Abstract

The windowed Fourier transform (WFT) has been recognized as an effective tool for extracting phase map from a single fringe pattern. This paper presents a new WFT-based algorithm, which is guided by the principal component analysis (PCA) of the fringe pattern. With it, the principal direction of frequency at each pixel is determined first by using the PCA of the gradients of the fringe pattern, and then the windowed Fourier ridge for each pixel is detected by searching a one-dimensional space of frequency, so that the computational burden for phase extraction is alleviated significantly. In addition, this technique enables automatically identifying and excluding the regions of singular points from the fringe pattern by using the eigenvalues resulting from the PCA just mentioned. Numerical simulation and experiment are carried out to demonstrate the validity of this method.

© 2013 Optical Society of America

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References

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  1. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  2. Q. Kemao, “Two dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser Eng. 45, 304–317 (2007).
    [CrossRef]
  3. Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408–5419 (2008).
    [CrossRef]
  4. H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
    [CrossRef]
  5. W. Gao and Q. Kemao, “Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis,” Appl. Opt. 51, 328–337 (2012).
    [CrossRef]
  6. I. T. Jolliffe, Principal Component Analysis, 2nd ed., Springer Series in Statistics (Springer, 2002).
  7. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  8. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. 3, 847–855 (1986).
    [CrossRef]
  9. Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40, 1649–1657 (2001).
    [CrossRef]
  10. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2007), Chap. 10.
  11. H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
    [CrossRef]
  12. H. Liu, A. N. Cartwright, and C. Basaran, “Mioré interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
    [CrossRef]
  13. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560–2562 (2005).
    [CrossRef]

2012 (1)

2011 (1)

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

2010 (1)

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

2008 (1)

2007 (1)

Q. Kemao, “Two dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser Eng. 45, 304–317 (2007).
[CrossRef]

2005 (1)

2004 (2)

2001 (1)

1986 (1)

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. 3, 847–855 (1986).
[CrossRef]

1982 (1)

Anand, A.

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

Basaran, C.

Bing, P.

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

Cartwright, A. N.

Gao, W.

Ge, Z.

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2007), Chap. 10.

Guo, H.

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

Ina, H.

Jolliffe, I. T.

I. T. Jolliffe, Principal Component Analysis, 2nd ed., Springer Series in Statistics (Springer, 2002).

Kemao, Q.

Kobayashi, F.

Kobayashi, S.

Kreis, T.

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. 3, 847–855 (1986).
[CrossRef]

Lei, H.

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

Liu, H.

Matsuda, S.

Qian, K.

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

Takeda, M.

Wang, H.

Weng, J.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2007), Chap. 10.

Zhong, J.

Appl. Opt. (5)

IEEE Signal Process. Mag. (1)

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. 3, 847–855 (1986).
[CrossRef]

Opt. Laser Eng. (2)

H. Lei, K. Qian, P. Bing, and A. Anand, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Laser Eng. 48, 141–148 (2010).
[CrossRef]

Q. Kemao, “Two dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser Eng. 45, 304–317 (2007).
[CrossRef]

Opt. Lett. (1)

Other (2)

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2007), Chap. 10.

I. T. Jolliffe, Principal Component Analysis, 2nd ed., Springer Series in Statistics (Springer, 2002).

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Figures (16)

Fig. 1.
Fig. 1.

Simulated interferogram with closed fringes, where the background and modulation are Gaussian-shaped with a 50% decrease in magnitude at the corners, and the SD of additive Gaussian noise is 0.05.

Fig. 2.
Fig. 2.

(a) and (b) show the intensity gradients of the interferogram in Fig. 1, in x and y directions, respectively.

Fig. 3.
Fig. 3.

Results of the PCA of the interferogram in Fig. 1 with (a) being the distribution of large eigenvalues of the covariance matrices and (b) being that of small ones.

Fig. 4.
Fig. 4.

Results of PCA of the interferogram in Fig. 1, with (a) and (b) showing the distributions of the components of the normalized eigenvectors corresponding to the large eigenvalues in Fig. 3(a), in x and y directions, respectively. The discontinuities in them are caused by the undetermined signs of the vectors.

Fig. 5.
Fig. 5.

(a) shows the contribution percentages of the large eigenvalues, calculated from Fig. 3 and (b) is the segmentation result of (a), when using a threshold of 0.6, with the black denoting the saddle or extreme regions of the phase map.

Fig. 6.
Fig. 6.

(a) and (b) show the continuous distributions of the normalized vector components regarding the principal frequency directions of Fig. 1, in x and y directions, respectively. These distributions are obtained from Fig. 4 by determining their signs of vectors. The singular points have been excluded by use of the segmentation results in Fig. 5(b).

Fig. 7.
Fig. 7.

Angle (in radians) of principal frequency directions from x axis, calculated from Fig. 6 using a four-quadrant arctangent function.

Fig. 8.
Fig. 8.

(a) is the phase map (in radians) recovered from Fig. 1 using the proposed method, with the WFR for each pixel being detected in a 1D frequency space along the direction determined by Fig. 6 and (b) is the unwrapped phase map (in radians) of (a).

Fig. 9.
Fig. 9.

Practical interferogram captured using a Zygo GPI XP/D 4 interferometer.

Fig. 10.
Fig. 10.

Results of PCA of the interferogram in Fig. 9, with (a) being the distribution of large eigenvalues of the covariance matrices and (b) being that of small ones.

Fig. 11.
Fig. 11.

(a) shows the contribution percentages of the large eigenvalues, calculated from Fig. 10 and (b) is the segmentation result of (a) using a threshold of 0.65, with the black denoting the saddle or extreme regions of the phase map.

Fig. 12.
Fig. 12.

(a) and (b) show the continuous distributions of the vector components regarding the principal frequency directions of Fig. 9, in x and y directions, respectively. They are calculated from the PCA results of Fig. 9 by determining the signs of eigenvectors. From them the singular points have been excluded by use of the segmentation results in Fig. 11(b).

Fig. 13.
Fig. 13.

Angle (in radians) of principal frequency directions from x axis, calculated from Fig. 12 using a four-quadrant arctangent function.

Fig. 14.
Fig. 14.

(a) is the phase map (in radians) recovered from Fig. 9, using the proposed method, with the WFR for each pixel being detected in a 1D frequency space along the direction determined by Fig. 12. (b) is the unwrapped phase map (in radians) of (a). As a comparison, (c) shows the wrapped phase map (in radians) recovered using the build-in phase-shifting algorithm of the interferometer, and (d) is its unwrapped result (in radians). (e) is the distribution of phase differences (in radians) between (b) and (d).

Fig. 15.
Fig. 15.

Percentage of area of the valid region decreases as the noise SD increases, when using a fixed threshold 0.55 for the contribution percentages of the large eigenvalues in PCA of fringe pattern.

Fig. 16.
Fig. 16.

RMS error of the recovered phase map increases as the noise SD increases.

Equations (24)

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sf(u,v,ξ,η)=++f(x,y)g(xu,y,v)exp[j(ξx+ηy)]dxdy,
f(x,y)=14π2++++[Sf(u,v;ξ,η)×g(xu,yv)exp(jξx+jηy)dξdηdudv].
f(x,y)=a(x,y)+b(x,y)cos[φ(x,y)],
f(x,y)xfx(x,y)=f(x+1,y)f(x1,y)2
f(x,y)yfy(x,y)=f(x,y+1)f(x,y1)2.
σx2=1(2m+1)2u=mmv=mm[fx(xu,yv)]2
σy2=1(2m+1)2u=mmv=mm[fy(xu,yv)]2.
σxy=1(2m+1)2u=mmv=mmfx(xu,yv)fy(xu,yv).
C=[σx2σxyσxyσy2].
|λIC|=0
λ1=[(σx2+σy2)+(σx2σy2)2+4σxy2]/2,
λ2=[(σx2+σy2)(σx2σy2)2+4σxy2]/2.
(Cλ1I)W=0,
{w1=σxy/σxy2+(λ1σx2)2w2=(λ1σx2)/σxy2+(λ1σx2)2,
{w1=(λ1σy2)/σxy2+(λ1σy2)2w2=σxy/σxy2+(λ1σy2)2.
P=λ1λ1+λ2.
W^=(w^1,w^2)T=(cosθ,sinθ)T.
θ=arctanw^2w^1.
[u,v]φ(u,v)=argmaxξ,η|Sf(u,v;ξ,η)|.
φ(u,v)=Im[ln(Sf(u,v;ξ,n)|[ξ,η]=[u,v]φ(u,v))],
ξ=ζcosθ=ζw^1η=ζsinθ=ζw^2,
|φW^|=argmaxζ|Sf(u,v;ζ)|.
φ(u,v)=Im[ln(sf(u,v;ζ)|ζ=|ϕW^|)].
|ϕW^|=argmaxζ|x=unu+ny=vnv+n[f(x,y)f¯]×h(xu,yv)exp[jζ(xw^1+yw^2)]|,

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