Abstract

In optical measurement, spatial carrier fringe pattern analysis is suitable for measuring dynamic events in real-time. This paper presents a novel technique for analyzing a spatial carrier fringe pattern. It estimates the local phase gradients at a pixel from its neighborhood, by use of statistics of the intensity gradients. Using the estimated phase gradients, the phase map of the fringe pattern is recovered by solving numerical partial derivative equations or using an adaptive spatial carrier phase shifting (SCPS) algorithm. Simulation and experimental results demonstrate this algorithm to be valid.

© 2014 Optical Society of America

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References

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    [Crossref]
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2014 (1)

2013 (1)

2012 (3)

2010 (2)

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

2009 (1)

2008 (2)

2007 (1)

2006 (1)

2005 (1)

2004 (1)

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

2001 (1)

1999 (1)

1997 (1)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

1995 (3)

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

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]

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

1994 (1)

1992 (1)

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

1991 (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

1990 (1)

1987 (1)

1986 (3)

1985 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

1983 (3)

1982 (1)

1981 (1)

1980 (1)

1979 (1)

1972 (1)

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Bachor, H.-A.

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Barnes, T. H.

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]

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, M.

Chen, W.

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]

Du, Y.

Federico, A.

Feng, G.

Ghiglia, D. C.

Guo, H.

Heppner, J.

Herrmann, J.

Hung, Y. Y.

Hunt, B. R.

Ichioka, Y.

Ina, H.

Inuiya, M.

Iwaasa, Y.

Jiang, M.

Józwicki, R.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

Kaufmann, G. H.

Kimbrough, B. T.

Kobayashi, S.

Kokal, J. V.

Kujawinska, M.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

Li, H.

Li, S.

Lin, J.-F.

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Macy, W. W.

Marks, R. J.

Massig, J. H.

Mertz, L.

Mutoh, K.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Nugent, K. A.

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]

Peng, H.

Qian, K.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

Ransom, P. L.

Roddier, C.

Roddier, F.

Romero, L. A.

Salbut, L.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

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]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Su, X.

Su, X.-Y.

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Takeda, M.

Tan, S. M.

Tang, S.

Toyooka, S.

Vargas, J.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Watkins, L. R.

Weng, J.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

Xu, J.

Xu, Q.

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Yang, Q.

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Zhang, Z.

Zhang, Z. H.

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Zhong, J.

Zhong, M.

Zhou, S.

Appl. Opt. (19)

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]

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]

A. Federico and G. H. Kaufmann, “Phase retrieval of singular scalar light fields using a two-dimensional directional wavelet transform and a spatial carrier,” Appl. Opt. 47(28), 5201–5207 (2008).
[Crossref] [PubMed]

S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48(36), 6893–6906 (2009).
[Crossref] [PubMed]

M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012).
[Crossref] [PubMed]

Y. Ichioka and M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972).
[Crossref] [PubMed]

S. Tang and Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29(20), 3012–3018 (1990).
[Crossref] [PubMed]

L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22(10), 1535–1539 (1983).
[Crossref] [PubMed]

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

S. Toyooka and Y. Iwaasa, “Automatic profilometry of 3-D diffuse objects by spatial phase detection,” Appl. Opt. 25(10), 1630–1633 (1986).
[Crossref] [PubMed]

B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
[Crossref] [PubMed]

P. L. Ransom and J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25(22), 4199–4204 (1986).
[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]

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]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20(10), 1815–1820 (1981).
[Crossref] [PubMed]

J. Mod. Opt. (2)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

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. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (4)

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Opt. Lasers Eng. (4)

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

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. (2)

Other (1)

M. Kujawińska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61–67 (1991).

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Figures (6)

Fig. 1
Fig. 1 (a) is a simulated spatial carrier fringe pattern and (b) is its phase map.
Fig. 2
Fig. 2 (a) and (d) show the theoretical phase gradients in x and y directions, respectively. (b) and (e) are the estimated phase gradients along the two directions, respectively. (c) and (f) show the differences between the estimated and the theoretical phase gradients in x and y directions, respectively.
Fig. 3
Fig. 3 Phase recovering results of the simulated fringe pattern. The left column, from top to bottom, shows the phase maps extracted using the numerical integration method, 1D SCPS method, 2D SCPS method, and FT method, respectively. The right column illustrates the phase errors corresponding to the above methods in turn.
Fig. 4
Fig. 4 A practical interferogram with spatial carrier.
Fig. 5
Fig. 5 The phase gradients in (a) the horizontal and (b) vertical directions estimated from the interferogram in Fig. 4.
Fig. 6
Fig. 6 The phase maps of the interferogram in Fig. 4, recovered using (a) the numerical integration method, (b) 1D SCPS method, (c) 2D SCPS method, and (d) FT method.

Tables (2)

Tables Icon

Table 1 Simulation Results of Estimating Phase gradients in the Presence of Gaussian Noise

Tables Icon

Table 2 Simulation Results of Phase Recovery in the Presence of Gaussian Noise

Equations (33)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ u C x + υ C y + ϕ ( x , y ) ] ,
u ( x , y ) = [ u C x + υ C y + ϕ ( x , y ) ] x = u C + ϕ ( x , y ) x
υ ( x , y ) = [ u C x + υ C y + ϕ ( x , y ) ] y = υ C + ϕ ( x , y ) y ,
h ( + ) I ( x , y ) = I ( x + 1 , y ) I ( x , y ) ,
h ( ) I ( x , y ) = I ( x , y ) I ( x 1 , y ) ,
H I ( x , y ) = I ( x + 1 , y ) I ( x 1 , y ) 2 .
σ h ( + ) ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ h ( + ) I ( x + s , y + t ) ] 2 ,
σ h ( ) ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ h ( ) I ( x + s , y + t ) ] 2 ,
σ H ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ H I ( x + s , y + t ) ] 2 ,
ω h ( x , y ) = 2 arc cos [ σ h ( ) ( x , y ) ] 2 + [ 2 σ H ( x , y ) ] 2 [ σ h ( + ) ( x , y ) ] 2 2 σ h ( ) ( x , y ) [ 2 σ H ( x , y ) ] .
ω h ( x , y ) = 2 arc cos σ H ( x , y ) σ h ( x , y ) ,
v I ( x , y ) = I ( x , y + 1 ) I ( x , y )
V I ( x , y ) = I ( x , y + 1 ) I ( x , y 1 ) 2 .
σ v ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ v I ( x + s , y + t ) ] 2
σ V ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ V I ( x + s , y + t ) ] 2 .
ω v ( x , y ) = 2 arc cos σ V ( x , y ) σ v ( x , y ) .
d I ( x , y ) = I ( x + 1 , y + 1 ) I ( x , y )
D I ( x , y ) = I ( x + 1 , y + 1 ) I ( x 1 , y 1 ) 2 .
σ d ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ d I ( x + s , y + t ) ] 2
σ D ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ D I ( x + s , y + t ) ] 2 .
ω d ( x , y ) = 2 arc cos σ D ( x , y ) σ d ( x , y ) .
{ u ( x , y ) = ω h ( x , y ) v ( x , y ) = ω v ( x , y )
{ u ( x , y ) = ω h ( x , y ) v ( x , y ) = ω d ( x , y ) ω h ( x , y )
{ u ( x , y ) = ω d ( x , y ) ω v ( x , y ) v ( x , y ) = ω v ( x , y )
φ ( x + 1 , y ) φ ( x , y ) u ( x + 1 , y ) + u ( x , y ) 2 ,
φ ( x , y + 1 ) φ ( x , y ) v ( x + 1 , y ) + v ( x , y ) 2 .
E = x y [ φ ( x + 1 , y ) φ ( x , y ) u ( x + 1 , y ) + u ( x , y ) 2 ] 2 + x y [ φ ( x , y + 1 ) φ ( x , y ) v ( x , y + 1 ) + v ( x , y ) 2 ] 2 .
[ φ ( x + 1 , y ) 2 φ ( x , y ) + φ ( x 1 , y ) ] + [ φ ( x , y + 1 ) 2 φ ( x , y ) + φ ( x , y 1 ) ] = ρ ( x , y )
ρ ( x , y ) = u ( x + 1 , y ) u ( x 1 , y ) 2 + v ( x , y + 1 ) v ( x , y 1 ) 2 .
[ 2 x 2 + 2 y 2 ] ϕ ( x , y ) = ρ ( x , y )
Φ ( i , j ) = P ( i , j ) 2 ( cos π i M + 2 + cos π j N + 2 2 ) ,
i = L L j = L L [ 1 cos [ u ( x , y ) i + v ( x , y ) j ] 0 cos [ u ( x , y ) i + v ( x , y ) j ] cos 2 [ u ( x , y ) i + v ( x , y ) j ] 0 0 0 sin 2 [ u ( x , y ) i + v ( x , y ) j ] ] [ c 0 ( x , y ) c 1 ( x , y ) c 2 ( x , y ) ] = i = L L j = L L [ I ( x + i , y + j ) I ( x + i , y + j ) cos [ u ( x , y ) i + v ( x , y ) j ] I ( x + i , y + j ) sin [ u ( x , y ) i + v ( x , y ) j ] ] .
φ ( x , y ) = arc tan c 2 ( x , y ) c 1 ( x , y )

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