Abstract

This paper presents a phase-shifting algorithm based on Hough transform for sinusoidal curves. Firstly, the background intensities of phase-shifting fringe patterns are removed by calculating their differences, thus we get purely sinusoidal intensity data for each pixel; and then we implement Hough transform to the intensity difference data of each pixel. As a result, the sinusoidal parameters, including phase and amplitude, of each pixel are extracted. The simulation and experimental results demonstrate that this algorithm enables eliminating the impacts of some gross errors such as saturation of camera and impulse noise in fringe patterns, and exactly recovering the phase map from fringe patterns.

©2012 Optical Society of America

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References

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  6. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13(10), 3743–3753 (2005).
    [Crossref] [PubMed]
  7. I. Yamaguchi, “Phase-Shifting Digital Holography,” in Digital Holography and Three-Dimensional Display, T-C. Poon, ed. (Springer, 2006), 145–171.
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  14. W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
    [Crossref]
  15. S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
    [Crossref]
  16. J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
    [Crossref]
  17. J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. 16(12), 2525–2533 (2005).
    [Crossref]
  18. N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
    [Crossref]
  19. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007), Chap. 10.
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    [Crossref] [PubMed]
  21. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997).
    [Crossref] [PubMed]
  22. K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36(10), 2084–2093 (1997).
    [Crossref] [PubMed]
  23. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley-Interscience, 1998).
  24. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, 1993), pp. 141–193.
  25. H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. 46(7), 1057–1065 (2007).
    [Crossref] [PubMed]
  26. K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24(21), 3631–3637 (1985).
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  27. H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).
  28. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011).
    [Crossref] [PubMed]
  29. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
    [Crossref]
  30. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
    [Crossref] [PubMed]

2011 (2)

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

2007 (2)

H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[Crossref]

2006 (1)

W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
[Crossref]

2005 (3)

J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. 16(12), 2525–2533 (2005).
[Crossref]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13(10), 3743–3753 (2005).
[Crossref] [PubMed]

2004 (1)

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

2003 (1)

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

2002 (1)

N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
[Crossref]

1997 (2)

1990 (1)

1987 (1)

1985 (2)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1982 (1)

1974 (1)

Booth, N. A.

N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
[Crossref]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Chen, M.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[Crossref]

H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

Chernov, A. A.

N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
[Crossref]

Cywiak, M.

Decraemer, W. F.

Dirckx, J. J. J.

Endo, J.

Estrada, J. C.

Gallagher, J. E.

Gao, P.

Geist, E.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Guo, H.

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011).
[Crossref] [PubMed]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[Crossref]

H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

Halioua, M.

Harder, I.

Hasegawa, S.

He, H.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

Herriott, D. R.

Hibino, K.

Hong, G. S.

W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
[Crossref]

Iwasaki, S.

Lee, J.-R.

J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. 16(12), 2525–2533 (2005).
[Crossref]

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

Lindlein, N.

Liu, H. C.

Mantel, K.

Molimard, J.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

Morgan, C. J.

Mosiño, J. F.

Nercissian, V.

Ohmura, K.

Prakash, S.

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

Quiroga, J. A.

Rana, S.

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

Rosenfeld, D. P.

Sasaki, O.

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

Servin, M.

Singh, S.

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

Soloviev, O.

Srinivasan, V.

Stetson, K. A.

Surrel, Y.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997).
[Crossref] [PubMed]

Tonomura, A.

Vautrin, A.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

Vdovin, G.

Vekilov, P. G.

N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
[Crossref]

Wang, W. H.

W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
[Crossref]

White, A. D.

Wong, Y. S.

W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
[Crossref]

Yang, Q.

Yao, B.

Yatagai, T.

Yu, Y.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

Zhao, Z.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[Crossref]

Appl. Opt. (8)

Composites: Part A (1)

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A 35(7-8), 849–859 (2004).
[Crossref]

J. Cryst. Growth (1)

N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth 237–239, 1818–1824 (2002).
[Crossref]

Meas. Sci. Technol. (1)

J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. 16(12), 2525–2533 (2005).
[Crossref]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (1)

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[Crossref]

Opt. Lett. (3)

Optik (Stuttg.) (1)

S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) 122(18), 1666–1671 (2011).
[Crossref]

Proc. SPIE (1)

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

Wear (1)

W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear 261(2), 164–171 (2006).
[Crossref]

Other (7)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis Group, 2005).

I. Yamaguchi, “Phase-Shifting Digital Holography,” in Digital Holography and Three-Dimensional Display, T-C. Poon, ed. (Springer, 2006), 145–171.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, 1993), pp. 94–140.

H. Schreiber and J. H. Bruning, “Phase Shifting Interferometry,” in Optical Shop Test, D. Malacara, ed. (Wiley-Interscience, 2007), 547–666.

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

D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley-Interscience, 1998).

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, 1993), pp. 141–193.

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Figures (8)

Fig. 1
Fig. 1 Hough transform of a sinusoid under the noise free condition. (a) xy-plane, (b) vu-plane, and (c) the accumulator.

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Fig. 2
Fig. 2 Hough transform of a sinusoid in the presence of noise. (a) xy-plane, (b) vu-plane, (c) the accumulator, and (d) the accumulator smoothed using an 11 × 11 Gaussian spatial filter.

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Fig. 3
Fig. 3 (a) shows, from top to bottom, one of eight phase-shifting fringe patterns with noise, the wrapped phase map (radians) recovered using the least-squares phase-shifting algorithm, and the phase errors (radians). (b) and (c) are parallel to (a), but the fringe patterns have been denoised in image preprocess stages using a 3 × 3 averaging smoothing spatial filter and a 3 × 3 median filter, respectively. (d) is the result of the proposed technique without preprocessing to the fringe patterns.

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Fig. 4
Fig. 4 (a) shows the probabilities of that at least one intensity value in an intensity sequence is contaminated by impulse noise, and (b) shows the probabilities of that more than half number of intensities in this sequence are contaminated by impulse noise, with the noise probabilities for a single pattern being 0.01, 0.05, 0.1, and 0.2. (The length of sequence equals the number of phase shifts.)

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Fig. 5
Fig. 5 (a) is a fringe pattern with 10% impulse noise. (b), (c), and (d) show the recovered phase maps when the numbers of phase shifts are 4, 8, and 16, respectively, where the first row is calculated using the least-squares phase-shifting algorithm, and the second row is obtained using the proposed technique. (e) The unwrapped phase map (radians) of the bottom panel of (d) with the carrier component having been removed.

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Fig. 6
Fig. 6 Simulation results. (a) The top panel is a fringe pattern distorted by camera saturation, and the bottom one plots its cross-section in horizontal direction with the horizontal axis being pixel position. (b) The top panel is the phase map recovered using the least-squares phase-shifting algorithm, and its error distribution is shown in the bottom panel, where the colorbars are in radians. (c) is parallel to (b), but the proposed algorithm is employed.

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Fig. 7
Fig. 7 Experimental results. (a) The top panel is a fringe pattern distorted by camera saturation, and the bottom one plots its cross-section in horizontal direction with the horizontal axis being pixel position. (b) The top panel is the phase map recovered using the least-squares phase-shifting algorithm, and the bottom panel shows the unwrapped phase map with the carrier component having been removed. (c) is parallel to (b) with the proposed algorithm being employed. In (b) and (c), the colorbars are in radians.

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Fig. 8
Fig. 8 Durations of computational time, depending on the noise probabilities, are functions of the number of phase shifts.

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Tables (1)

Tables Icon

Table 1 Phase Errors (radians) of the Least-squares Phase-shifting Algorithm with Different Image Preprocessing Methods and the Proposed Algorithm

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Equations (12)

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

y=Acos(φ+x).
y=ucosx−vsinx,
A= u 2 + v 2
φ=arctan(v/u).
I k (i,j)=a(i,j)+b(i,j)cos[φ(i,j)+ α k ] k=0,1,⋯,K−1 ,
Δ I k,l = I k − I l =bcosφ(cos α k −cos α l )−bsinφ(sin α k −sin α l ) with k>l ,
( K 2 )= K! (K−2)!2! = K(K−1) 2 .
Δ I k,l =u(cos α k −cos α l )−v(sin α k −sin α l ).
I k =a+ucos α k −vsin α k .
E= ∑ k=0 K−1 [ I k −(a+ucos α k −vsin α k )] 2 .
φ=arctan(v/u),
P N = ∑ k=N K ( K k ) p k (1−p) K−k .

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