## Spatial carrier phase-shifting algorithm based on principal component analysis method |

Optics Express, Vol. 20, Issue 15, pp. 16471-16479 (2012)

http://dx.doi.org/10.1364/OE.20.016471

Acrobat PDF (1109 KB)

### Abstract

A non-iterative spatial phase-shifting algorithm based on principal component analysis (PCA) is proposed to directly extract the phase from only a single spatial carrier interferogram. Firstly, we compose a set of phase-shifted fringe patterns from the original spatial carrier interferogram shifting by one pixel their starting position. Secondly, two uncorrelated quadrature signals that correspond to the first and second principal components are extracted from the phase-shifted interferograms by the PCA algorithm. Then, the modulating phase is calculated from the arctangent function of the two quadrature signals. Meanwhile, the main factors that may influence the performance of the proposed method are analyzed and discussed, such as the level of random noise, the carrier-frequency values and the angle of carrier-frequency of fringe pattern. Numerical simulations and experiments are given to demonstrate the performance of the proposed method and the results show that the proposed method is fast, effectively and accurate. The proposed method can be used to on-line detection fields of dynamic or moving objects.

© 2012 OSA

## 1. Introduction

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**(1), 156–160 (1982). [CrossRef]

9. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. **22**(24), 3977–3982 (1983). [CrossRef] [PubMed]

10. M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE **1508**, 61–67 (1991). [CrossRef]

17. J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. **47**(29), 5446–5453 (2008). [PubMed]

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**(1), 156–160 (1982). [CrossRef]

3. W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. **22**(23), 3898–3901 (1983). [CrossRef] [PubMed]

6. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. **26**(9), 1668–1673 (1987). [CrossRef] [PubMed]

18. D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. **30**(5), 492–494 (2005). [CrossRef] [PubMed]

19. D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. **46**(34), 8305–8314 (2007). [CrossRef] [PubMed]

9. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. **22**(24), 3977–3982 (1983). [CrossRef] [PubMed]

5. D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. **25**(10), 1653–1660 (1986). [CrossRef] [PubMed]

7. J. B. Liu and P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt. **36**(25), 6231–6241 (1997). [CrossRef] [PubMed]

8. J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. **40**(13), 2081–2088 (2001). [CrossRef] [PubMed]

20. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. **43**(13), 2695–2702 (2004). [CrossRef] [PubMed]

22. W. Gao, N. T. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express **17**(25), 23147–23152 (2009). [CrossRef] [PubMed]

23. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. **24**(13), 905–907 (1999). [CrossRef] [PubMed]

24. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. **30**(19), 2560–2562 (2005). [CrossRef] [PubMed]

10. M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE **1508**, 61–67 (1991). [CrossRef]

12. M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. **42**(9), 1853–1862 (1995). [CrossRef]

11. P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. **23**(5), 343–354 (1995). [CrossRef]

14. S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. **36**(23), 4677–4679 (2011). [CrossRef] [PubMed]

15. 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]

16. A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity contrast determination,” Appl. Opt. **46**(21), 4613–4624 (2007). [CrossRef] [PubMed]

17. J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. **47**(29), 5446–5453 (2008). [PubMed]

## 2. Theory Analysis

### 2.1. Constructing phase-shifting interferograms

*x*and

*y*are integers denoting the pixel position in the fringe pattern;

*A*(

*x*,

*y*),

*B*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) are the background illumination, the modulation signal and the modulating phase at pixel (

*x*,

*y*). Additionally, κ

_{x}and κ

_{y}are the spatial carrier-frequency along

*x*and

*y*axes respectively. Typically,

*A*,

*B*, and

*ϕ*maps are assumed to be smooth signals, so we can assume that

*A*,

*B*, and

*ϕ*have approximately the same quantity for four adjacent pixels on the carrier-frequency pattern (see Fig. 1(a) ). Thus, from four consecutive pixels on the original spatial carrier interferogram, we can construct four phase-shifted interferograms,

*I*

_{1}=

*I*(

*x*,

*y*),

*I*

_{2}=

*I*(

*x*+ 1,

*y*),

*I*

_{3}=

*I*(

*x*,

*y*+ 1) and

*I*

_{4}=

*I*(

*x*+ 1,

*y*+ 1) that are given by where

*δ*

_{1},

*δ*

_{2}and

*δ*

_{3}are the relatively phase-shift at (

*x*,

*y*) pixel between the constructed randomly phase-shifted fringe pattern

*I*

_{1},

*I*

_{2},

*I*

_{3}and

*I*

_{4}. Similarly, referring to Fig. 1(b), we also can construct nine randomly phase-shifting interferogram

*I*

_{1}~

*I*

_{n}(

*n*= 9) from nine consecutive neighboring pixels of the original carrier-frequency interferogram

*I*(

*x*,

*y*). For the convenient of the subsequent analysis, the former constructed method is denoted as mode a while the latter is denoted as mode b.

### 2.2. Determination phase with principal component analysis method

## 3. Numerical Simulation

*A*(

*x*,

*y*) = 50,

*B*(

*x*,

*y*) = 50exp[-(

*x*

^{2}+

*y*

^{2})/8.0], and phase

*ϕ*(

*x*,

*y*) = 2πPeak (

*x*/8,

*y*/8) + 2π(κ

_{x}

*x*+ κ

_{y}

*y*); where (

*x*,

*y*)∈[-2,2] and they are discretized into an array of 400 × 400, Peak is a multi-peak built-in function of Matlab shown in Fig. 2(a) , and κ

_{x}and κ

_{y}are the carrier-frequencies along x and y direction, respectively. Moreover, additive Gaussian white noise which goes from 0% to 10% of Signal-to-Noise-Ratio (SNR) is added to the different spatial carrier interferogram. Then the intensity of each pattern is assumed to be captured by a CCD with 8-bit quantization. The image size is 400 × 400. In all following cases, the patterns are processed with a dual-core 2.2 GHz desktop and using Matlab.

_{x}and κ

_{y}equals to 10 (px

^{−1}) and we add additive Gaussian white noise (2%) to the fringe pattern. In Fig. 2 we show the resultant interferogram (right) and the phase distribution (left). Then we use the proposed SPS-PCA and LSI-SPS methods to demodulate the interferogram. Figure 3(a) and 3(c) shows the extracted phase using the proposed SPS-PCA method with mode a and mode b, respectively. The obtained phase by Xu’s LSI-SPS method is shown in Fig. 3(e). Here, we also show the corresponding residual phase error distributions in Fig. 3(b), 3(d), and 3(f). The obtained root mean square error (RMS) between the theoretical and reconstructed phases by the proposed SPS-PCA method are 0.114 (rad) with mode a, and 0.052 (rad) with mode b. The obtained RMS error by the Xu’s LSI-SPS is 0.098 (rad). Additionally, the processing times are 0.121 s (mode a), 0.242s (mode b) and 26.97 s for the proposed SPS-PCA method and Xu’s LSI-SPS method, respectively. As can be seen from the results shown above, the proposed method with mode a has accuracy similar to the Xu’s LSI-SPS algorithm. On the other hand, the obtained accuracy is higher than Xu’s method when we use mode b. In both mode a and b, the preprocessing times are approximately two orders of magnitude faster than LSI-SPS method.

_{x}= κ

_{y}= 10 (px

^{−1}) and we add different levels of random noise. We show the obtained results in Table 1 . As can be seen from the Table 1, the proposed method has accuracy similar to the Xu’s LSI-SPS algorithm and is approximately two orders of magnitude faster which maintain the same results to above analysis.

## 4. Experiment and results

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. Malacara, M. Servín, and Z. Malacara, |

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. |

3. | W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. |

4. | K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. |

5. | D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. |

6. | C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. |

7. | J. B. Liu and P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt. |

8. | J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. |

9. | M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. |

10. | M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE |

11. | P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. |

12. | M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. |

13. | J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. |

14. | S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett. |

15. | H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. |

16. | A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity contrast determination,” Appl. Opt. |

17. | J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. |

18. | D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. |

19. | D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. |

20. | Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. |

21. | Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. |

22. | W. Gao, N. T. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express |

23. | L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. |

24. | J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. |

25. | R. C. Gonzalez and R. E. Woods, |

26. | J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. |

27. | J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. |

28. | |

29. | J. Xu, L. Sun, Y. Li, and Y. Li, “Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts,” Opt. Express |

30. | J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express |

31. | |

32. | |

33. | http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem. |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 11, 2012

Revised Manuscript: June 20, 2012

Manuscript Accepted: June 25, 2012

Published: July 5, 2012

**Citation**

Yongzhao Du, Guoying Feng, Hongru Li, J. Vargas, and Shouhuan Zhou, "Spatial carrier phase-shifting algorithm based on principal component analysis method," Opt. Express **20**, 16471-16479 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16471

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### References

- D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
- 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(1), 156–160 (1982). [CrossRef]
- W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt.22(23), 3898–3901 (1983). [CrossRef] [PubMed]
- K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt.24(18), 3101–3105 (1985). [CrossRef] [PubMed]
- D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt.25(10), 1653–1660 (1986). [CrossRef] [PubMed]
- C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt.26(9), 1668–1673 (1987). [CrossRef] [PubMed]
- J. B. Liu and P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt.36(25), 6231–6241 (1997). [CrossRef] [PubMed]
- J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt.40(13), 2081–2088 (2001). [CrossRef] [PubMed]
- M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt.22(24), 3977–3982 (1983). [CrossRef] [PubMed]
- M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE1508, 61–67 (1991). [CrossRef]
- P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng.23(5), 343–354 (1995). [CrossRef]
- M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt.42(9), 1853–1862 (1995). [CrossRef]
- J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun.271(1), 59–64 (2007). [CrossRef]
- S. K. Debnath and Y. Park, “Real-time quantitative phase imaging with a spatial phase-shifting algorithm,” Opt. Lett.36(23), 4677–4679 (2011). [CrossRef] [PubMed]
- 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]
- A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity contrast determination,” Appl. Opt.46(21), 4613–4624 (2007). [CrossRef] [PubMed]
- J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt.47(29), 5446–5453 (2008). [PubMed]
- D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett.30(5), 492–494 (2005). [CrossRef] [PubMed]
- D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt.46(34), 8305–8314 (2007). [CrossRef] [PubMed]
- Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt.43(13), 2695–2702 (2004). [CrossRef] [PubMed]
- Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt.47(29), 5408–5419 (2008). [CrossRef] [PubMed]
- W. Gao, N. T. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express17(25), 23147–23152 (2009). [CrossRef] [PubMed]
- L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett.24(13), 905–907 (1999). [CrossRef] [PubMed]
- J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett.30(19), 2560–2562 (2005). [CrossRef] [PubMed]
- R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).
- J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett.36(8), 1326–1328 (2011). [CrossRef] [PubMed]
- J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett.36(12), 2215–2217 (2011). [CrossRef] [PubMed]
- http://en.wikipedia.org/wiki/Principal_component_analysis
- J. Xu, L. Sun, Y. Li, and Y. Li, “Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts,” Opt. Express19(15), 14464–14472 (2011). [CrossRef] [PubMed]
- J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express19(21), 20483–20492 (2011). [CrossRef] [PubMed]
- http://en.wikipedia.org/wiki/Principal_component_analysis .
- http://en.wikipedia.org/wiki/Singular_value_decomposition .
- http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem .

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