Abstract

A non-iterative spatial phase-shifting algorithm based on principal component analysis (PCA) is proposed to directly extract the phase from only a single spatial carrier interferogram. Firstly, we compose a set of phase-shifted fringe patterns from the original spatial carrier interferogram shifting by one pixel their starting position. Secondly, two uncorrelated quadrature signals that correspond to the first and second principal components are extracted from the phase-shifted interferograms by the PCA algorithm. Then, the modulating phase is calculated from the arctangent function of the two quadrature signals. Meanwhile, the main factors that may influence the performance of the proposed method are analyzed and discussed, such as the level of random noise, the carrier-frequency values and the angle of carrier-frequency of fringe pattern. Numerical simulations and experiments are given to demonstrate the performance of the proposed method and the results show that the proposed method is fast, effectively and accurate. The proposed method can be used to on-line detection fields of dynamic or moving objects.

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References

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2011 (5)

2009 (1)

2008 (2)

2007 (4)

2005 (2)

2004 (1)

2001 (1)

1999 (1)

1997 (1)

1995 (2)

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[CrossRef]

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[CrossRef]

1991 (1)

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

1987 (1)

1986 (1)

1985 (1)

1983 (2)

1982 (1)

Bachor, H. A.

Barnes, T. H.

Belenguer, T.

Bone, D. J.

Bryanston-Cross, P. J.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[CrossRef]

Chai, L.

Chan, P. H.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[CrossRef]

Chen, H.

Chen, M.

Cuevas, F. J.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[CrossRef]

Debnath, S. K.

Ferrari, J. A.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[CrossRef]

Frins, E. M.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[CrossRef]

Gao, W.

Guo, H.

Heppner, J.

Huyen, N. T.

Ina, H.

Jin, W.

Kemao, Q.

Kobayashi, S.

Kujawinska, M.

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

Li, D.

Li, X.

Li, Y.

Liu, D.

Liu, J. B.

Loi, H. S.

Macy, W. W.

Massig, J. H.

Mutoh, K.

Nugent, K. A.

Park, Y.

Parker, S. C.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[CrossRef]

Patorski, K.

Peng, H.

Quiroga, J. A.

Roddier, C.

Roddier, F.

Ronney, P. D.

Sandeman, R. J.

Servin, M.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[CrossRef]

Styk, A.

Sun, L.

Takeda, M.

Tan, S. M.

Vargas, J.

Wang, H.

Wang, L.

Wang, P.

Watkins, L. R.

Weng, J.

Wójciak, J.

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

Xu, J.

Xu, Q.

Yang, H.

Yang, Q.

Yang, Y.

Zhong, J.

Zhuo, Y.

Appl. Opt. (13)

W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
[CrossRef] [PubMed]

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
[CrossRef] [PubMed]

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985).
[CrossRef] [PubMed]

D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986).
[CrossRef] [PubMed]

J. B. Liu and P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt. 36(25), 6231–6241 (1997).
[CrossRef] [PubMed]

C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987).
[CrossRef] [PubMed]

J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001).
[CrossRef] [PubMed]

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

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[CrossRef] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity contrast determination,” Appl. Opt. 46(21), 4613–4624 (2007).
[CrossRef] [PubMed]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
[CrossRef] [PubMed]

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47(29), 5408–5419 (2008).
[CrossRef] [PubMed]

J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47(29), 5446–5453 (2008).
[PubMed]

J. Mod. Opt. (1)

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[CrossRef]

Opt. Lett. (6)

Proc. SPIE (1)

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

Other (6)

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

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).

http://en.wikipedia.org/wiki/Principal_component_analysis

http://en.wikipedia.org/wiki/Principal_component_analysis .

http://en.wikipedia.org/wiki/Singular_value_decomposition .

http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem .

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Figures (5)

Fig. 1
Fig. 1

Phase-shifting interferogram constructed schematic: (a) divided mode a; and (b) divided mode b.

Fig. 2
Fig. 2

Theoretical reference phase map (left) and the simulated fringe pattern (right)

Fig. 3
Fig. 3

Simulation results: Extracted phase (a), (c) and residual error (b), (d) by mode a, b of the proposed SPS-PCA method, respectively; and extracted phase (e) and residual error (f) by Xu’s LSI-SPS method.

Fig. 4
Fig. 4

Relationship phase residual errors with the direction of carrier-frequency of interferogram.

Fig. 5
Fig. 5

Experiment results: (a) Real interferogram; Reconstructed phases map by (b) and (c) mode a, b of the proposed SPS-PCA method, and (d) Xu’s LSI-SPS method from (a).

Tables (3)

Tables Icon

Table 1 Results Obtained by the Proposed SPS-PCA and LSI-SPS Methods for Different Level of Noise

Tables Icon

Table 2 Results Obtained by the Proposed SPS-PCA and LSI-SPS Methods for Different Carrier-Frequency

Tables Icon

Table 3 PV and RMS of the Reconstructed Phases and Processing Times Obtained by the Proposed SPS-PCA and Xu’s LSI-SPS Methods with Real Interferogram

Equations (13)

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I( x,y )=A( x,y )+B( x,y )cos[ 2π( κ x x+ κ y y )+ϕ( x,y ) ]
I 1 ( x,y )=I( x,y )=A( x,y )+B( x,y )cos[ 2π( κ x x+ κ y y )+ϕ( x,y ) ]
I 2 ( x,y )=I( x+1,y )=A( x,y )+B( x,y )cos[ 2π( κ x x+ κ y y )+ϕ( x,y )+ δ 1 ]
I 3 ( x,y )=I( x,y+1 )=A( x,y )+B( x,y )cos[ 2π( κ x x+ κ y y )+ϕ( x,y )+ δ 2 ]
I 4 ( x,y )=I( x+1,y+1 )=A( x,y )+B( x,y )cos[ 2π( κ x x+ κ y y )+ϕ( x,y )+ δ 3 ]
X= [ I 1 , I 2 ,, I n , I N ] T
X m = 1 N n=1 n=N I n A
C=[ X X m ] [ X X m ] T
C Q i = V i Q i ,i=1,2,...,N
Q= [ Q 1 , Q 2 , Q n , Q N ] T
D= Q T CQ
Φ={ Φ 1 Φ 2 Φ N }=Q( X X m )={ Q 1 Q 2 Q N }( X X m )
ϕ( x,y )=arctan( I c I s )2π( κ x x+ κ y y )=±arctan( Φ 1 Φ 2 )2π( κ x x+ κ y y )

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