Abstract

The optimum condition of fringe spacing in a flatness test using digital fringe analysis is described. This condition is based on information theory and maximizes the amount of measurement information in a fringe image. First, the relationship between resolution and data sampling number is discussed. Then the optimum fringe spacing is derived using the concept of average information—entropy. Numerical analysis to evaluate fringe spacing in FFT fringe analysis is performed using the derived condition. An experiment is carried out with a liquid level flatness tester and its results show high space frequency resolution superiority at the condition.

© 1985 Optical Society of America

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References

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  1. W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
    [CrossRef] [PubMed]
  2. A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957), p. 3.
  3. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1980), p. 89.
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier-Transform Method of Fringe-Pattern Analysis for Computer-Based Topography and Interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  5. Y. Yokoyama, J. Matsuda, “Trial Manufacture of Flatness Tester,” Proc. Jpn. Soc. Prec. Eng. S.46, 305 (1971), in Japanese.

1983 (1)

1982 (1)

1971 (1)

Y. Yokoyama, J. Matsuda, “Trial Manufacture of Flatness Tester,” Proc. Jpn. Soc. Prec. Eng. S.46, 305 (1971), in Japanese.

Ina, H.

Khinchin, A. I.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957), p. 3.

Kobayashi, S.

Macy, W. W.

Matsuda, J.

Y. Yokoyama, J. Matsuda, “Trial Manufacture of Flatness Tester,” Proc. Jpn. Soc. Prec. Eng. S.46, 305 (1971), in Japanese.

Shannon, E.

E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1980), p. 89.

Takeda, M.

Weaver, W.

E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1980), p. 89.

Yokoyama, Y.

Y. Yokoyama, J. Matsuda, “Trial Manufacture of Flatness Tester,” Proc. Jpn. Soc. Prec. Eng. S.46, 305 (1971), in Japanese.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. Jpn. Soc. Prec. Eng. (1)

Y. Yokoyama, J. Matsuda, “Trial Manufacture of Flatness Tester,” Proc. Jpn. Soc. Prec. Eng. S.46, 305 (1971), in Japanese.

Other (2)

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957), p. 3.

E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1980), p. 89.

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Figures (7)

Fig. 1
Fig. 1

Example of Fizeau fringes for the flatness test using digital images having M × N pixels.

Fig. 2
Fig. 2

(a) Relationship between fringe spacing and resolution in a horizontal raster having N pixels. (b) Relationship between fringe spacing and the number of sampling points.

Fig. 3
Fig. 3

(a) Peak position determination of the fringe intensity distribution, which corresponds to phase shift detection. (b) Probability density functions: (i) uniform distribution with equivalent occurrence probability 1/l; (ii) Gaussian distribution with standard deviation σ (6σ ≦ λ/2).

Fig. 4
Fig. 4

Computer simulation flow chart for examining the relationship between fringe spacing and average shape estimation error.

Fig. 5
Fig. 5

Results of numerical analysis for N = 256: (a) is the rms error for fringe spacing and it indicates that 4 pixels/fringe minimize the error; (b) represents the rms error for carrier order and 64 fringes in a 256-pixel field of view minimize the error.

Fig. 6
Fig. 6

Schematic construction of the optical flatness measuring system: L1,L2 works as a collimator for the He–Ne laser beam. Fizeau fringe spacing can be changed by adjusting the micrometer head and fringes are observed through a TV camera–frame memory system. A microcomputer analyzes these fringes using a FFT fringe analysis algorithm.

Fig. 7
Fig. 7

Experimental results of the flatness test: (a) 4 pixels-fringe, (b) 8 pixels/fringe, and (c) 12 pixels/fringe. Upper left part of the liquid level is deformed for the spatial resolution test.

Equations (14)

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z ( x , y ) = d ( x , y ) D × λ 2 ,
I ( x , y ) = a ( x , y ) + b ( x , y ) × cos [ 2 π f c x + φ ( x , y ) ] ,
φ ( x , y ) = 2 π λ / 2 × z ( x , y ) ,
A = [ 1 2 i l p 1 p 2 p i p l ] ,
H 0 = i = 1 l p i × log 1 p i .
H = N l × H 0 = N l × i = 1 l p i × log 1 p i ,
H 0 = i = 1 l 1 l × log l = log l .
H = N l × log l .
H l = c N l 2 × ( 1 ln l ) ,
p ( x ) = 1 / ( 2 π σ ) × exp ( x 2 / 2 σ 2 ) .
H = log 2 π e σ .
H 0 = log ( 2 π e l / 6 ) ,
H = N l × log ( 2 π e l / 6 ) .
H l = c N j 2 [ 1 ln ( 2 π e l / 6 ) ] = 0 ,

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