Abstract

Principal component analysis phase shifting (PCA) is a useful tool for fringe pattern demodulation in phase shifting interferometry. The PCA has no restrictions on background intensity or fringe modulation, and it is a self-calibrating phase sampling algorithm (PSA). Moreover, the technique is well suited for analyzing arbitrary sets of phase-shifted interferograms due to its low computational cost. In this work, we have adapted the standard phase shifting algorithm based on the PCA to the particular case of photoelastic fringe patterns. Compared with conventional PSAs used in photoelasticity, the PCA method does not need calibrated phase steps and, given that it can deal with an arbitrary number of images, it presents good noise rejection properties, even for complicated cases such as low order isochromatic photoelastic patterns.

© 2016 Optical Society of America

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References

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  1. K. Ramesh, Digital Photoelasticity: Advanced Techniques and Applications, Volume 1, (Springer-Verlag, 2000). Available at: https://books.google.com/books/about/Digital_Photoelasticity.html?id=f8hRAAAAMAAJ&pgis=1 [Accessed August 12, 2015].
    [Crossref]
  2. M. Servín, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications, (Wiley, 2014).
  3. K. Ramesh, T. Kasimayan, and B. Neethi Simon, “Digital photoelasticiy - a comprehensive review,” J. Strain Analysis 46(4), 245–266 (2011).
    [Crossref]
  4. J. A. Quiroga and A. González-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36(32), 8397–8402 (1997).
    [Crossref] [PubMed]
  5. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [Crossref] [PubMed]
  6. J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009).
    [Crossref] [PubMed]
  7. J. C. Estrada, M. Servin, and J. A. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18(3), 2632–2638 (2010).
    [Crossref] [PubMed]
  8. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
    [Crossref]
  9. 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]
  10. 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]
  11. J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
    [Crossref]
  12. 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]
  13. J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 355–364 (2014).
    [Crossref]
  14. J. A. Quiroga and A. González-Cano, “Separation of isoclinics and isochromatics from photoelastic data with a regularized phase-tracking technique,” Appl. Opt. 39(17), 2931–2940 (2000).
    [Crossref] [PubMed]
  15. J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
    [Crossref]
  16. A. Asundi, L. Tong, and C. G. Boay, “Dynamic phase-shifting photoelasticity,” Appl. Opt. 40(22), 3654–3658 (2001).
    [Crossref] [PubMed]
  17. J. A. Quiroga and J. A. Gomez-Pedrero, (2016): “Photoleastic fringe pattern demodulation through PCA”, figshare (2016) [Retreived February 18, 2016], https://dx.doi.org/10.6084/m9.figshare.2353396.v1

2014 (1)

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

2013 (1)

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

2011 (4)

2010 (1)

2009 (2)

2008 (1)

J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
[Crossref]

2001 (1)

2000 (1)

1997 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Asundi, A.

Belenguer, T.

Boay, C. G.

Carazo, J. M.

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

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Chai, L.

Estrada, J. C.

González-Cano, A.

Jin, W.

Kasimayan, T.

K. Ramesh, T. Kasimayan, and B. Neethi Simon, “Digital photoelasticiy - a comprehensive review,” J. Strain Analysis 46(4), 245–266 (2011).
[Crossref]

Neethi Simon, B.

K. Ramesh, T. Kasimayan, and B. Neethi Simon, “Digital photoelasticiy - a comprehensive review,” J. Strain Analysis 46(4), 245–266 (2011).
[Crossref]

Pascual, E.

J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
[Crossref]

Quiroga, J. A.

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]

J. C. Estrada, M. Servin, and J. A. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18(3), 2632–2638 (2010).
[Crossref] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[Crossref] [PubMed]

J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009).
[Crossref] [PubMed]

J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
[Crossref]

J. A. Quiroga and A. González-Cano, “Separation of isoclinics and isochromatics from photoelastic data with a regularized phase-tracking technique,” Appl. Opt. 39(17), 2931–2940 (2000).
[Crossref] [PubMed]

J. A. Quiroga and A. González-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36(32), 8397–8402 (1997).
[Crossref] [PubMed]

Ramesh, K.

K. Ramesh, T. Kasimayan, and B. Neethi Simon, “Digital photoelasticiy - a comprehensive review,” J. Strain Analysis 46(4), 245–266 (2011).
[Crossref]

Servin, M.

Sorzano, C. O. S.

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

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Tong, L.

Vargas, J.

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

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. 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]

Villa-Hernández, J.

J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
[Crossref]

Xu, J.

Xu, Q.

Appl. Opt. (3)

Appl. Phys. B (1)

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

J. Strain Analysis (1)

K. Ramesh, T. Kasimayan, and B. Neethi Simon, “Digital photoelasticiy - a comprehensive review,” J. Strain Analysis 46(4), 245–266 (2011).
[Crossref]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Opt. Commun. (1)

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Proc. SPIE (1)

J. A. Quiroga, E. Pascual, and J. Villa-Hernández, “Robust isoclinic calculation for automatic analysis of photoelastic fringe patterns,” Proc. SPIE 7155, 715530 (2008).
[Crossref]

Other (3)

J. A. Quiroga and J. A. Gomez-Pedrero, (2016): “Photoleastic fringe pattern demodulation through PCA”, figshare (2016) [Retreived February 18, 2016], https://dx.doi.org/10.6084/m9.figshare.2353396.v1

K. Ramesh, Digital Photoelasticity: Advanced Techniques and Applications, Volume 1, (Springer-Verlag, 2000). Available at: https://books.google.com/books/about/Digital_Photoelasticity.html?id=f8hRAAAAMAAJ&pgis=1 [Accessed August 12, 2015].
[Crossref]

M. Servín, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications, (Wiley, 2014).

Supplementary Material (1)

NameDescription
» Code 1       Code for processing photoelastic fringe patterns through the PCA algorithm

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Figures (6)

Fig. 1
Fig. 1 Diagram of a polariscope in the linear bright field (LBF) configuration with arbitrary orientation of the polarizers. The polariscope is formed by two linear polarizers P(β) whose transmission axis (labeled with T) forms an angle β with the X axis, while the sample, located between the two polarizers, is equivalent to a retarder plate R(θ,δ) with spatially variable retardation δ and fast axis (labeled with F in the figure) direction θ.
Fig. 2
Fig. 2 Diagram of a polariscope in the circular configuration. The polariscope is formed by a circular polarizer (CP) plus a quarter wave plate Q(φ) whose fast axis F forms an angle φ with the X axis and a linear polarizer P(ψ) whose transmission axis T forms an angle ψ with the horizontal.
Fig. 3
Fig. 3 Wrapped isoclinics phase map W[4θ] obtained for a diametrically compressed disc after applying (a) the proposed phase shifting algorithm (abbreviated as PCA), and (b) a standard 4-step phase shifting algorithm (abbreviated as PSA4).
Fig. 4
Fig. 4 (a) Wrapped isochromatics phase map W[δ] obtained with the proposed algorithm (abbreviated as PCA) for a diametrically compressed disk and (b) the same isochromatics phase map obtained after applying an standard eight step phase shifting algorithm (abbreviated as PSA8). In this latter case, a broken fringe can be observed at the lower left part of the disk indicating an error in the recovery of the phase W[δ].
Fig. 5
Fig. 5 (a) Photoelastic fringe pattern with intermingled isochromatics and isoclinics with the polariscope in the LBF configuration illuminated with polychromatic light and (b) fringe pattern obtained for the same object after changing the configuration of the polariscope from linear to a circular one and switching from color to monochromatic illumination.
Fig. 6
Fig. 6 (a) Wrapped isoclinics phase map W[4θ] for an ophthalmic lens under stress obtained after the application of the proposed PCA technique and (b) the same map obtained after applying a conventional four step PS algorithm, (c) wrapped isochromatics phase map W[δ] obtained for the same sample using the PCA method and (d) wrapped isochromatics phase resulting after applying an standard eight step phase shifting algorithm. In order to highlight the ability of the algorithm proposed for recovering correctly the phase map we have represented in (e) and (f) the sinus of the retardation angle δ for the PCA and the PSA8, respectively, note the better recovery of retardation angle δ for the case of PCA as the sin(δ) map is more similar to the fringe map shown in Fig. 5(b).

Equations (22)

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I k ( r )=a( r )+b( r )cos( φ( r )+ α k ),
I k ( r )=a+cos( α k ) g 1 ( r )+sin( α k ) g 2 ( r ),
X= [ I 1 ( n ) I 2 ( n ) I K ( n ) ] T ,
C X =( X m X ) ( X m X ) T ,
Y=A·( X m X ),
φ= tan 1 ( g 2 g 1 ).
δ= 2π λ C( σ 1 σ 2 )d,
I( r )=1 sin 2 ( δ( r ) 2 ) sin 2 ( 2θ( r ) )=1 1 4 [ ( 1cosδ( r ) )( 1cos( 4θ( r ) ) ) ].
I k ( r )=a( r )+b( r )cos ( 4( θ β k ) ) k=1,2,...,K ,
a( r )=1 1 2 sin 2 ( δ( r ) 2 ), b( r )= 1 2 sin 2 ( δ( r ) 2 ).
I ' k = I k I k1 =b ( cos( 4( θ β k ) )cos( 4( θ β k.1 ) ) ) k=2,3,...,K .
W[ 4θ ]= tan 1 ( g 2 g 1 ),
I=1sin2( ψϕ )cosδsin2( ϕθ )cos2( ψϕ )sinδ,
I k =1+sin( 2θ )sinδcos( 2 ψ k )cosδsin( 2 ψ k ) k=1,2,...,K .
f 1 =cosδ and f 2 =sin2θsinδ.
I k =1+cosδcos( 2 ψ k )sin( 2θ )sinδsin( 2 ψ k ) k=1,2,...,K .
h 1 =cosδ and h 2 =cos2θsinδ.
z= 1 2 ( f 1 + h 1 )+i( f 2 sin2θ+ h 2 cos2θ ),
W[δ]=arctan( Im( z ) Re( z ) ).
z 1,2 = 1 2 ( f 1 ± h 1 )+i·( f 2 sin( 2θ )± h 2 cos( 2θ ) ), z 3,4 = 1 2 ( f 1 ± h 1 )i·( f 2 sin( 2θ )± h 2 cos( 2θ ) ), z 5,6 = 1 2 ( f 1 ± h 1 )+i·( f 2 cos( 2θ )± h 2 sin( 2θ ) ), z 7,8 = 1 2 ( f 1 ± h 1 )i·( f 2 cos( 2θ )± h 2 sin( 2θ ) ).
m= r z( r )·z*( r ) .
W[δ]=arctan( Im( z max ) Re( z max ) ).

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