Abstract

A refinement of the Fourier transform fringe-pattern analysis technique which uses a 2-D Fourier transform is described. The 2-D transform permits better separation of the desired information components from unwanted components than a 1-D transform. The accuracy of the technique when applied to real data recorded by a system with a nonlinear response function is investigated. This leads to simple techniques for optimizing an interferogram for analysis by these Fourier transform methods and to an estimate of the error in the retrieved fringe shifts. This estimate is tested on simulated data and found to be reliable.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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 (1981).
    [CrossRef]
  2. W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
    [CrossRef] [PubMed]
  3. D. S. Rozhdestvenskii, “Anomalous Dispersion in Sodium Vapor,” Ann. Phys. 39, 307 (1912).
  4. W. C. Marlow, “Hakenmethode,” Appl. Opt. 6, 1715 (1967).
    [CrossRef] [PubMed]
  5. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Spectral-Line Interferometry with Temporal and Spatial Resolution,” to be published in Opt. Comm.57, (1986).
    [CrossRef]
  6. For example, J. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  7. M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl Opt. 22,3977 (1983).
    [CrossRef] [PubMed]
  8. See, for example,P. Z. Peebles, Communications System Principles (Addison-Wesley, Reading, Mass., 1976).
  9. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  10. R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice Hall, Englewood Cliffs, NJ, 1983).

1983 (2)

M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl Opt. 22,3977 (1983).
[CrossRef] [PubMed]

W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
[CrossRef] [PubMed]

1981 (1)

1967 (1)

1912 (1)

D. S. Rozhdestvenskii, “Anomalous Dispersion in Sodium Vapor,” Ann. Phys. 39, 307 (1912).

Bachor, H.-A.

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Spectral-Line Interferometry with Temporal and Spatial Resolution,” to be published in Opt. Comm.57, (1986).
[CrossRef]

Bone, D. J.

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Spectral-Line Interferometry with Temporal and Spatial Resolution,” to be published in Opt. Comm.57, (1986).
[CrossRef]

Crochiere, R. E.

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice Hall, Englewood Cliffs, NJ, 1983).

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

Ina, H.

Kobayashi, S.

Macy, W. W.

Marlow, W. C.

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

Mutoh, K.

M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl Opt. 22,3977 (1983).
[CrossRef] [PubMed]

Peebles, P. Z.

See, for example,P. Z. Peebles, Communications System Principles (Addison-Wesley, Reading, Mass., 1976).

Pratt, J. K.

For example, J. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Rabiner, L. R.

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice Hall, Englewood Cliffs, NJ, 1983).

Rozhdestvenskii, D. S.

D. S. Rozhdestvenskii, “Anomalous Dispersion in Sodium Vapor,” Ann. Phys. 39, 307 (1912).

Sandeman, R. J.

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Spectral-Line Interferometry with Temporal and Spatial Resolution,” to be published in Opt. Comm.57, (1986).
[CrossRef]

Takeda, M.

M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl Opt. 22,3977 (1983).
[CrossRef] [PubMed]

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 (1981).
[CrossRef]

Ann. Phys. (1)

D. S. Rozhdestvenskii, “Anomalous Dispersion in Sodium Vapor,” Ann. Phys. 39, 307 (1912).

Appl Opt. (1)

M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl Opt. 22,3977 (1983).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (5)

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Spectral-Line Interferometry with Temporal and Spatial Resolution,” to be published in Opt. Comm.57, (1986).
[CrossRef]

For example, J. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

See, for example,P. Z. Peebles, Communications System Principles (Addison-Wesley, Reading, Mass., 1976).

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice Hall, Englewood Cliffs, NJ, 1983).

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 (7)

Fig. 1
Fig. 1

Rule of thumb. The interferogram (a) for which the mean intensity contours are shown has a 2-D Fourier transform for which the contour plot of the power of the components is (b). The contour interval in (b) is 1 order of magnitude with contours below 10−4 of the peak power omitted for clarity. The line separating the Q(νν0) and Q*(−νν0) components is indicated.

Fig. 2
Fig. 2

Relative rms error in the fringe shifts as a function of Δϕ for the fringe shift data of Fig. 7 with ν0 = (0,10.5) fr/fd in the presence of the spurious components indicated. The data are extended with a border of five points to form a field of 88 × 50 points, and the Hanning window is used. Filter 1 is the half-plane filter, which includes components in the range νx = −nx/2 to nx/2 fr/fd, νy = 0 to ny/2 fr/fd. Filter 2 includes all components in the range νx = −11 to 11 fr/fd, νy = 7 to 19 fr/fd. For □, s0 = 0%, and filter 1 is used, the rest use filter 2 with ○ s0 = 0%, ◊ s0 = 1%, × s0 = 5%, + s0 = 20%, where s0 measures the level of spurious components and is defined in Appendix C.

Fig. 3
Fig. 3

Interferogram produced by spectral line interferometry with temporal and spatial resolution.5

Fig. 4
Fig. 4

Results of evaluating the test interferogram which simulates the interferogram of Fig. 3. (a) The fringe shift distribution used to generate the test interferogram. The contour interval is 0.25 fringes, (b) The recovered fringe shift distribution, (c) The fringe shift along the line indicated. The points indicate the recovered fringe shifts, and the curve is the generated test fringe shift distribution.

Fig. 5
Fig. 5

Results of analyzing the interferogram of Fig. 3. (a) Contour plot of the fringe shift distribution. The contour interval is 0.25 fringes, (b) Fringe shift along the line indicated in (a).

Fig. 6
Fig. 6

Aliasing in two dimensions. The region N bounded by the Nyquist frequencies and the region R containing the baseband of the desired informations spectral components are indicated. Even in the presence of unwanted components (complex conjugate in R*, low-frequency background in B, second harmonic in H and H*), the heterodyning frequency and sampling frequency can still be chosen to avoid aliasing error, even though these data are by the generally accepted criterion undersampled.

Fig. 7
Fig. 7

Contour plot of the fringe shift distributions used for Fig. 2. The contour interval is Δϕ/10.

Equations (17)

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

i ( r ) = m ( r ) cos { 2 π [ ν 0 r + ϕ ( r ) ] } + h ( r ) cos { 4 π [ ν 0 r + ϕ ( r ) ] } + b ( r ) + n ( r ) ,
I ( ν ) = ½ M Q ( ν ν 0 ) + ½ M Q * ( ν ν 0 ) + ½ H Q Q ( ν 2 ν 0 ) + ½ H Q * Q * ( ν 2 ν 0 ) + B ( ν ) + N ( ν ) ,
g ( r ) = ½ m ( r ) exp { 2 π i [ ν 0 r + ϕ ( r ) ] } .
Δ ν θ = Δ f θ + η Δ f θ m / ( π Δ ϕ ) + ( Δ ν m ) θ + ( Δ ν w ) θ ,
( d y / d x ) ϑ ( r ) = const = [ f x ( r ) ] / [ f y ( r ) ] .
E ( δ ϕ rms ) = ( α / 2 ) 1 / 2 σ n / ( π m ̅ ) ,
( δ ϕ rms ) est = ( 2 α ) 1 / 2 σ n / ( π m ̅ ) .
s ( r ) = j δ ( r N 1 j ) S ( v ) = | Det ( N ) | j δ ( ν N j ) ,
G ( ν ) = Γ ( ν ) W f ( ν ) ,
p ( r ) = j δ ( r j ) P ( ν ) = j δ ( ν j ) .
Q ( ν x , ν y ) = P ( ν x , y ) exp [ 2 π i ( y ν y ) d y ] ,
P ( ν x , y ) = q ( x , y ) exp [ 2 π i ( x ν x d x ] .
f ( r ) = ϑ ( r ) .
Δ f ± = π μ Δ ϕ .
Δ ν ± = Δ f ± + [ η Δ f m / ( π Δ ϕ ) ] .
Δ ν θ = Δ f θ + η Δ f θ m / ( π Δ ϕ ) ,
n ( r ) = n 0 ρ , b ( r ) = b 0 ( 2 y 3 4 y 2 + 2 ) ( 2 x 2 2 x ) , m ( r ) = [ 1 m 0 ( y 2 2 y + 1 ) ] [ 1 m 0 ( 4 x 2 4 x + 1 ) ] , h ( r ) = h 0 m ( r ) .

Metrics