Abstract

The regularized phase tracker (RPT) is one of the most powerful approaches for demodulation of a single fringe pattern. However, two disadvantages limit the applications of the RPT in practice. One is the necessity of a normalized fringe pattern as input and the other is the sensitivity to critical points. To overcome these two disadvantages, a generalized regularized phase tracker (GRPT) is presented. The GRPT is characterized by two novel improvements. First, a general local fringe model that includes a linear background, a linear modulation and a quadratic phase is adopted in the proposed enhanced cost function. Second, the number of iterations in the optimization process is proposed as a comprehensive measure of fringe quality and used to guide the demodulation path. With these two improvements, the GRPT can directly demodulate a single fringe pattern without any pre-processing and post-processing and successfully get rid of the problem of the sensitivity to critical points. Simulation and experimental results are presented to demonstrate the effectiveness and robustness of the GRPT.

© 2012 OSA

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References

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    [CrossRef] [PubMed]
  4. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), Chap. 4, pp. 94–140.
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    [CrossRef]
  8. J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. E. Robin and V. Valle, “Phase Demodulation from a Single Fringe Pattern Based on a Correlation Technique,” Appl. Opt. 43(22), 4355–4361 (2004).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  16. O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. 41(26), 5519–5526 (2002).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  22. C. Tian, Y. Y. Yang, D. Liu, Y. J. Luo, and Y. M. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
    [CrossRef] [PubMed]
  23. H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
    [CrossRef]
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    [CrossRef]
  26. J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
    [CrossRef]
  27. J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30(22), 3018–3020 (2005).
    [CrossRef] [PubMed]
  28. N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
    [CrossRef]
  29. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
    [CrossRef] [PubMed]
  30. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
    [CrossRef] [PubMed]
  31. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
    [CrossRef]
  32. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999).
  33. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, Inc, 1998).
  34. M. Zhao, L. Huang, Q. Zhang, X. Y. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
    [CrossRef] [PubMed]
  35. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.

2011 (2)

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[CrossRef]

M. Zhao, L. Huang, Q. Zhang, X. Y. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[CrossRef] [PubMed]

2010 (1)

2009 (2)

2008 (1)

2007 (4)

Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15(5), 2288–2298 (2007), http://www. opticsinfobase.org/ oe/abstract. cfm?URI = oe-15–5-2288 .
[CrossRef] [PubMed]

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[CrossRef]

2006 (1)

2005 (3)

2004 (2)

2003 (2)

2002 (1)

2001 (3)

1998 (1)

1997 (2)

1986 (1)

1982 (1)

1974 (1)

Antonio Gómez-Pedrero, J.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. Garcı́a-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[CrossRef]

Antonio Quiroga, J.

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

Asundi, A.

Bernini, M. B.

Bone, D. J.

Brangaccio, D. J.

Brémand, F.

Bruning, J. H.

Cuevas, F. J.

Dalmau-Cedeño, O.

Estrada, J. C.

Federico, A.

Gallagher, J. E.

Garci´a-Botella, Á.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. Garcı́a-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[CrossRef]

Guerrero, J. A.

Herriott, D. R.

Hock Soon, S.

Huang, L.

Ina, H.

Jüptner, W.

Kaufmann, G. H.

Kemao, Q.

Kobayashi, S.

Kreis, T.

Larkin, K. G.

Legarda-Saenz, R.

Legarda-Sáenz, R.

Li, K.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[CrossRef]

Liu, D.

Luo, Y. J.

Marroquin, J. L.

Marroquín, J. L.

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[CrossRef]

Oldfield, M. A.

Osten, W.

Quiroga, J. A.

Rivera, M.

Robin, E.

Rodriguez-Vera, R.

Rosenfeld, D. P.

Servin, M.

Silva-Moreno, A. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[CrossRef]

Su, X. Y.

Takeda, M.

Tian, C.

Valle, V.

Wang, H.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[CrossRef]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-17-15118 .
[CrossRef] [PubMed]

White, A. D.

Yang, Y. Y.

Zhang, Q.

Zhao, M.

Zhuo, Y. M.

Appl. Opt. (9)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[CrossRef] [PubMed]

E. Robin and V. Valle, “Phase Demodulation from a Single Fringe Pattern Based on a Correlation Technique,” Appl. Opt. 43(22), 4355–4361 (2004).
[CrossRef] [PubMed]

E. Robin, V. Valle, and F. Brémand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44(34), 7261–7269 (2005).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
[CrossRef] [PubMed]

R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. 41(26), 5519–5526 (2002).
[CrossRef] [PubMed]

C. Tian, Y. Y. Yang, D. Liu, Y. J. Luo, and Y. M. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[CrossRef] [PubMed]

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

M. Zhao, L. Huang, Q. Zhang, X. Y. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986).
[CrossRef]

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14(8), 1742–1753 (1997).
[CrossRef]

J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001).
[CrossRef] [PubMed]

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[CrossRef] [PubMed]

R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A 23(11), 2724–2731 (2006).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001).
[CrossRef]

M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22(6), 1170–1175 (2005).
[CrossRef] [PubMed]

O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008).
[CrossRef]

Opt. Commun. (3)

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[CrossRef]

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. Garcı́a-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[CrossRef]

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

Opt. Express (2)

Opt. Lasers Eng. (2)

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[CrossRef]

Opt. Lett. (2)

Other (7)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), Chap. 4, pp. 94–140.

G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1995).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Marcel Deker, 2003).

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 1999).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, Inc, 1998).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.

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Figures (6)

Fig. 1
Fig. 1

Demodulation of a computer-generated fringe pattern by the GRPT. (a) computer-generated fringe pattern; (b) the true phase of the fringe pattern; (c) the ideal fringe pattern; (d) the NI map obtained along with the demodulation of (a) by the GRPT; (e)-(g) three snapshots of the estimated phase results obtained by the GRPT; (h) rewrapped phase result obtained by the GRPT.

Fig. 2
Fig. 2

Some comparative demodulation results of Fig. 1(a). (a)-(b) two snapshots of the estimated phase results obtained by the GRPT using the frequency guided strategy instead of the NI guided strategy; (c) rewrapped phase result obtained by the GRPT using a rectangular window instead of the Gaussian window; (d) rewrapped phase result obtained by the GRPT with the regularizing parameter λ=0 ; (e) rewrapped phase result obtained by the GRPT with the regularizing parameter λ=100 ; (f) normalized fringe pattern from the fringe pattern in Fig. 1(a); (g) the frequency map obtained along with the demodulation of (f) by the QFGRPT [23]; (h) rewrapped phase result obtained by the QFGRPT guided by the frequency map in (g).

Fig. 3
Fig. 3

Four complex computer-generated fringe patterns demodulated by the GRPT. First column shows the four computer-generated fringe patterns; second column shows the true phase distributions of the fringe patterns; third column is the NI maps obtained by the GRPT along with the fringe patterns demodulation; fourth column shows the rewrapped phase results obtained by the GRPT.

Fig. 4
Fig. 4

Experimental fringe pattern demodulated by the GRPT. (a) noisy fringe pattern from speckle shearography; (b) the NI map obtained by the GRPT from the fringe pattern in (a); (c-f) four snapshots of the estimated phase results obtained by the GRPT; (g) rewrapped phase result obtained by the GRPT; (h) cosine value of (g).

Fig. 5
Fig. 5

Experimental fringe pattern demodulated by the GRPT. (a) noisy fringe pattern from electronic speckle pattern interferometry; (b) the NI map obtained by the GRPT from the fringe pattern in (a); (c-f) four snapshots of the estimated phase results obtained by the GRPT; (g) rewrapped phase result obtained by the GRPT; (h) cosine value of (g).

Fig. 6
Fig. 6

Fringe pattern with phase discontinuities demodulated by the GRPT. (a) fringe pattern with phase discontinuities; (b) the NI map obtained by the GRPT from the fringe pattern in (a); (c-d) two snapshots of the estimated phase results obtained by the GRPT from the fringe pattern in (a); (e) rewrapped phase result obtained by the GRPT from the fringe pattern in (a); (f) a noisy fringe pattern with the same phase distribution as the fringe pattern in (a); (g) the NI map obtained by the GRPT from the fringe pattern in (f); (h) rewrapped phase result obtained by the GRPT from the fringe pattern in (f).

Tables (1)

Tables Icon

Table 1 Demodulation Results

Equations (12)

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

f( x,y )=a( x,y )+b( x,y )cos[ φ( x,y ) ]+n( x,y ),
U( x,y )= ( ε,η ) N x,y { [ f n ( ε,η ) f e ( x,y;ε,η ) ] 2 +λ [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 m( ε,η ) } ,
f n ( ε,η )cos[ φ( ε,η ) ],
f e ( x,y;ε,η )=cos[ φ e ( x,y;ε,η ) ],
φ e ( x,y;ε,η )= φ 0 ( x,y )+ ω x ( x,y )( εx )+ ω y ( x,y )( ηy ),
f e ( x,y;ε,η )= a e ( x,y;ε,η )+ b e ( x,y;ε,η )cos[ φ e ( x,y;ε,η ) ],
a e ( x,y;ε,η )= a 0 ( x,y )+ a x ( x,y )( εx )+ a y ( x,y )( ηy ),
b e ( x,y;ε,η )= b 0 ( x,y )+ b x ( x,y )( εx )+ b y ( x,y )( ηy ),
φ e ( x,y;ε,η )= φ 0 ( x,y )+ ω x ( x,y )( εx )+ ω y ( x,y )( ηy )+ c xx ( x,y ) 2 ( εx ) 2 + c yy ( x,y ) 2 ( ηy ) 2 + c xy ( x,y )( εx )( ηy ),
U( x,y )= ( ε,η ) N x,y ( w( x,y;ε,η ){ [ f( ε,η ) f e ( x,y;ε,η ) ] 2 +λ [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 m( ε,η ) } )
w( x,y;ε,η )=exp[ ( xε ) 2 + ( yη ) 2 2 σ 2 ].
a 0 ( x,y )= a 0 ( x s , y s )+ a x ( x s , y s )( x x s )+ a y ( x s , y s )( y y s ) a x ( x,y )= a x ( x s , y s ) a y ( x,y )= a y ( x s , y s ) b 0 ( x,y )= b 0 ( x s , y s )+ b x ( x s , y s )( x x s )+ b y ( x s , y s )( y y s ) b x ( x,y )= b x ( x s , y s ) b y ( x,y )= b y ( x s , y s ) φ 0 ( x,y )= φ 0 ( x s , y s )+ ω x ( x s , y s )( x x s )+ ω y ( x s , y s )( y y s ) + c xx ( x s , y s ) ( x x s ) 2 /2+ c yy ( x s , y s ) ( y y s ) 2 /2 + c xy ( x s , y s )( x x s )( y y s ) ω x ( x,y )= ω x ( x s , y s )+ c xx ( x s , y s )( x x s )+ c xy ( x s , y s )( y y s ) ω y ( x,y )= ω y ( x s , y s )+ c yy ( x s , y s )( y y s )+ c xy ( x s , y s )( x x s ) c xx ( x,y )= c xx ( x s , y s ) c yy ( x,y )= c yy ( x s , y s ) c xy ( x,y )= c xy ( x s , y s ).

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