Abstract

Focus on the phase reconstruction from three phase-shifting interferograms with unknown phase shifts, an advanced principal component analysis method is proposed. First, use a simple subtraction operation among interferograms, two intensity difference images are obtained easily. Second, set the center region of the data of intensity difference images to zero, and then construct a covariance matrix to obtain a transformation matrix. Third, two principal components of interferograms can be determined by the Hotelling transform and then phase can be calculated from the two normalized principal components by an arctangent function. By means of the simulation calculation and the experimental research, it is proved that the phase with high precision can be obtained rapidly by the proposed algorithm.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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  20. H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2015 (1)

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

2014 (4)

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

J. Vargas, J. Carazo, and C. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

C. S. Guo, B. Sha, Y. Y. Xie, and X. J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (2014).
[Crossref] [PubMed]

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]

2012 (2)

2011 (5)

2009 (1)

2004 (1)

2003 (1)

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

1992 (1)

T. M. Kreis and W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

1974 (1)

Antonio Quiroga, J.

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Belenguer, T.

Brangaccio, D. J.

Bruning, J. H.

Carazo, J.

J. Vargas, J. Carazo, and C. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
[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]

Carazo, J. M.

Chai, L.

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]

Estrada, J. C.

Fan, J.

Gallagher, J. E.

Guo, C. S.

Guo, H.

Han, B.

Hao, Q.

Herriott, D. R.

Hu, Y.

Jin, W.

Juptner, W. P. O.

T. M. Kreis and W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Kreis, T. M.

T. M. Kreis and W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Li, Y.

Lu, X.

Luo, C.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

Ma, S.

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Quiroga, J. A.

Rosenfeld, D. P.

Servin, M.

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Sha, B.

Sorzano, C.

J. Vargas, J. Carazo, and C. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
[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 and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Sorzano, C. O.

Sun, L.

Vargas, J.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

J. Vargas, J. Carazo, and C. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
[Crossref]

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, C. O. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the gram-schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[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]

J. Vargas, J. A. Quiroga, C. O. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36(17), 3485–3487 (2011).
[Crossref] [PubMed]

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]

Wang, H.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

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]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

Wang, Z.

White, A. D.

Xie, Y. Y.

Xu, J.

Xu, Q.

Zhang, D.

Zhang, F.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

Zhang, W.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Zhang, X. J.

Zhang, Z.

Zhong, L.

Zhu, Q.

Appl. Opt. (2)

Appl. Phys. B (1)

J. Vargas, J. Carazo, and C. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
[Crossref]

Opt. Commun. (4)

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]

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (1)

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

Opt. Lett. (9)

C. S. Guo, B. Sha, Y. Y. Xie, and X. J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (2014).
[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]

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, C. O. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36(17), 3485–3487 (2011).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, C. O. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the gram-schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[Crossref] [PubMed]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

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]

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]

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[Crossref] [PubMed]

Proc. SPIE (1)

T. M. Kreis and W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (the Chemical Rubber Company, 2005).

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Figures (9)

Fig. 1
Fig. 1 Three simulated phase-shifting fringe patterns with spherical wavefront and low noise (SNR: 3%).
Fig. 2
Fig. 2 (a) one of two intensity difference images; (b) the intensity difference image whose center region is set to zero.
Fig. 3
Fig. 3 Reconstructed wrapped phase maps with different algorithms. (a) theoretical phase (REF), (b) APCA, (c) PCA, and (d) GS3.
Fig. 4
Fig. 4 Three simulated phase-shifting fringe patterns with high noise (SNR: 15%).
Fig. 5
Fig. 5 Two simulated fringe patterns of different phase distribution. (a) plane wavefront; (b) complex wavefront.
Fig. 6
Fig. 6 Three experimental phase-shifting interferograms.
Fig. 7
Fig. 7 (a) one of two intensity difference images; (b) the intensity difference image whose center region is set to zero.
Fig. 8
Fig. 8 Reconstructed wrapped phase maps with different algorithms. (a) Four-step (REF), (b) APCA, (c) PCA, (d) GS3.
Fig. 9
Fig. 9 (a) one of interferograms with phase distribution of plane wavefront; (b)-(e) are the reconstructed wrapped phase maps with different algorithms. (b) Four-step (REF), (c) APCA, (d) PCA, (e) GS3.

Tables (2)

Tables Icon

Table 1 RMS Errors and the Processing Time of Phase (Spherical Wavefront) Extraction with Different Algorithms (Simulation)

Tables Icon

Table 2 RMS Errors and the Processing Time of Phase (Spherical Wavefront) Extraction with Different Algorithms (Experiment)

Equations (16)

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I n,k = a k + b k cos( φ k + θ n )
D 1,k = I 1,k I 2,k = b k cos( φ k ) b k cos( φ k + θ 2 ) =2 b k sin( θ 2 /2 )sin( φ k + θ 2 /2 )=2 b k sin( θ 2 /2 )sin( Φ k )
D 2,k = I 1,k I 3,k = b k cos( φ k ) b k cos( φ k + θ 3 ) =2 b k sin( θ 3 /2 )sin( φ k + θ 3 /2 )=2 b k sin( θ 3 /2 )sin( Φ k +Δ )
Q=[ D 1,1 D 1,2 D 1,K D 2,1 D 2,2 D 2,K ]
C=Q Q T =[ k=1 K D 1,k 2 k=1 K D 1,k D 2,k k=1 K D 1,k D 2,k k=1 K D 2,k 2 ]
B=UC U T
U=[ 1 0 C 1,2 / C 1,1 1 ]
C 1,2 / C 1,1 sin( θ 3 /2 )cos( Δ ) / sin( θ 2 /2 )
k=1 K b k 2 sin( Φ k )cos( Φ k ) k=1 K b k 2 sin 2 ( Φ k )
y=UQ=[ 1 0 C 1,2 / C 1,1 1 ][ D 1,k D 2,k ]=[ D 1,k D 2,k C 1,2 / C 1,1 D 1,k ]
y 1,k =2sin( θ 2 /2 ) b k sin( Φ k )
y 2,k =2sin( θ 3 /2 )sin( Δ ) b k cos( Φ k )
y 1,k * = y 1,k / y 1 = b k sin( Φ k ) k=1 K b k 2 sin 2 ( Φ k )
y 2,k * = y 2,k / y 2 = b k cos( Φ k ) k=1 K b k 2 cos 2 ( Φ k )
k=1 K b k 2 sin 2 ( Φ k ) k=1 K b k 2 cos 2 ( Φ k )
Φ k =arctan( y 1,k * / y 2,k * )

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