Abstract

A novel technique that uses a fan two-dimensional (2D) continuous wavelet transform (CWT) to phase demodulate fringe patterns is proposed. The fan 2D CWT algorithm is tested by using computer generated and real fringe patterns. The result of this investigation reveals that the 2D CWT technique is capable of successfully demodulating fringe patterns. The proposed algorithm demodulates fringe patterns without the requirement of removing their background illumination prior to the demodulation process. Also, the algorithm is exceptionally robust against speckle noise. The performance of the 2D CWT technique in fringe pattern demodulation is compared with that of the 1D CWT algorithms. This comparison indicates that the 2D CWT outperforms its 1D counterpart for this application.

© 2006 Optical Society of America

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2005

Q. Chenggen, C. Tay, and L. Chen, "Fringe-density estimation by continuous wavelet transform," Appl. Opt. 44, 2359-2365 (2005).
[CrossRef]

F. Kirby, "Which wavelet reproduces the Fourier power spectrum?" Comput. Geosci. 31, 846-864 (2005).
[CrossRef]

2004

Y. Fu, C. Tay, C. Quan, and L. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

H. Liu, A. Cartwright, and C. Basaran, "Moiré interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms," Appl. Opt. 43, 850-857 (2004).
[CrossRef] [PubMed]

A. Durson, S. Ozder, and N. Ecevit, "Continuous wavelet transform analysis of projected fringe patterns," Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

J. Zhong and J. Weng, "Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry," Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

K. Qian, "Windowed Fourier transform method for demodulation of carrier fringes," Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

J. Zhong and J. Weng, "Dilating Gabor transform for the fringe pattern analysis of 3D-shape measurement," Opt. Eng. 43, 895-899 (2004).
[CrossRef]

2003

A. Federico and G. Kaufmann, "Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution," Appl. Opt. 42, 7066-7071 (2003).
[CrossRef] [PubMed]

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

2002

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

A. Federico and G. Kaufmann, "Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes," Opt. Eng. 41, 3209-3216 (2002).
[CrossRef]

2001

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

P. Tomassino, A. Giulietti, L. Gizzi, M. Galimberti, D. Giuilietti, M. Borghesi, and O. Willi, "Analyzing laser plasma interferograms with a continuous wavelet transform ridge extraction technique: the method," Appl. Opt. 40, 6561-6568 (2001).
[CrossRef]

2000

F. Lilley, M. Lalor, and D. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. Gdeisat, D. Burton, and M. Lalor, "Real-time fringe pattern demodulation with a second-order digital phase-locked loop," Appl. Opt. 39, 5326-5336 (2000).
[CrossRef]

1997

1982

1972

Afif, M.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Ali, S.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge U. Press, 2004).
[CrossRef]

Antoine, J.

J. Antoine, D. Barache, R. Cesar, and L. da Fontoura Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

J. Antoine, R. Murenzi, P. Vandergheynst, and S. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge U. Press, 2004).
[CrossRef]

Barache, D.

J. Antoine, D. Barache, R. Cesar, and L. da Fontoura Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

Basaran, C.

Berger, E.

Borghesi, M.

Burton, D.

F. Lilley, M. Lalor, and D. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. Gdeisat, D. Burton, and M. Lalor, "Real-time fringe pattern demodulation with a second-order digital phase-locked loop," Appl. Opt. 39, 5326-5336 (2000).
[CrossRef]

Cartwright, A.

Cesar, R.

J. Antoine, D. Barache, R. Cesar, and L. da Fontoura Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

Chen, L.

Q. Chenggen, C. Tay, and L. Chen, "Fringe-density estimation by continuous wavelet transform," Appl. Opt. 44, 2359-2365 (2005).
[CrossRef]

Y. Fu, C. Tay, C. Quan, and L. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

Chenggen, Q.

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, and X. Colonna de Lega, "Wavelet processing of interferometric signals and fringe analysis," in Wavelet Applications in Signal and Image Processing VII, M.A.Unser, A.Aldroubi, and A.F.Laine, eds., Proc. SPIE 3813,692-702 (1999).

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Fringe 2001-- The 4th International Workshop on Automatic Processing of Fringe Patterns (Elsevier, 2001), pp. 154-162.

Colonna de Lega, X.

X. Colonna de Lega, "Continuous deformation measurement using dynamic phase-shifting and wavelet transform," in Applied Optics and Optoelectronics 1996 (Institute of Physics, 1996), pp. 261-267.

X. Colonna de Lega, "Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurement by holographic and speckle interferometry," Ph.D. dissertation 1666 (Swiss Federal Institute of Technology Lausanne, 1997).

M. Cherbuliez, P. Jacquot, and X. Colonna de Lega, "Wavelet processing of interferometric signals and fringe analysis," in Wavelet Applications in Signal and Image Processing VII, M.A.Unser, A.Aldroubi, and A.F.Laine, eds., Proc. SPIE 3813,692-702 (1999).

Creath, K.

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W.R.Robinson and G.T.Reid, eds. (Institute of Physics, 1993), pp. 94-140.

da Fontoura Costa, L.

J. Antoine, D. Barache, R. Cesar, and L. da Fontoura Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

Demoli, N.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Dose, V.

Durson, A.

A. Durson, S. Ozder, and N. Ecevit, "Continuous wavelet transform analysis of projected fringe patterns," Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Ecevit, N.

A. Durson, S. Ozder, and N. Ecevit, "Continuous wavelet transform analysis of projected fringe patterns," Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Fassi-Fihri, A.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Federico, A.

A. Federico and G. Kaufmann, "Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution," Appl. Opt. 42, 7066-7071 (2003).
[CrossRef] [PubMed]

A. Federico and G. Kaufmann, "Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes," Opt. Eng. 41, 3209-3216 (2002).
[CrossRef]

A. Federico and G. Kaufmann, "Phase retrieval in electronic speckle interferometry using the continuous wavelet transform," in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V.L.Brudny, S.A.Ledesma, and M.C.Marconi, eds., Proc. SPIE 4419,162-165 (2001).

Fu, Y.

Y. Fu, C. Tay, C. Quan, and L. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

Galimberti, M.

Gdeisat, M.

Ghiglia, C.

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

Giuilietti, D.

Giulietti, A.

Gizzi, L.

Gruber, H.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Hariharan, P.

P. Hariharan, "Applications of interferogram analysis," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W.R.Robinson and G.T.Reid, eds. (Institute of Physics, 1993), pp. 262-284.

Ichioka, Y.

Ina, H.

Inuiya, M.

Itoh, K.

Jacquot, P.

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Fringe 2001-- The 4th International Workshop on Automatic Processing of Fringe Patterns (Elsevier, 2001), pp. 154-162.

M. Cherbuliez, P. Jacquot, and X. Colonna de Lega, "Wavelet processing of interferometric signals and fringe analysis," in Wavelet Applications in Signal and Image Processing VII, M.A.Unser, A.Aldroubi, and A.F.Laine, eds., Proc. SPIE 3813,692-702 (1999).

Kadooka, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Kaufmann, G.

A. Federico and G. Kaufmann, "Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution," Appl. Opt. 42, 7066-7071 (2003).
[CrossRef] [PubMed]

A. Federico and G. Kaufmann, "Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes," Opt. Eng. 41, 3209-3216 (2002).
[CrossRef]

A. Federico and G. Kaufmann, "Phase retrieval in electronic speckle interferometry using the continuous wavelet transform," in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V.L.Brudny, S.A.Ledesma, and M.C.Marconi, eds., Proc. SPIE 4419,162-165 (2001).

Kayser, D.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Kirby, F.

F. Kirby, "Which wavelet reproduces the Fourier power spectrum?" Comput. Geosci. 31, 846-864 (2005).
[CrossRef]

Kobayashi, S.

Koch, A.

Krüger, S.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Kunoo, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Lalor, M.

F. Lilley, M. Lalor, and D. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. Gdeisat, D. Burton, and M. Lalor, "Real-time fringe pattern demodulation with a second-order digital phase-locked loop," Appl. Opt. 39, 5326-5336 (2000).
[CrossRef]

Lamberti, L.

C. A. Sciammarella, C. Patimo, P. D. Manicone, and L. Lamberti, "Fringe pattern information retrieval using wavelets," in Wavelets XI, M.Papadakis, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 5914,59140A-1-59140A-14 (2005).

Lilley, F.

F. Lilley, M. Lalor, and D. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

Linden, W.

Liu, H.

Malat, S.

S. Malat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

Manicone, P. D.

C. A. Sciammarella, C. Patimo, P. D. Manicone, and L. Lamberti, "Fringe pattern information retrieval using wavelets," in Wavelets XI, M.Papadakis, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 5914,59140A-1-59140A-14 (2005).

Marjane, M.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Murenzi, R.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge U. Press, 2004).
[CrossRef]

Nagayasu, T.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Nassim, K.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Ono, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Osten, W.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Ozder, S.

A. Durson, S. Ozder, and N. Ecevit, "Continuous wavelet transform analysis of projected fringe patterns," Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Patimo, C.

C. A. Sciammarella, C. Patimo, P. D. Manicone, and L. Lamberti, "Fringe pattern information retrieval using wavelets," in Wavelets XI, M.Papadakis, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 5914,59140A-1-59140A-14 (2005).

Pritt, M.

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

Qian, K.

K. Qian, "Windowed Fourier transform method for demodulation of carrier fringes," Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

Quan, C.

Y. Fu, C. Tay, C. Quan, and L. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

Rachafi, S.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Ruprecht, M.

Sandoz, S.

Sciammarella, C. A.

C. A. Sciammarella, C. Patimo, P. D. Manicone, and L. Lamberti, "Fringe pattern information retrieval using wavelets," in Wavelets XI, M.Papadakis, A.F.Laine, and M.A.Unser, eds., Proc. SPIE 5914,59140A-1-59140A-14 (2005).

Sidki, M.

M. Afif, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, "Paul wavelet-based algorithm for optical phase distribution evaluation," Opt. Commun. 21, 47-51 (2002).
[CrossRef]

Takeda, M.

Tay, C.

Q. Chenggen, C. Tay, and L. Chen, "Fringe-density estimation by continuous wavelet transform," Appl. Opt. 44, 2359-2365 (2005).
[CrossRef]

Y. Fu, C. Tay, C. Quan, and L. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

Tomassino, P.

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, "Strain analysis for Moiré interferometry using the two-dimensional continuous wavelet transform," Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Vandergheynst, P.

J. Antoine, R. Murenzi, P. Vandergheynst, and S. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge U. Press, 2004).
[CrossRef]

Weng, J.

J. Zhong and J. Weng, "Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry," Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

J. Zhong and J. Weng, "Dilating Gabor transform for the fringe pattern analysis of 3D-shape measurement," Opt. Eng. 43, 895-899 (2004).
[CrossRef]

Wernicke, G.

S. Krüger, G. Wernicke, W. Osten, D. Kayser, N. Demoli, and H. Gruber, "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," J. Electron. Imaging 10, 228-233 (2001).
[CrossRef]

Willi, O.

Zhong, J.

J. Zhong and J. Weng, "Dilating Gabor transform for the fringe pattern analysis of 3D-shape measurement," Opt. Eng. 43, 895-899 (2004).
[CrossRef]

J. Zhong and J. Weng, "Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry," Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

Appl. Opt.

M. Gdeisat, D. Burton, and M. Lalor, "Real-time fringe pattern demodulation with a second-order digital phase-locked loop," Appl. Opt. 39, 5326-5336 (2000).
[CrossRef]

Y. Ichioka and M. Inuiya, "Direct phase detection system," Appl. Opt. 11, 1507-1514 (1972).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

(Color online) Fringe pattern demodulated by using the complex Morlet 1D CWT (phase estimation).

Fig. 2
Fig. 2

(Color online) Fringe pattern demodulated by using the complex Morlet 1D CWT (frequency estimation).

Fig. 3
Fig. 3

(Color online) Fringe pattern demodulated by using the real Morlet 1D CWT (frequency estimation).

Fig. 4
Fig. 4

2D complex Morlet wavelet and its spectrum for two different values of θ.

Fig. 5
Fig. 5

Fan mother wavelet and its spectrum.

Fig. 6
Fig. 6

(Color online) Demodulating the fringe pattern shown in Fig. 1(b) by using the fan 2D CWT algorithm.

Fig. 7
Fig. 7

(Color online) Computer-generated fringe pattern demodulated by using 1D CWT and 2D CWT algorithms.

Fig. 8
Fig. 8

(Color online) Computer-generated fringe pattern demodulated by using 1D CWT and 2D CWT algorithms.

Fig. 9
Fig. 9

(Color online) Real fringe pattern demodulated by using 1D CWT and 2D CWT algorithms.

Equations (15)

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g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f o x + ϕ ( x , y ) ] ,
ψ ( x ) = π 1 / 4  exp ( i c x ) exp ( x 2 / 2 ) ,
ψ b , x ( x ) = 1 s ψ ( x b s ) .
W ( s , b ) = 1 s ψ * ( x b s ) g ( x ) d x .
φ ( s , b ) = tan - 1 [ { W ( s , b ) } { W ( s , b ) } ] ,
f ^ ( b ) = c + c 2 + 2 2 s max ( b ) 2 π f o .
S ( a , b , s , θ ) = ψ a , b , s , θ ,   g ( x , y ) = s 1 ψ ( x a s , y b s , r θ ) g ( x , y ) d x d y .
c ψ ( 2 π ) 2 ψ ̂ ( s , r ) s 2 + r 2  d s d r < ,
ψ M ( x , y ) = exp [ i k o ( x   cos   θ + y   sin   θ ) ] exp ( 1 2 x 2 y 2 ) ,
ψ ̂ M ( s , r ) = exp { - 1 2 [ ( r k o  cos   θ ) 2 + ( s k o  sin   θ ) 2 ] } .
ψ F ( x , y ) = j = 0 N θ 1 exp [ i k o ( x   cos   θ j + y   sin   θ j ) ] × exp ( - 1 2 x 2 + y 2 ) .
ψ ̂ F ( s , r ) = k = 0 N θ 1 exp { - 1 2 [ ( r k o   cos   θ k ) 2 + ( s k o   sin   θ k ) 2 ] } ,
ϕ ( m , n ) = tan - 1 ( { ρ } / { ρ } ) ,
ϕ ( x , y ) = 0.15 [ ( x 256 ) 2 + ( y 256 ) 2 ] 1 / 2 ,
I ( x , y ) = 0.3 ϕ ( x , y ) + cos [ 2 π f o x + ϕ ( x , y ) ] + NOISE .

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