Abstract

An analysis is conducted to determine the effect of random microdensitometer positioning errors on the computed Fourier transform (FT) of measured data. Three statistical measures of the FT are derived: the first two moments and the mean-square error (MSE). Results show that the expected FT is multiplied by the conjugate of the characteristic function of the position errors, whereas the variance and the MSE are multiplied by linear combinations of the same function. Working relationships are developed for both the continuous and discrete cases. The continuous errors were Gaussian and uniformly distributed. A simple model of a counter was used to develop expressions for the discrete case. These simple equations relate the error probability density function, or counter parameters to the relative root-mean-square error of the measured FT. An example drawn from reflection microdensitometry suggests that continuous position error tolerances should be about the order of 0.5 μm, with counter failure probabilities less than 9 × 10−9, for a 10% relative rms error in the FT.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. S. Blackman, “Effects of noise on the determination of photographic system modulation transfer function,” Photog. Sci. Eng. 12, 244–250 (1968).
  2. D. Dutton, “Noise and other artifacts of OTF derived from image scanning,” Appl. Opt. 14, 513–521 (1975).
    [Crossref] [PubMed]
  3. M. Takeda and T. Ose, “Influence of noise on the measurement of optical transfer functions by the digital Fourier transform method,” J. Opt. Soc. Am. 65, 502 (1975).
    [Crossref]
  4. S. S. Wilks, Mathematical Statistics (Wiley, New York, 1962).

1975 (2)

1968 (1)

E. S. Blackman, “Effects of noise on the determination of photographic system modulation transfer function,” Photog. Sci. Eng. 12, 244–250 (1968).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Photog. Sci. Eng. (1)

E. S. Blackman, “Effects of noise on the determination of photographic system modulation transfer function,” Photog. Sci. Eng. 12, 244–250 (1968).

Other (1)

S. S. Wilks, Mathematical Statistics (Wiley, New York, 1962).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Relative rms error for Gaussian return-to-position (RTP) errors versus spatial frequency. The three values of the parameter sigma shown are 0.1, 1.0, and 10.0 μm.

Fig. 2
Fig. 2

Relative rms error for uniform RTP errors versus spatial frequency for three values of the parameter theta: 0.1, 1.0, and 10.0 μm.

Fig. 3
Fig. 3

Probability density-function for a single step in the positive x direction. P is the probability of making the step.

Fig. 4
Fig. 4

Relative rms error for discrete RTP errors versus normalized spatial frequency uΔx. The probability of making a step is 0.01.

Fig. 5
Fig. 5

Relative rms error for discrete RTP errors with probability P (=0.0001) versus normalized spatial frequency uΔx.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

f ( x + k ) ,
f j Δ x = j = - f ( x + k ) δ ( x - j Δ x ) ,
F ( u , k ) = F ( u ) exp ( - 2 π i u k ) .
E [ g ( a ) ] = - g ( a ) P ( a ) d a ,
E [ F ( u , ) ] = F ( u ) C * ( u ) ,
Var [ F ( u , ) ] = E [ F ( u , ) 2 ] - E [ F ( u , ) ] 2 .
Var [ F ( u , ) ] = F ( u ) 2 [ 1 - C ( u ) 2 ] .
MSE ( u ) = E [ F ( u , ) - F ( u ) 2 ] .
MSE ( u ) = F ( u ) 2 { 2 - 2 Re [ C ( u ) ] } ,
var nor [ F ( u , ) ] = var [ F ( u , ) ] F ( u ) 2 = 1 - C ( u ) 2 .
RMSE ( u ) = MSE ( u ) F ( u ) 2 = 2 [ 1 - Re C ( u ) ] .
P g ( ) = 1 σ 2 π [ exp ( - 2 / 2 σ 2 ] ,
C g ( u ) = exp ( - 2 π 2 u 2 σ 2 ) .
RRMS ( u ) = { 2 [ 1 - exp ( - 2 π 2 u 2 σ 2 ) ] } 1 / 2 .
σ RRMS ( u ) 2 π u .
T g Δ x = 3 π RRMS ( 1 2 Δ x ) = 0.955 RRMS ( 1 2 Δ x ) .
P u ( ) = 1 / θ , θ / 2 0 , > θ / 2 .
C u ( u ) = sin ( π θ u ) π θ u .
RRMS ( u ) = { 2 [ 1 - sin ( π θ u ) ( π θ u ) ] } 1 / 2 .
T u Δ x = 2 3 π RRMS ( 1 / 2 Δ x ) = 1.103 RRMS ( 1 / 2 Δ x ) .
C + 1 ( u ) = q 1 + p 1 exp ( i 2 π u Δ x ) .
C + N ( u ) = [ q 1 + p 1 exp ( i 2 π u Δ x ) ] N ,
C - N ( u ) = q 2 + p 2 exp ( - i 2 π u Δ x N ) ] .
C D ( u ) = C + N ( u ) C - N ( u ) = [ q 1 + p 1 exp ( i 2 π u Δ x ) ] N × [ q 2 + p 2 exp ( - i 2 π u Δ x ) ] N .
C D ( u ) = { 1 - 2 p ( 1 - p ) [ 1 - cos ( 2 π u Δ x ) ] } N .
RMSE ( u ) = 2 - 2 { 1 - 2 p ( 1 - p ) [ 1 - cos ( 2 π u Δ x ) ] } N .
RMSE ( u ) 8 N p ( 1 - p ) .
q [ RRMS ( 1 / 2 Δ x ) ] 2 8 N .