Abstract

Phase extraction methods based on the principal component analysis (PCA) can extract objective phase from phase-shifted fringes without any prior knowledge about their shift steps. Although it is fast and easy to implement, many fringe images are needed for extracting the phase accurately from noisy fringes. In this paper, a simple extension of the PCA method for reducing extraction error is proposed. It can effectively reduce influence from random noise, while most of the advantages of the PCA method is inherited because it only modifies the construction process of the data matrix from fringes. Although it takes more time because size of the data matrix to be decomposed is larger, computational time of the proposed method is shown to be reasonably fast by using the iterative singular value decomposition algorithm. Numerical experiments confirmed that the proposed method can reduce extraction error even when the number of interferograms is small.

© 2016 Optical Society of America

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References

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  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).
    [Crossref]
  2. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007).
    [Crossref]
  3. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [Crossref] [PubMed]
  4. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
    [Crossref]
  5. D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
    [Crossref]
  6. K. Ishikawa, K. Yatabe, N. Chitanont, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “High-speed imaging of sound using parallel phase-shifting interferometry,” Opt. Express 24(12), 12922–12932 (2016).
    [Crossref] [PubMed]
  7. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
    [Crossref]
  8. C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
    [Crossref]
  9. I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
    [Crossref]
  10. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [Crossref] [PubMed]
  11. A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5)475–490 (2005).
    [Crossref]
  12. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
    [Crossref]
  13. J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
    [Crossref] [PubMed]
  14. H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
    [Crossref] [PubMed]
  15. F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
    [Crossref] [PubMed]
  16. C. Meneses-Fabian and F. A. Lara-Cortes, “Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps,” Opt. Express 23(10), 13589–13604 (2015).
    [Crossref] [PubMed]
  17. Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
    [Crossref]
  18. X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
    [Crossref]
  19. 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]
  20. 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]
  21. 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 19(21), 20483–20492 (2011).
    [Crossref] [PubMed]
  22. J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
    [Crossref]
  23. J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
    [Crossref]
  24. J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
    [Crossref] [PubMed]
  25. K. Yatabe and Y. Oikawa, “Convex optimization based windowed Fourier filtering with multiple windows for wrapped phase denoising,” Appl. Opt. 55(17), 4632–4641 (2016).
    [Crossref] [PubMed]
  26. K. Yatabe, K. Ishikawa, and Y. Oikawa, “Compensation of fringe distortion for phase-shifting three-dimensional shape measurement by inverse map estimation,” Appl. Opt. 55(22), 6017–6024 (2016).
    [Crossref] [PubMed]

2016 (5)

2015 (3)

2014 (1)

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

2013 (4)

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

2011 (3)

2007 (1)

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
[Crossref]

2005 (1)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5)475–490 (2005).
[Crossref]

2004 (2)

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

1997 (1)

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
[Crossref]

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Belenguer, T.

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007).
[Crossref]

Carazo, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chai, L.

Chen, M.

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
[Crossref]

Chen, W.

Chitanont, N.

Deng, J.

Estrada, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Fan, J.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Guo, H.

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
[Crossref]

Han, B.

Han, H.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Hao, J.

Ikeda, Y.

Ishikawa, K.

Ji, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Jin, W.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

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 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Lara-Cortes, F. A.

Li, C.

Liu, F.

Lu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Lv, X.

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

Martínez-García, A.

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

Meneses-Fabian, C.

Niwa, H.

Oikawa, Y.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
[Crossref]

Onuma, T.

Otani, Y.

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

Patil, A.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5)475–490 (2005).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rastogi, P.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5)475–490 (2005).
[Crossref]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
[Crossref]

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007).
[Crossref]

Serrano-García, D. I.

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

Shou, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Sorzano, C.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Tian, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Toto-Arellano, N.-I.

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
[Crossref]

Vargas, J.

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

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]

Wang, H.

Wang, K.

Wang, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Wang, Z.

Wu, D.

Wu, F.

Wu, Y.

Xu, J.

Xu, Q.

Xu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Xu, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Yamaguchi, I.

Yatabe, K.

Yoshii, M.

Yu, Y.

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
[Crossref]

Zhang, D.

Zhang, T.

Zhang, Z.

Zheng, D.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Zhong, L.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

J. Opt. Soc. Am. A. (1)

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase shifting interferometry,” J. Opt. Soc. Am. A. 24, (1)25–33 (2007).
[Crossref]

Meas. Sci. Technol. (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Opt. Commun. (3)

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3), 118–124 (1991).
[Crossref]

Opt. Eng. (2)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

D. I. Serrano-García, A. Martínez-García, N.-I. Toto-Arellano, and Y. Otani, “Dynamic temperature field measurements using a polarization phase-shifting technique,” Opt. Eng. 53(11), 112202 (2014).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (3)

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5)475–490 (2005).
[Crossref]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Opt. Lett. (5)

Other (2)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 A schematic diagram of adjacent pixels defined in Eqs. (18)(22), where pixels included in d ˜ i , j are colored by green.
Fig. 2
Fig. 2 An example of extracted phase and extraction error for fringes obtained from equispaced shifting steps. The first three figures in the first row represent one of the three fringe images and phase maps of the true phase ϕ and extracted phase ϕ ^ obtained by the 3-step phase-shifting algorithm. Error maps of extracted phase obtained by the 3-step phase-shifting algorithm and the PCA method are also illustrated in the first row. The second row shows error maps of extracted phase obtained by the proposed method, where each result corresponds to each case defined in Eqs. (18)(22).
Fig. 3
Fig. 3 RMSE of each method with different noise levels. Each line corresponds to different σ ∈ {0.1, 0.2, 0.3, 0.4, 0.5} as in the legend. The horizontal axis represents the number of fringe images N used for the phase extraction. The vertical axis denotes RMSE, where every axis is illustrated by the same scale.
Fig. 4
Fig. 4 An example of extracted phase and extraction error for randomly shifted fringes. The first three figures in the first row represent one of the three fringe images and phase maps of the true phase ϕ and extracted phase ϕ ^ obtained by the self-calibrating method (AIA). Error maps of extracted phase obtained by AIA and the PCA method are also illustrated in the first row. The second row shows error maps of extracted phase obtained by the proposed method, where each result corresponds to each case defined in Eqs. (18)(22).
Fig. 5
Fig. 5 Mean of RMSE calculated from 100 random trials, where some outliers, which were greater than a standard deviation from the average values, were excluded from the calculation. Each line corresponds to different σ ∈ {0.1, 0.2, 0.3, 0.4, 0.5} as in the legend. The horizontal axis represents the number of fringe images N used for the phase extraction. The vertical axis denotes the mean RMSE, where every axis is illustrated by the same scale.

Tables (1)

Tables Icon

Table 1 Computational time of the phase extraction from five fringe images (N = 5) whose size was 300×300, where constructions of the extended data matrices were included. M = 1 corresponds to the ordinary PCA method, while M > 1 are the proposed method.

Equations (25)

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

I [ n ] ( x ) = B ( x ) + A ( x ) cos ( ϕ ( x ) + δ [ n ] ) ,
d i , j = [ I i , j [ 1 ] , I i , j [ 2 ] , , I i , j [ N ] ] T ,
D = [ d 1 , 1 , d 2 , 1 , , d N v , 1 , d 1 , 2 , d 2 , 2 , d N v , N h ] ,
I i , j [ n ] = B i , j + A i , j [ cos ( ϕ i , j ) cos ( δ [ n ] ) sin ( ϕ i , j ) sin ( δ [ n ] ) ] ,
d i , j = B i , j o + C i , j c + S i , j s ,
c = [ cos ( δ [ 1 ] ) , cos ( δ [ 2 ] ) , , cos ( δ [ N ] ) ] T ,
s = [ sin ( δ [ 1 ] ) , sin ( δ [ 2 ] ) , , sin ( δ [ N ] ) ] T .
D = U V T ,
D = n = 1 N σ n u n v n T ,
ϕ i , j = Arg [ C i , j ι S i , j ] = Arg [ A i , j { cos ( ϕ i , j ) + ι sin ( ϕ i , j ) } ] ,
D D T = U V T V U T = U 2 U T ,
I [ n ] = B + A cos ( ϕ + δ [ n ] ) A ϕ sin ( ϕ + δ [ n ] ) = B + A [ cos ( ϕ ) cos ( δ [ n ] ) sin ( ϕ ) sin ( δ [ n ] ) ] A ϕ [ sin ( ϕ ) cos ( δ [ n ] ) + cos ( ϕ ) sin ( δ [ n ] ) ] ,
d i , j = [ I i , j [ 1 ] , I i , j [ 2 ] , , I i , j [ N ] ] T ,
d i , j = B i , j o + C i , j c + S i , j s ,
d ˜ i , j = [ d i , j T , d i + 1 , j T ] T ,
D ˜ = [ d ˜ 1 , 1 , d ˜ 2 , 1 , , d ˜ N v , 1 , d ˜ 1 , 2 , d ˜ 2 , 2 , d ˜ N v , N h ] ,
d ˜ i , j = [ d i , j d i + 1 , j ] = [ d i , j d i , j + d ^ i , j ] [ d i , j d i , j + d i , j ] = [ d i , j d i , j ] + [ 0 d i , j ] ,
d ˜ i , j = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T ] T ,
d ˜ i , j = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T ] T ,
d ˜ i , j = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T , d i + 2 , j T , d i 2 , j T , d i , j + 2 T , d i , j 2 T ] T ,
d ˜ i , j = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T , d i + 2 , j T , d i 2 , j T , d i , j + 2 T , d i , j 2 T , d i + 2 , j + 1 T , d i + 2 , j 1 T , d i 2 , j + 1 T , d i 2 , j 1 T , d i + 1 , j + 2 T , d i 1 , j + 2 T , d i + 1 , j 2 T , d i 1 , j 2 T ] T ,
d ˜ i , j = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T , d i + 2 , j T , d i 2 , j T , d i , j + 2 T , d i , j 2 T , d i + 2 , j + 1 T , d i + 2 , j 1 T , d i 2 , j + 1 T , d i 2 , j 1 T , d i + 1 , j + 2 T , d i 1 , j + 2 T , d i + 1 , j 2 T , d i 2 , j 2 T , d i + 2 , j + 2 T , d i + 2 , j 2 T , d i 2 , j + 2 T , d i 2 , j 2 T ] T ,
I [ n ] ( x ) = B ( x ) + A ( x ) cos ( ϕ ( x ) + 2 π n / N ) + σ ε ( x ) ,
ϕ ^ = Arg [ ( n I [ n ] cos ( 2 π n / N ) ) ι ( n I [ n ] sin ( 2 π n / N ) ) ] .
I [ n ] ( x ) = B ( x ) + A ( x ) cos ( ϕ ( x ) + δ [ n ] + σ ε ( x ) ,

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