Abstract

A generalized regularized phase tracker (GRPT) for demodulation of a single fringe pattern was recently proposed. It is very successful for many fringe patterns. However, the GRPT has poor performance in the area where the fringe pattern is sparse. An improved GRPT (iGRPT) with two novel improvements is proposed to overcome the problem. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. With these two improvements, the proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT. Simulation and experimental results are presented to verify the performance of the iGRPT.

© 2013 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 (Marcel Dekker, 1998).
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    [CrossRef]
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    [CrossRef]
  4. 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. A18(8), 1862–1870 (2001).
    [CrossRef] [PubMed]
  5. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A20(5), 925–934 (2003).
    [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] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express20(11), 12579–12592 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  21. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng.45(2), 304–317 (2007).
    [CrossRef]
  22. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun.224(4-6), 221–227 (2003).
    [CrossRef]
  23. W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]

2012 (1)

2011 (1)

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]

2010 (2)

2009 (2)

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express17(17), 15118–15127 (2009).
[CrossRef] [PubMed]

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[CrossRef]

2008 (1)

2007 (3)

2006 (1)

2005 (2)

2003 (2)

2002 (1)

2001 (2)

1999 (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol.10(3), R33–R55 (1999).
[CrossRef]

1998 (1)

1997 (2)

1982 (1)

Bone, D. J.

Brémand, F.

Cuevas, F. J.

Dalmau-Cedeño, O.

Dorrío, B. V.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol.10(3), R33–R55 (1999).
[CrossRef]

Estrada, J. C.

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol.10(3), R33–R55 (1999).
[CrossRef]

Gao, W. J.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[CrossRef]

Hock Soon, S.

Ina, H.

Jüptner, W.

Kai, L.

Kemao, Q.

L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express20(11), 12579–12592 (2012).
[CrossRef] [PubMed]

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]

L. Kai and Q. Kemao, “Fast frequency-guided sequential demodulation of a single fringe pattern,” Opt. Lett.35(22), 3718–3720 (2010).
[CrossRef] [PubMed]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express17(17), 15118–15127 (2009).
[CrossRef] [PubMed]

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[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]

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]

Kobayashi, S.

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]

Lin, F.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[CrossRef]

Liu, D.

Luo, Y. J.

Marroquin, J. L.

Marroquín, J. L.

Oldfield, M. A.

Osten, W.

Quiroga, J. A.

Rivera, M.

Robin, E.

Rodriguez-Vera, R.

Servin, M.

Soon, S. H.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[CrossRef]

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. Express17(17), 15118–15127 (2009).
[CrossRef] [PubMed]

Wang, H. X.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[CrossRef]

Yang, Y. Y.

Zhuo, Y. M.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A18(3), 689–695 (2001).
[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. A18(8), 1862–1870 (2001).
[CrossRef] [PubMed]

R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A23(11), 2724–2731 (2006).
[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. A20(5), 925–934 (2003).
[CrossRef] [PubMed]

M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A22(6), 1170–1175 (2005).
[CrossRef] [PubMed]

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. A15(6), 1536–1544 (1998).
[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. A14(8), 1742–1753 (1997).
[CrossRef]

O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A25(6), 1361–1370 (2008).
[CrossRef]

Meas. Sci. Technol. (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol.10(3), R33–R55 (1999).
[CrossRef]

Opt. Commun. (1)

J. A. 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. (3)

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng.47(11), 1286–1292 (2009).
[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]

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]

Opt. Lett. (2)

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

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Figures (10)

Fig. 1
Fig. 1

Demodulation of a simulated fringe pattern by the GRPT. (a) simulated fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).

Fig. 2
Fig. 2

Demodulation of a simulated fringe pattern by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: σ min =7.5 , σ max =15 , λ a =0 , λ b =0 , λ φ =20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λ a =10 and λ b =10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).

Fig. 3
Fig. 3

Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).

Fig. 4
Fig. 4

Demodulation of Fig. 3(a) by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: σ min =7.5 , σ max =15 , λ a =0 , λ b =0 , λ φ =20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λ a =10 and λ b =10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).

Fig. 5
Fig. 5

Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).

Fig. 6
Fig. 6

Demodulation of Fig. 5(a) by the iGRPT with σ min =3 , σ max =12 , λ a =10 , λ b =10 , λ φ =20 and the window size is determined by Eq. (10). (a) phase result obtained by the iGRPT, (b) NI quality map of (a), (c) window size.

Fig. 7
Fig. 7

Demodulation of fringe patterns by the iGRPT without background estimation. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: demodulation results obtained from the simulated fringe pattern shown in Fig. 1(a); the second row: demodulation results obtained from the experimental fringe pattern shown in Fig. 3(a); the third row: demodulation results obtained from the experimental fringe pattern shown in Fig. 5(a).

Fig. 8
Fig. 8

Demodulation of the simulated fringe pattern by the RPT [12] with different window sizes. (a) the same simulated fringe pattern as shown in Fig. 1(a), (b)-(f) the phase results obtained by the RPT with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

Fig. 9
Fig. 9

Demodulation of the experimental fringe pattern shwon in Fig. 3(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 3(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

Fig. 10
Fig. 10

Demodulation of the experimental fringe pattern shwon in Fig. 5(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 5(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

Equations (15)

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f( x,y )=a( x,y )+b( x,y )cos[ φ( x,y ) ]+n( x,y ),
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 ) },
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 )+ 1 2 c xx ( x,y ) ( εx ) 2 + 1 2 c yy ( x,y ) ( ηy ) 2 + c xy ( x,y )( εx )( ηy ),
ω TLF = ω x 2 + ω y 2 ,
σ ˜ =2/ ω TLF ,
σ={ σ min if σ ˜ σ min σ ˜ if σ min < σ ˜ < σ max σ max if σ ˜ σ max ,
σ={ σ min if σ ˜ ( σ min + σ max )/2 σ max if σ ˜ >( σ min + σ max )/2 .
U( x,y )= ( ε,η ) N x,y ( w( x,y;ε,η ){ [ f( ε,η ) f e ( x,y;ε,η ) ] 2 + ( λ a R a + λ b R b + λ φ R φ )m( ε,η ) } ),
R a = [ a 0 ( ε,η ) a e ( x,y;ε,η ) ] 2 ,
R b = [ b 0 ( ε,η ) b e ( x,y;ε,η ) ] 2 ,
R φ = [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 ,

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