1984 Optical Society of America

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References

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  1. S. K. Park, R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).
    [CrossRef] [PubMed]
  2. R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the Two-Dimensional Sampling and Reconstruction of Images,” Int. J. Remote Sensing, to be published.
  3. W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
    [CrossRef]
  4. D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
    [CrossRef]
  5. O. H. Schade, in Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 233–278.
    [CrossRef]
  6. R. Legault, Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 279–312.
    [CrossRef]
  7. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  8. W. D. Montgomery, J. Opt. Soc. Am., 65, 700 (1975).
    [CrossRef]
  9. D. F. Barbe, S. B. Campana, in Advances in Image Pickup and Display, Vol. 3, B. Kazan, Ed. (Academic Press, New York, 1977), pp. 171–296.
  10. Advanced Scanners and Imaging Systems for Earth Observations, NASA Spec. Publ. 335, (1973), pp. 104–109.
  11. P. N. Slater, Opt. Acta 22, 277 (1975).
    [CrossRef]
  12. J. C. Lansing, R. W. Cline, Opt. Eng. 14, 312 (1975).
  13. R. J. Arguello, Proc. Soc. Photo-Opt. Instrum. Eng. 271, 86 (1981).
  14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  15. S. K. Park, R. A. Schowengerdt, Comput. Vision Graphics and Image Process. 23, 258 (1983).
    [CrossRef]
  16. P. N. Slater, Photogramm. Eng. Remote Sensing 45, 1479 (1979).
  17. A. P. Colvocoresses, Photogramm. Eng. Remote Sensing 46, 765 (1980).
  18. L. L. Thompson, Photogramm. Eng. Remote Sensing 46, 766 (1980).
  19. P. N. Slater, Photogramm. Eng. Remote Sensing 46, 767 (1980).
  20. D. E. Friedmann, Photogramm. Eng. Remote Sensing 46, 1541 (1980).
  21. D. E. Johnson, Introduction to Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  22. S. J. Katzberg, F. O. Huck, S. D. Wall, Appl. Opt. 12, 1054 (1973).
    [CrossRef] [PubMed]
  23. F. O. Huck, S. K. Park, N. Halyo, Appl. Opt. 19, 2174 (1980).
    [CrossRef] [PubMed]
  24. D. S. Simonett, Manual of Remote Sensing (American Society of Photogrammetry, 1983), Chap. 1, pp. 21, 22.

1983

S. K. Park, R. A. Schowengerdt, Comput. Vision Graphics and Image Process. 23, 258 (1983).
[CrossRef]

1982

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

S. K. Park, R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).
[CrossRef] [PubMed]

1981

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

1980

A. P. Colvocoresses, Photogramm. Eng. Remote Sensing 46, 765 (1980).

L. L. Thompson, Photogramm. Eng. Remote Sensing 46, 766 (1980).

P. N. Slater, Photogramm. Eng. Remote Sensing 46, 767 (1980).

D. E. Friedmann, Photogramm. Eng. Remote Sensing 46, 1541 (1980).

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

1979

P. N. Slater, Photogramm. Eng. Remote Sensing 45, 1479 (1979).

1975

W. D. Montgomery, J. Opt. Soc. Am., 65, 700 (1975).
[CrossRef]

P. N. Slater, Opt. Acta 22, 277 (1975).
[CrossRef]

J. C. Lansing, R. W. Cline, Opt. Eng. 14, 312 (1975).

1973

Advanced Scanners and Imaging Systems for Earth Observations, NASA Spec. Publ. 335, (1973), pp. 104–109.

S. J. Katzberg, F. O. Huck, S. D. Wall, Appl. Opt. 12, 1054 (1973).
[CrossRef] [PubMed]

1962

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

Arguello, R. J.

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

Baars, J.

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

Barbe, D. F.

D. F. Barbe, S. B. Campana, in Advances in Image Pickup and Display, Vol. 3, B. Kazan, Ed. (Academic Press, New York, 1977), pp. 171–296.

Campana, S. B.

D. F. Barbe, S. B. Campana, in Advances in Image Pickup and Display, Vol. 3, B. Kazan, Ed. (Academic Press, New York, 1977), pp. 171–296.

Cline, R. W.

J. C. Lansing, R. W. Cline, Opt. Eng. 14, 312 (1975).

Colvocoresses, A. P.

A. P. Colvocoresses, Photogramm. Eng. Remote Sensing 46, 765 (1980).

Fontanella, J. C.

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

Friedmann, D. E.

D. E. Friedmann, Photogramm. Eng. Remote Sensing 46, 1541 (1980).

Gaskill, J. D.

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

Gray, R.

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the Two-Dimensional Sampling and Reconstruction of Images,” Int. J. Remote Sensing, to be published.

Halyo, N.

Huck, F. O.

Johnson, D. E.

D. E. Johnson, Introduction to Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Katzberg, S. J.

Lansing, J. C.

J. C. Lansing, R. W. Cline, Opt. Eng. 14, 312 (1975).

Legault, R.

R. Legault, Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 279–312.
[CrossRef]

Middleton, D.

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

Montgomery, W. D.

Newberry, A. R.

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

Park, S. K.

S. K. Park, R. A. Schowengerdt, Comput. Vision Graphics and Image Process. 23, 258 (1983).
[CrossRef]

S. K. Park, R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).
[CrossRef] [PubMed]

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

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the Two-Dimensional Sampling and Reconstruction of Images,” Int. J. Remote Sensing, to be published.

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).

Schade, O. H.

O. H. Schade, in Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 233–278.
[CrossRef]

Schowengerdt, R. A.

S. K. Park, R. A. Schowengerdt, Comput. Vision Graphics and Image Process. 23, 258 (1983).
[CrossRef]

S. K. Park, R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).
[CrossRef] [PubMed]

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the Two-Dimensional Sampling and Reconstruction of Images,” Int. J. Remote Sensing, to be published.

Simonett, D. S.

D. S. Simonett, Manual of Remote Sensing (American Society of Photogrammetry, 1983), Chap. 1, pp. 21, 22.

Slater, P. N.

P. N. Slater, Photogramm. Eng. Remote Sensing 46, 767 (1980).

P. N. Slater, Photogramm. Eng. Remote Sensing 45, 1479 (1979).

P. N. Slater, Opt. Acta 22, 277 (1975).
[CrossRef]

Thompson, L. L.

L. L. Thompson, Photogramm. Eng. Remote Sensing 46, 766 (1980).

Wall, S. D.

Wittenstein, W.

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

Advanced Scanners and Imaging Systems for Earth Observations

Advanced Scanners and Imaging Systems for Earth Observations, NASA Spec. Publ. 335, (1973), pp. 104–109.

Appl. Opt.

Comput. Vision Graphics and Image Process.

S. K. Park, R. A. Schowengerdt, Comput. Vision Graphics and Image Process. 23, 258 (1983).
[CrossRef]

Inf. Control

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

J. Opt. Soc. Am.

Opt. Acta

W. Wittenstein, J. C. Fontanella, A. R. Newberry, J. Baars, Opt. Acta 29, 41 (1982).
[CrossRef]

P. N. Slater, Opt. Acta 22, 277 (1975).
[CrossRef]

Opt. Eng.

J. C. Lansing, R. W. Cline, Opt. Eng. 14, 312 (1975).

Photogramm. Eng. Remote Sensing

P. N. Slater, Photogramm. Eng. Remote Sensing 45, 1479 (1979).

A. P. Colvocoresses, Photogramm. Eng. Remote Sensing 46, 765 (1980).

L. L. Thompson, Photogramm. Eng. Remote Sensing 46, 766 (1980).

P. N. Slater, Photogramm. Eng. Remote Sensing 46, 767 (1980).

D. E. Friedmann, Photogramm. Eng. Remote Sensing 46, 1541 (1980).

Proc. Soc. Photo-Opt. Instrum. Eng.

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

Other

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

O. H. Schade, in Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 233–278.
[CrossRef]

R. Legault, Perception of Displayed Information, L. M. Biberman, Ed. (Plenum, New York, 1973), pp. 279–312.
[CrossRef]

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

D. S. Simonett, Manual of Remote Sensing (American Society of Photogrammetry, 1983), Chap. 1, pp. 21, 22.

D. E. Johnson, Introduction to Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1976).

R. A. Schowengerdt, S. K. Park, R. Gray, “Topics in the Two-Dimensional Sampling and Reconstruction of Images,” Int. J. Remote Sensing, to be published.

D. F. Barbe, S. B. Campana, in Advances in Image Pickup and Display, Vol. 3, B. Kazan, Ed. (Academic Press, New York, 1977), pp. 171–296.

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Figures (14)

Fig. 1
Fig. 1

General sampled image system.

Fig. 2
Fig. 2

(a) Illustration that the sampling process is not shift-invariant. A (binary) image g(x,y) with the sampling grid and sampled image gs(x,y) superimposed. The sampled image is δ(x,y). (b) The (binary) image g(xu, yv) shifted relative to the sampling grid; the sampled image is not δ(xu, yv).

Fig. 3
Fig. 3

Illustration of the combined effects of imaging, sampling, and reconstruction. The targets in (a) (upper left) are identical as are their images in (b) (upper right). Image (c) (lower left) is a reconstruction of a sampled version of (b). Image (d) (lower right) is the difference of (b) and (c); it illustrates the shift-variant image degradation associated with sampling and reconstruction.

Fig. 4
Fig. 4

Average system PSF (c) corresponding to an idealized system with a square-imaging subsystem PSF (a) and a bilinear image reconstruction PSF (b). All three PSFs are scaled relative to a common sampling grid.

Fig. 5
Fig. 5

General sampled image system is amenable to MTF analysis provided the imaging and sampling subsystems are combined to form an imaging–sampling subsystem whose sample–scene phase-averaged MTF is | t ^ (μ,ν)|.

Fig. 6
Fig. 6

Comparison of the image subsystem OTF ĥ(ν) and the average MTF of the imaging–sampling subsystem | t ^ (ν)|. Below the Nyquist frequency, 0.5 cycles/sample interval, the combined effects of imaging, sampling, and sample-scene phase averaging reduce the MTF from |ĥ(ν)| to | t ^ (ν)j|. Above the Nyquist frequency, spectrum replication causes a spurious MTF increase.

Fig. 7
Fig. 7

Comparison of the optical subsystem OTF (a) with the associated imaging–sampling subsystem average MTF [(b) and (c)]. The dashed lines indicate in (b) the approximation | t ^ (ν)| ≈ |ĥ(ν)| for νc = 0.2, 0.3 and in (c) the approximation | t ^ (ν)| ≈ |cos(πν)|. In (b) and (c) only the frequency band 0 ≤ ν ≤ 0.5 is illustrated; at higher frequencies | t ^ (ν)| is periodic as indicated in Fig. 6.

Fig. 8
Fig. 8

(a) Comparison of the average imaging–sampling subsystem MTF | t ^ (ν)| and the reconstruction system MTF | r ^ (ν)| corresponding (in 2-D) to bilinear interpolation. (b) Resulting average system MTF, which is the product of | t ^ (ν)| and | r ^ (ν)|.

Fig. 9
Fig. 9

Comparison of the average system MTF for two reconstruction filters: linear and parametric cubic convolution (PCC) with α = −0.5. For comparison, the ASMTF associated with ideal reconstruction [r(x) = sinc(x)] is also shown. Below the Nyquist frequency (0.5), PCC is clearly superior to linear interpolation.

Fig. 10
Fig. 10

Imaging subsystem MTF of the Landsat MSS. In the along-track direction, the frequency response is significantly undersampled.

Fig. 11
Fig. 11

Average imaging-sampling subsystem MTF of the Landsat MSS. In the along-track direction, undersampling has produced large MTF reductions; in the along-scan direction that is not the case.

Fig. 12
Fig. 12

Average system MTF of the Landsat MSS, including the effects of bilinear image reconstruction. Frequency response is significantly better in the along-scan direction.

Fig. 13
Fig. 13

Simulated image of a target as it would be formed by a system whose EIFOV is as indicated. The additional blur associated with the larger asymmetric EIFOV is much more evident than the associated asymmetry.

Fig. 14
Fig. 14

(Sample-scene) phase-averaged EIFOV of the Landsat MSS: (a) based on only the scanning aperture; (b) based on the scanning aperture, image-forming optics, and electronic filter; (c) including also the effects of sampling; (d) the system EIFOV with bilinear image reconstruction; (e) the system EIFOV using PCC (α = 0.5) in place of bilinear image reconstruction.

Equations (25)

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comb ( x , y ) = m n δ ( x - m , y - n ) ,
g r ( x , y ) = { [ f ( x , y ) * h ( x , y ) ] comb ( x , y ) } * r ( x , y ) ,
SPSF ( x , y ; u , v ) = [ h ( x - u , y - v ) comb ( x , y ) ] * r ( x , y ) .
ASPSF ( x , y ) = 0 1 0 1 SPSF ( x , y ; u , v ) d u d v .
ASPSF ( x , y ) = { [ rect ( x - 1 2 , y - 1 2 ) * h ( x , y ) ] × comb ( x , y ) } * r ( x , y ) .
rect ( x - 1 2 , y - 1 2 ) = { 1 0 < x < 1 , 0 < y < 1 0 elsewhere .
SOTF ( μ , ν ; u , v ) = - - SPSF ( x , y ; u , v ) × exp [ - 2 π i ( μ x + ν y ) ] d x d y ,
SOFT ( μ , ν ; u , v ) = r ^ ( μ , ν ) m n exp { - 2 π i [ u ( μ - m ) + v ( ν - n ) ] } h ^ ( μ - m , ν - n ) ,
ASOTF ( μ , ν ) = 0 1 0 1 SOTF ( μ , ν ; u , v ) d u d v .
0 1 exp ( - 2 π u ξ i ) d u = exp ( - π ξ i ) sinc ( ξ ) ,
sinc ( ξ ) = sin π ξ π ξ .
ASOTF ( μ , ν ) = r ^ ( μ , ν ) m n × exp [ - π ( μ - m + ν - n ) i ] sinc ( μ - m ) × sinc ( ν - n ) h ^ ( μ - m , ν - n ) ,
ASOTF ( μ , ν ) = t ^ ( μ , ν ) r ^ ( μ , ν ) exp [ - π ( μ + ν ) i ] ,
t ^ ( μ , ν ) = m n ( - 1 ) m + n sinc ( μ - m ) × sinc ( ν - n ) h ^ ( μ - m , ν - n )
ASMTF ( μ , ν ) = t ^ ( μ , ν ) r ^ ( μ , ν ) .
t ^ ( μ , ν ) h ^ ( μ , ν )
t ^ ( μ , ν ) = exp [ π ( μ + ν ) i ] m n [ rect ( x - ½ , y - ½ ) * h ( x , y ) ] m , n × exp [ - 2 π ( m μ + n ν ) i ] ,
t ^ ( μ , ν ) ¼ exp [ π ( μ + ν ) i ] [ 1 + exp ( - 2 π μ i ) ] [ 1 + exp ( - 2 π ν i ) ] ,
t ^ ( μ , ν ) cos ( π μ ) cos ( π ν ) .
h ^ ( μ , ν ) = exp [ - ( k x μ ) 2 ] sinc ( s x μ ) h ^ b ( k b μ ) × exp [ - ( k y ν ) 2 ] sinc ( s y ν ) ,
h ^ b ( ω ) = ( 1 - 2 ω 2 ) - i ω ( 2 - ω 2 ) 1 + ω 6             ( ω = k b μ ) ,
k x = 31.1 58 , s x = 76.2 58 , k b = 138 58 , k y = 31.1 81.5 , s y = 76.2 81.5 .
r ^ ( μ , ν ) = sinc 2 ( μ ) sinc 2 ( ν ) .
( EIFOV ) along - scan = 1 2 μ c = 104 m ,
( EIFOV ) along - track = 1 2 ν c = 148 m ,

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