Abstract

We present what we believe to be a novel complex phase tracing method for fringe pattern analysis related to the phase-locked loop idea. The image with deformed complex fringes is analyzed with lexicographic scansion that leads directly to the investigated phase without unwrapping. Robustness of the procedure is ensured by the delay mechanism in the process of calculating the reference value. A numerical model and examples of application of the presented method are given.

© 1999 Optical Society of America

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References

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  1. M. Takeda, “Spatial-carrier fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
    [CrossRef]
  2. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 13.
  3. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-differences measurements,” J. Opt. Soc. Am. 67, 370–378 (1977).
    [CrossRef]
  4. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. 11, 107–117 (1994).
    [CrossRef]
  5. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
    [CrossRef]
  6. M. Servin, J. L. Marroquin, D. Malacara-Hernandez, “Some applications of quadratic cost functionals in fringe analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 24–33 (1996).
    [CrossRef]
  7. M. Rivera, M. Servin, J. L. Marroquin, “Fourier transform technique for phase unwrapping with minimized boundary effects,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds. Proc. SPIE2860, 54–60 (1996).
    [CrossRef]
  8. R. Rodriguez-Vera, M. Servin, “Phase locked loop profilometry,” Opt. Laser Technol. 26, 393–397 (1994).
    [CrossRef]
  9. M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
    [CrossRef]
  10. J. Kozłowski, G. Serra, “A new modified PLL method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
    [CrossRef]
  11. J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
    [CrossRef]
  12. J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfield, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  13. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986/1987).
    [CrossRef]
  14. Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
    [CrossRef] [PubMed]
  15. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 56–160 (1982).
    [CrossRef]
  16. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  17. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  18. M. Kujawińska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
    [CrossRef]
  19. T. Kreis, Holographic Interferometry (Akademie, Berlin, 1996).
  20. J. Kozłowski, G. Serra, “Analysis of the complex phase error introduced by the application of the Fourier transform method,” J. Mod. Opt. (1999), in press.
  21. A. Papoulis, Circuits and Systems: A Modern Approach (Holt, Rinehart & Wilson, New York, 1980).

1997 (1)

J. Kozłowski, G. Serra, “A new modified PLL method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

1995 (2)

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

1994 (2)

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. 11, 107–117 (1994).
[CrossRef]

R. Rodriguez-Vera, M. Servin, “Phase locked loop profilometry,” Opt. Laser Technol. 26, 393–397 (1994).
[CrossRef]

1991 (1)

M. Kujawińska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

1990 (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

1987 (1)

1983 (1)

1982 (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, 56–160 (1982).
[CrossRef]

1977 (2)

D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-differences measurements,” J. Opt. Soc. Am. 67, 370–378 (1977).
[CrossRef]

J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
[CrossRef]

1974 (1)

1972 (1)

Brangaccio, D. J.

Bruning, J. H.

Fried, D. L.

Gallagher, J. E.

Ghiglia, D. C.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. 11, 107–117 (1994).
[CrossRef]

Herriot, D. R.

Ichioka, Y.

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 56–160 (1982).
[CrossRef]

Inuiya, M.

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 56–160 (1982).
[CrossRef]

Kozlowski, J.

J. Kozłowski, G. Serra, “A new modified PLL method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

J. Kozłowski, G. Serra, “Analysis of the complex phase error introduced by the application of the Fourier transform method,” J. Mod. Opt. (1999), in press.

Kreis, T.

T. Kreis, Holographic Interferometry (Akademie, Berlin, 1996).

Kujawinska, M.

M. Kujawińska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Macy, W. W.

Malacara, D.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Malacara-Hernandez, D.

M. Servin, J. L. Marroquin, D. Malacara-Hernandez, “Some applications of quadratic cost functionals in fringe analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 24–33 (1996).
[CrossRef]

Marguin, J. L.

J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
[CrossRef]

Marroquin, J. L.

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

M. Rivera, M. Servin, J. L. Marroquin, “Fourier transform technique for phase unwrapping with minimized boundary effects,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds. Proc. SPIE2860, 54–60 (1996).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara-Hernandez, “Some applications of quadratic cost functionals in fringe analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 24–33 (1996).
[CrossRef]

Papoulis, A.

A. Papoulis, Circuits and Systems: A Modern Approach (Holt, Rinehart & Wilson, New York, 1980).

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 13.

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986/1987).
[CrossRef]

Rivera, M.

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

M. Rivera, M. Servin, J. L. Marroquin, “Fourier transform technique for phase unwrapping with minimized boundary effects,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds. Proc. SPIE2860, 54–60 (1996).
[CrossRef]

Roddier, C.

Roddier, F.

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

R. Rodriguez-Vera, M. Servin, “Phase locked loop profilometry,” Opt. Laser Technol. 26, 393–397 (1994).
[CrossRef]

J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
[CrossRef]

Romero, L. A.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. 11, 107–117 (1994).
[CrossRef]

Rosenfield, D. P.

Serra, G.

J. Kozłowski, G. Serra, “A new modified PLL method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

J. Kozłowski, G. Serra, “Analysis of the complex phase error introduced by the application of the Fourier transform method,” J. Mod. Opt. (1999), in press.

Servin, M.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

R. Rodriguez-Vera, M. Servin, “Phase locked loop profilometry,” Opt. Laser Technol. 26, 393–397 (1994).
[CrossRef]

J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
[CrossRef]

M. Rivera, M. Servin, J. L. Marroquin, “Fourier transform technique for phase unwrapping with minimized boundary effects,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds. Proc. SPIE2860, 54–60 (1996).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara-Hernandez, “Some applications of quadratic cost functionals in fringe analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 24–33 (1996).
[CrossRef]

Takeda, M.

M. Takeda, “Spatial-carrier fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 56–160 (1982).
[CrossRef]

White, A. D.

Wójciak, J.

M. Kujawińska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Appl. Opt. (4)

Ind. Metrol. (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

J. Opt. Soc. Am. (5)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 56–160 (1982).
[CrossRef]

D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-differences measurements,” J. Opt. Soc. Am. 67, 370–378 (1977).
[CrossRef]

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. 11, 107–117 (1994).
[CrossRef]

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

J. L. Marguin, M. Servin, R. Rodriguez-Vera, “Adaptative quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. 14, 1742–1753 (1977).
[CrossRef]

Opt. Eng. (1)

J. Kozłowski, G. Serra, “A new modified PLL method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

Opt. Laser Technol. (1)

R. Rodriguez-Vera, M. Servin, “Phase locked loop profilometry,” Opt. Laser Technol. 26, 393–397 (1994).
[CrossRef]

Opt. Lasers Eng. (3)

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986/1987).
[CrossRef]

M. Kujawińska, J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Other (6)

T. Kreis, Holographic Interferometry (Akademie, Berlin, 1996).

J. Kozłowski, G. Serra, “Analysis of the complex phase error introduced by the application of the Fourier transform method,” J. Mod. Opt. (1999), in press.

A. Papoulis, Circuits and Systems: A Modern Approach (Holt, Rinehart & Wilson, New York, 1980).

M. Servin, J. L. Marroquin, D. Malacara-Hernandez, “Some applications of quadratic cost functionals in fringe analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 24–33 (1996).
[CrossRef]

M. Rivera, M. Servin, J. L. Marroquin, “Fourier transform technique for phase unwrapping with minimized boundary effects,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds. Proc. SPIE2860, 54–60 (1996).
[CrossRef]

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 13.

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Figures (14)

Fig. 1
Fig. 1

Block diagram of the proposed CPT method.

Fig. 2
Fig. 2

Complex amplitude of the reference phase-modulation distribution r(x) compared with the complex amplitude of the traced phase-modulating function ϕ(x).

Fig. 3
Fig. 3

Multiplication of the positive (right-hand) part of the spectrum by a linear term i ν and the negative (left-hand) part by 1.

Fig. 4
Fig. 4

One-dimensional convolution of the left- and right-hand parts of the modified spectrum, its windowing, and its inverse Fourier transform.

Fig. 5
Fig. 5

Numerically generated model of a fringe pattern.

Fig. 6
Fig. 6

Modulus of the left-hand part of the Fourier transform of the distribution shown in Fig. 5 and the slope term by which it has to be multiplied (see Fig. 3).

Fig. 7
Fig. 7

Modulus of the right-hand part of the Fourier transform of the distribution presented in Fig. 5.

Fig. 8
Fig. 8

Modulus of the left-hand part of the Fourier transform multiplied by the slope distribution.

Fig. 9
Fig. 9

Modulus of a convolution of two modified parts of the Fourier transform F c (ν).

Fig. 10
Fig. 10

Modulus of the convolution of two modified parts of the Fourier transform F c (ν), windowed and shifted in frequency (necessary operations when standard fast Fourier transform procedures are applied).

Fig. 11
Fig. 11

Original phase-modulating distribution ϕ(x) and estimated difference between ϕ(x) and tracing, reference distribution r(x) [see Δϕ n (x), Eq. (21)]. Assumed feedback value τ n was equal to 0.05 (note that max|Δϕ n (x)| = 5.2 is greater than π).

Fig. 12
Fig. 12

Elements of a full CPT reconstruction of ϕ(x), with applied value of τ e of 0.1021 (as calculated according to the presented description). The real value of max|Δϕ(x)| is smaller than π [equal to ∼2.5 (for 360th pixel)].

Fig. 13
Fig. 13

Vectorial three-dimensional graph of the reconstructed form of the human back.

Fig. 14
Fig. 14

Contour map with a contour interval of 4 mm of the human back presented in Fig. 13.

Equations (36)

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

frcx=B0 expiw0·x+ϕx,
frcx=B0 expiw0x+ϕx.
frx=B0 expiϕx.
Rx=expirx.
expirx0=frx0/B0.
rx0=arctanImfrx0/Refrx0.
Mx0=frx1/Rx0,
Mx0=expiϕx1/expirx0/B0,
Mx0=expiϕx1-rx0/B0.
Δϕx0=ϕx1-rx0.
Δϕx0=arctanImMx0/ReMx0.
ϕRECx0=rx0+Δϕx0,
rx1=1-τrx0+τϕRECx0.
rx+rxdx=1-τrx+τϕRECx.
rx+rxdx=1-τrx+τϕx.
τrx+rxdx=τϕx.
τrν+iνrν=τϕν,
rν=γϕντ/τ+iν.
ϕν-rν=iνϕν/τ+iν.
Δϕx=ϕx-rx,
|Δϕx|<π.
frx=iνFcν,
frx=iϕx+w0frcx,
ϕxfrcx=νFcν,
ϕx+w0frcx]=-1νFcν.
B02ϕx+w0=-1νFcνfrc*x,
ϕx=-1νFcνfrc*x/B02-w0.
Δϕx=iνϕν/τ+iν.
Δϕx=ϕx/τ+iν.
exp-τx|0=1/τ+iν.
Δϕx=ϕxTν.
ϕx=νFcνFc*-ν/B02-w0δ0.
Δϕx=B0-2νFcνFc*-ν-w0δ0Tν.
Δϕx=1/2πτB0-2νFcνFc*-ν-w0δ0.
τe>τnπ/|Δϕnx|max,
e>nπ/|Δϕnx|max.

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