Abstract

A rigorous and simple method for the determination of the modulation transfer function (MTF) of a sampled imaging system is presented. One calculates the MTF by imaging bar patterns and calculating the reduction in amplitude of the fundamental frequency components. The optimal set of bar-pattern frequencies that reduce errors from aliased frequency components is derived. Theoretical and experimental data are presented.

© 1995 Optical Society of America

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References

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  1. W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
    [CrossRef]
  2. L. J. Pinson, Electro-Optics (Wiley, New York, 1985), pp. 118–127.
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 113–133.
  4. S. K. Park, R. Schowengerdt, M. Kaczynski, “Modulation-transfer function analysis for sampled image systems,” Appl. Opt. 23, 2572–2582 (1984).
    [CrossRef] [PubMed]
  5. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 26–30.
  6. W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 345–363.
  7. Ref. 5, Chap. 11.

1984 (1)

1982 (1)

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Baars, J.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Fontanella, J. C.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 113–133.

Kaczynski, M.

Newbery, A. R.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 26–30.

Park, S. K.

Pinson, L. J.

L. J. Pinson, Electro-Optics (Wiley, New York, 1985), pp. 118–127.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 26–30.

Schowengerdt, R.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 345–363.

Wittenstein, W.

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Appl. Opt. (1)

Opt. Act. (1)

W. Wittenstein, J. C. Fontanella, A. R. Newbery, J. Baars, “The definition of the OTF and the measurement of aliasing for sampled imaging systems,” Opt. Act. 29, 41–50 (1982).
[CrossRef]

Other (5)

L. J. Pinson, Electro-Optics (Wiley, New York, 1985), pp. 118–127.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 113–133.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 26–30.

W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 345–363.

Ref. 5, Chap. 11.

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Figures (6)

Fig. 1
Fig. 1

Linear sampled system that is modeled as a continuous linear filtering operation and that is followed by a sampling operator.

Fig. 2
Fig. 2

Schematic diagram of the square-wave input with a period d and a duty cycle L/d that was used for the MTF test.

Fig. 3
Fig. 3

Illustration of the fundamental frequency component and aliased harmonic components of the imaged-bar-pattern input in the spatial frequency domain. Only the fundamental frequency component and the aliased mth and (m + 1)th harmonic components are shown. The dashed curves illustrate the envelope of the MTF of the system.

Fig. 4
Fig. 4

Set of spatial frequencies ( k ¯ = 1 , 2 , 3 , 4 , 5 , 6 ) that introduce minimal aliasing effects during the measurement of the MTF. The filled circles indicate the set of points k ¯ = 7 , 8 , 9 , , which cannot be represented adequately in the figure.

Fig. 5
Fig. 5

Bar-target chart for measurement of the MTF of a sampled imaging system. The chart includes six sets of bar patterns for measuring the MTF in the vertical and horizontal directions. The long solid black bars are used to verify the proper magnification of the system. Fiduciary markings (pluses) border the chart to permit location and verification of the proper orientation of the target in the digital image.

Fig. 6
Fig. 6

Plot of the computed MTF in the vertical and horizontal directions of a commercial scanner. The bar-target chart shown in Fig. 5 was used along with the analysis described in this paper.

Equations (22)

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I ( u , ν ) = H ( u , ν ) O ( u , ν ) = | H ( u , ν ) | exp [ j θ H ( u , ν ) ] O ( u , ν ) ,
H ( u , ν ) = + h ( x , y ) exp [ j 2 π ( u x + ν y ) ] d x d y .
m = + n = + h ( m Δ , n Δ ) exp [ j 2 π Δ ( u m + ν n ) ] = 1 Δ 2 m = + n = + H ( u m Δ , ν n Δ ) ,
o ( x ) = c + a n = + rect ( x n d L ) ,
rect ( x ) = { 1 if | x | < 0 . 5 0 otherwise .
O ( u ) = c δ ( u ) + a L d m = + sinc ( m L d ) δ ( u m d ) ,
sinc ( x ) = sin ( π x ) π x
I ¯ [ k ] = n = 0 N 1 1 ¯ [ n ] exp ( j 2 π k n N ) ,
1 ¯ [ n ] = i ( n Δ ) , n = 0 , 1 , , N 1 ,
I ¯ [ k ] = [ 1 Δ n = + O ( u n Δ ) H ( u n Δ ) ] * W ( u ) | u = k N Δ ,
W ( u ) = N Δ sinc ( N Δ u ) ,
f ( x ) * g ( x ) = + f ( ξ ) g ( x ξ ) d ξ .
1 d = 1 ( m + 1 ) Δ , m = 1 , 2 , 3 , , .
1 d ( 1 Δ m + 1 d ) = ( 1 Δ m d ) 1 d .
1 d = 2 ( 2 m + 3 ) Δ , m = 1 , 2 , 3 , .
k ¯ = 2 N ( 2 m + 3 ) , m = 1 , 2 , 3 , ,
| H ( 1 d ) | = | d I ¯ [ k ¯ ] N a L sinc ( L / d ) | .
L d = I ¯ [ 0 ] / N c a ,
u = 1 d = k ¯ N Δ ,
u = ( 1 + ε ) d = k ¯ ( 1 + ε ) N Δ .
k ¯ ε N Δ .
sinc ( ε k ¯ ) .

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