Abstract

We describe a technique termed multichannel Fourier fringe analysis and its application to the problem of automatic phase unwrapping in the presence of surface discontinuities. The technique is especially useful for the analysis of fringe projection contour maps in order to measure surface height distributions. Use is made of multiple fringe patterns that are separated in the frequency space of the Fourier transform by means of a set of bandpass filters. We also describe the design of a special fiber-optic interferometer with features particularly important in the case of this technique: easily adjustable fringe spacing and rotation. Fringe production by the interferometer is analyzed, and the relationship between the fringe phase and the height distribution of an illuminated surface is derived. A method for measuring phase in the case of multichannel Fourier fringe analysis is presented. The application of the technique to automatic phase unwrapping is shown. An example of the technique in operation is given, and a discussion of implementation of the technique is included.

© 1994 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. T. M. Kreis, “Digital holographic interference phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–856 (1986).
    [CrossRef]
  7. A. A. Malcolm, D. R. Burton, “The relationship between Fourier fringe analysis and the FFT,” in Lase’r Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 286–297 (1991).
  8. T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. V. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 116–125 (1990).
  9. D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 149–160 (1989).
  10. D. R. Burton, M. J. Lalor, “Precision measurement of engineering form by computer analysis of optically generated contours,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 17–24 (1989).
  11. J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
    [CrossRef] [PubMed]
  12. W. Macy, “Two-dimensional fringe pattern analysis,” Appl. Opt. 22, 3898–3901 (1984).
    [CrossRef]
  13. W. H. Carter, “On unwrapping two-dimensional phase data in contour maps,” Opt. Commun. 94, 1–7 (1992).
    [CrossRef]
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    [CrossRef]
  15. M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” J. Magn. Reson. 24, 177–181 (1992).
    [CrossRef]
  16. T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  21. E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988), pp. 30–31.
  22. R. J. Schalkoff, Digital Image Processing and Computer Vision, 1st ed. (Wiley, New York, 1989), pp. 149–150.
  23. M. Takeda, H. Iijima, “Spatio-temporal heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 210–216 (1990).

1993 (1)

1992 (6)

M. Takeda, M. Kitoh, “Spatiotemporal frequency multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 9, 1607–1614(1992).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

W. H. Carter, “On unwrapping two-dimensional phase data in contour maps,” Opt. Commun. 94, 1–7 (1992).
[CrossRef]

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spacing tree algorithm,” IEEE Trans. Image Proc. 1, 355–365 (1992).
[CrossRef]

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” J. Magn. Reson. 24, 177–181 (1992).
[CrossRef]

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
[CrossRef]

1991 (1)

1988 (1)

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

1984 (1)

1983 (1)

1982 (1)

1974 (1)

Atkinson, J. T.

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–261.

Brangaccio, J.

Braun, M.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spacing tree algorithm,” IEEE Trans. Image Proc. 1, 355–365 (1992).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988), pp. 30–31.

Bruning, J. H.

Bryanston-Cross, P. J.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
[CrossRef]

Burton, D. R.

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

A. A. Malcolm, D. R. Burton, “The relationship between Fourier fringe analysis and the FFT,” in Lase’r Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 286–297 (1991).

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 149–160 (1989).

D. R. Burton, M. J. Lalor, “Precision measurement of engineering form by computer analysis of optically generated contours,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 17–24 (1989).

Carter, W. H.

W. H. Carter, “On unwrapping two-dimensional phase data in contour maps,” Opt. Commun. 94, 1–7 (1992).
[CrossRef]

Ching, N. H.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spacing tree algorithm,” IEEE Trans. Image Proc. 1, 355–365 (1992).
[CrossRef]

Creath, K.

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Gallagher, J. E.

Hedley, M.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” J. Magn. Reson. 24, 177–181 (1992).
[CrossRef]

Herriot, D. R.

Huntley, J. M.

Iijima, H.

M. Takeda, H. Iijima, “Spatio-temporal heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 210–216 (1990).

Ina, H.

Judge, T. R.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
[CrossRef]

Juptner, W. P. O.

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. V. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 116–125 (1990).

Kitoh, M.

Kobayashi, S.

Kreis, T. M.

T. M. Kreis, “Digital holographic interference phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–856 (1986).
[CrossRef]

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. V. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 116–125 (1990).

Lalor, M. J.

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 149–160 (1989).

D. R. Burton, M. J. Lalor, “Precision measurement of engineering form by computer analysis of optically generated contours,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 17–24 (1989).

Macy, W.

Malcolm, A. A.

A. A. Malcolm, D. R. Burton, “The relationship between Fourier fringe analysis and the FFT,” in Lase’r Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 286–297 (1991).

Mutoh, K.

O’Donovan, P. C.

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

Player, M. A.

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Quan, C.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
[CrossRef]

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spacing tree algorithm,” IEEE Trans. Image Proc. 1, 355–365 (1992).
[CrossRef]

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” J. Magn. Reson. 24, 177–181 (1992).
[CrossRef]

Rosenfeld, D. P.

Saldner, H.

Schalkoff, R. J.

R. J. Schalkoff, Digital Image Processing and Computer Vision, 1st ed. (Wiley, New York, 1989), pp. 149–150.

Takeda, M.

White, A. D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–261.

Appl. Opt. (6)

Eng. Med. (1)

J. T. Atkinson, D. R. Burton, M. J. Lalor, P. C. O’Donovan, “Opto-computer methods applied to the evaluation of a range of acetabular cups,” Eng. Med. 17, 105–110 (1988).
[CrossRef] [PubMed]

IEEE Trans. Image Proc. (1)

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spacing tree algorithm,” IEEE Trans. Image Proc. 1, 355–365 (1992).
[CrossRef]

J. Magn. Reson. (1)

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” J. Magn. Reson. 24, 177–181 (1992).
[CrossRef]

J. Meas. Sci. Technol. (1)

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

W. H. Carter, “On unwrapping two-dimensional phase data in contour maps,” Opt. Commun. 94, 1–7 (1992).
[CrossRef]

Opt. Eng. (1)

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic unwrapping,” Opt. Eng. 31, 533–543(1992).
[CrossRef]

Other (8)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–261.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988), pp. 30–31.

R. J. Schalkoff, Digital Image Processing and Computer Vision, 1st ed. (Wiley, New York, 1989), pp. 149–150.

M. Takeda, H. Iijima, “Spatio-temporal heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 210–216 (1990).

A. A. Malcolm, D. R. Burton, “The relationship between Fourier fringe analysis and the FFT,” in Lase’r Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 286–297 (1991).

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms, R. V. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 116–125 (1990).

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 149–160 (1989).

D. R. Burton, M. J. Lalor, “Precision measurement of engineering form by computer analysis of optically generated contours,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 17–24 (1989).

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Figures (3)

Fig. 1
Fig. 1

Optical arrangement.

Fig. 2
Fig. 2

Geometry of fringe projection contouring.

Fig. 3
Fig. 3

Illustration of the technique in operation.

Equations (33)

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p 0 = λ D δ ,
h ( i , j ) = f [ ϕ ( i , j ) ] ,
i ( i , j ) = a ( i , j ) cos [ ϕ ( i , j ) ] + b ( i , j ) ,
ϕ ( i , j ) = 2 π p 0 [ h p ( i , j ) cos α - x p ( i , j ) sin α ] .
h p = h s sin θ + y s cos θ ,
ϕ ( i , j ) = 2 π p 0 { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] × cos α - x s ( i , j ) sin α } .
D = r + y p .
p 0 = λ δ ( r + y p ) .
ϕ ( i , j ) = 2 π δ λ [ r + y p ( i , j ) ] { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] × cos α - x s ( i , j ) sin α } .
y p ( i , j ) = y s ( i , j ) sin θ - h s ( i , j ) cos θ ,
ϕ ( i , j ) = 2 π δ λ × { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] cos α - x s ( i , j ) sin α [ r + y s ( i , j ) sin θ - h s ( i , j ) cos θ ] } .
ϕ ( i , j ) = 2 π δ sin θ λ × { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] cos α - x s ( i , j ) sin α s cos θ + y s ( i , j ) sin 2 θ - h s ( i , j ) cos θ sin θ } .
h s ( i , j ) = λ ϕ ( i , j ) [ s cos θ + y s ( i , j ) sin 2 θ ] - π δ [ y s ( i , j ) sin 2 θ cos α + 2 x s ( i , j ) sin θ sin α ] λ ϕ ( i , j ) cos θ sin θ + 2 π δ sin 2 θ cos α .
i 1 ( i , j ) = a 1 ( i , j ) cos [ ϕ 1 ( i , j ) ] + b 1 ( i , j ) .
ϕ 1 ( i , j ) = 2 π δ sin θ λ × { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] cos α 1 - x s ( i , j ) sin α 1 s cos θ + y s ( i , j ) sin 2 θ - h s ( i , j ) cos θ sin θ } .
i 2 ( i , j ) = a 2 ( i , j ) cos [ ϕ 2 ( i , j ) + b 2 ( i , j ) ,
ϕ 2 ( i , j ) = 2 π δ sin θ λ × { [ h s ( i , j ) sin θ + y s ( i , j ) cos θ ] cos α 2 - x s ( i , j ) sin α 2 s cos θ + y s ( i , j ) sin 2 θ - h s ( i , j ) cos θ sin θ } .
i s ( i , j ) = i 1 ( i , j ) + i 2 ( i , j ) .
F [ i s ( i , j ) ] = F [ i 1 ( i , j ) + i 2 ( i , j ) ] = F [ i 1 ( i , j ) ] + F [ i 2 ( i , j ) ] ,
f i = L i X m 2 π ϕ ( i , j ) i ,             f j = L j Y m 2 π ϕ ( i , j ) j ,
ϕ i = 2 π p 0 [ ( h s i sin θ + y s i cos θ ) cos α - x s i sin α ] .
y s = i Y m ,             x s = i X m ,
y s i = 0 ,             x s i = 1 X m .
h s i = 1 X m h s x s
ϕ i = 2 π p 0 X m ( sin θ cos α h s x s - sin α ) .
f i = L i p 0 ( sin θ cos α h s x s - sin α )
f i = L j p 0 ( sin θ cos α h s y s - sin α ) .
c 1 ( i , j ) = b 1 ( i , j ) 2 cos [ ϕ 1 ( i , j ) ] + j b 1 ( i , j ) 2 sin [ ϕ 1 ( i , j ) ] , c 2 ( i , j ) = b 2 ( i , j ) 2 cos [ ϕ 2 ( i , j ) ] + j b 2 ( i , j ) 2 sin [ ϕ 2 ( i , j ) ] ,
ϕ 1 ( i , j ) = tan - 1 [ Im 1 ( i , j ) Re 1 ( i , j ) ] ,             ϕ 2 ( i , j ) = tan - 1 [ Im 2 ( i , j ) Re 2 ( i , j ) ] .
Δ x = [ 1 0 - 1 2 0 - 2 1 0 - 1 ] ,             Δ y = [ - 1 - 2 - 1 0 0 0 1 2 1 ] .
ϕ e ( i , j ) = ( S x 2 + S y 2 ) 1 / 2 ,             ϕ o ( i , j ) = tan - 1 ( S y S x ) .
ϕ a ( i , j ) = ϕ 1 e ( i , j ) ϕ 2 e ( i , j ) .
X m = 50.0 ,             Y m = 50.0 , θ = 1.2 rad ,             α 1 = π / 4 ,             α 2 = - π / 4.

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