Abstract

We present a computationally efficient method for solving the method of excess fractions used in multi-frequency interferometry for absolute phase measurement. The Chinese remainder theorem, an algorithm from number theory is used to provide a unique solution for absolute distance via a set of congruence’s based on modulo arithmetic. We describe a modified version of this theorem to overcome its sensitivity to phase measurement noise. A comparison with the method of excess fractions has been performed to assess the performance of the algorithm and processing speed achieved. Experimental data has been obtained via a full-field fringe projection system for three projected fringe frequencies and processed using the modified Chinese remainder theorem algorithm.

© 2004 Optical Society of America

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    [CrossRef]
  2. F. H. Rolt, “The application of optics to engineering measurements,” Engineering 144, 162 (1937).
  3. M. Kujawinska and J. Wojciak, “High Accuracy Fourier Transform Fringe Pattern Analysis,” Opt. Lasers Eng. 14, 325 (1991).
    [CrossRef]
  4. M. R. Benoit, “Applications des phenomenes d’interference a des determinations metrologiques,” J. Phys. 3, 57 (1898).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. K. H. Rosen, Elementary number theory and its applications (Addison-Wesley publishing Co., 1988).
  14. I. Agurok, “The rigorous decision of the excess fraction method in absolute distance interferometry,” SPIE 3134, 504 (1997).
    [CrossRef]
  15. M Takeda, Q. Gu, M. Kinoshita, H Takai, and Y Takahash, “Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. 36, 5347 (1997).
    [CrossRef] [PubMed]
  16. J. Zhong and Y. Zhang, “Absolute phase measurement technique based on number theory in multi-frequency grating projection profilometry,” Appl. Opt. 40, 492 (2001).
    [CrossRef]
  17. J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131 (1997).
    [CrossRef]

2003 (3)

2001 (1)

1997 (3)

M Takeda, Q. Gu, M. Kinoshita, H Takai, and Y Takahash, “Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. 36, 5347 (1997).
[CrossRef] [PubMed]

I. Agurok, “The rigorous decision of the excess fraction method in absolute distance interferometry,” SPIE 3134, 504 (1997).
[CrossRef]

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

1994 (1)

1991 (1)

M. Kujawinska and J. Wojciak, “High Accuracy Fourier Transform Fringe Pattern Analysis,” Opt. Lasers Eng. 14, 325 (1991).
[CrossRef]

1985 (1)

1977 (1)

1937 (1)

F. H. Rolt, “The application of optics to engineering measurements,” Engineering 144, 162 (1937).

1902 (1)

C. Fabry and A. Perot, “Measures of absolute wavelengths in the solar spectrum and in the spectrum of iron,” Astrophysical J. 15, 73 (1902).
[CrossRef]

1898 (1)

M. R. Benoit, “Applications des phenomenes d’interference a des determinations metrologiques,” J. Phys. 3, 57 (1898).

Agurok, I.

I. Agurok, “The rigorous decision of the excess fraction method in absolute distance interferometry,” SPIE 3134, 504 (1997).
[CrossRef]

Benoit, M. R.

M. R. Benoit, “Applications des phenomenes d’interference a des determinations metrologiques,” J. Phys. 3, 57 (1898).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Seventh Edition, 2002).

Cheng, Y.

Creath, C.

C. Creath, “Temporal Phase Measurement,” in Interferogram Analysis editors D.W. Robinson and G. T. Reid (Bristol, Institute of Physics Publishing1993).

de Groot, P.J.

Fabry, C.

C. Fabry and A. Perot, “Measures of absolute wavelengths in the solar spectrum and in the spectrum of iron,” Astrophysical J. 15, 73 (1902).
[CrossRef]

Gu, Q.

Hand, D.P.

M. Reeves, A.J. Moore, D.P. Hand, and J.D.C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42, 2923 (2003).
[CrossRef]

Huntley, J. M.

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

Jones, J. D. C.

Jones, J.D.C.

M. Reeves, A.J. Moore, D.P. Hand, and J.D.C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42, 2923 (2003).
[CrossRef]

Kinoshita, M.

Kujawinska, M.

M. Kujawinska and J. Wojciak, “High Accuracy Fourier Transform Fringe Pattern Analysis,” Opt. Lasers Eng. 14, 325 (1991).
[CrossRef]

Moore, A.J.

M. Reeves, A.J. Moore, D.P. Hand, and J.D.C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42, 2923 (2003).
[CrossRef]

Perot, A.

C. Fabry and A. Perot, “Measures of absolute wavelengths in the solar spectrum and in the spectrum of iron,” Astrophysical J. 15, 73 (1902).
[CrossRef]

Pfoertner, A.

Reeves, M.

M. Reeves, A.J. Moore, D.P. Hand, and J.D.C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42, 2923 (2003).
[CrossRef]

Rolt, F. H.

F. H. Rolt, “The application of optics to engineering measurements,” Engineering 144, 162 (1937).

Rosen, K. H.

K. H. Rosen, Elementary number theory and its applications (Addison-Wesley publishing Co., 1988).

Schwider, J.

Takahash, Y

Takai, H

Takeda, M

Tilford, C. R.

Towers, C. E.

Towers, D. P.

Wojciak, J.

M. Kujawinska and J. Wojciak, “High Accuracy Fourier Transform Fringe Pattern Analysis,” Opt. Lasers Eng. 14, 325 (1991).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Seventh Edition, 2002).

Wyant, J.

Zhang, Y.

Zhong, J.

Appl. Opt. (6)

Astrophysical J. (1)

C. Fabry and A. Perot, “Measures of absolute wavelengths in the solar spectrum and in the spectrum of iron,” Astrophysical J. 15, 73 (1902).
[CrossRef]

Engineering (1)

F. H. Rolt, “The application of optics to engineering measurements,” Engineering 144, 162 (1937).

J. Phys. (1)

M. R. Benoit, “Applications des phenomenes d’interference a des determinations metrologiques,” J. Phys. 3, 57 (1898).

Opt. Eng. (1)

M. Reeves, A.J. Moore, D.P. Hand, and J.D.C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42, 2923 (2003).
[CrossRef]

Opt. Lasers Eng. (2)

M. Kujawinska and J. Wojciak, “High Accuracy Fourier Transform Fringe Pattern Analysis,” Opt. Lasers Eng. 14, 325 (1991).
[CrossRef]

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

Opt. Lett. (1)

SPIE (1)

I. Agurok, “The rigorous decision of the excess fraction method in absolute distance interferometry,” SPIE 3134, 504 (1997).
[CrossRef]

Other (3)

C. Creath, “Temporal Phase Measurement,” in Interferogram Analysis editors D.W. Robinson and G. T. Reid (Bristol, Institute of Physics Publishing1993).

K. H. Rosen, Elementary number theory and its applications (Addison-Wesley publishing Co., 1988).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Seventh Edition, 2002).

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Figures (5)

Fig. 1.
Fig. 1.

Comparison between Excess Fractions, CRT and Modified CRT algorithms with phase measurement noise of 2π/200 radians.

Fig. 2.
Fig. 2.

Comparison between Excess Fractions, CRT and Modified CRT algorithms with phase measurement noise of 2π/500 radians.

Fig. 3.
Fig. 3.

Experimental arrangement.

Fig. 4. (a)
Fig. 4. (a)

Wrapped phase map from an engine port model.

Fig. 4. (b)
Fig. 4. (b)

Unwrapped phase map from an engine port model.

Equations (3)

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x φ 1 ( mod λ 1 ) , x φ 2 ( mod λ 2 ) , , x φ r ( mod λ r ) ,
x = Λ 1 Λ 1 φ 1 + Λ 2 Λ 2 φ 2 + + Λ r Λ r φ r ,
Δ x = i = 1 r Λ i Λ i Δ φ i

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