Abstract

A robust two-dimensional continuous wavelet transform (2D-CWT) technique for interferogram analysis is presented. To cope with the phase determination ambiguity issue encountered in the analysis of complex interferograms, a phase determination rule is proposed according to the phase distribution continuity, and a frequency-guided scheme is employed to obtain the correct phase distribution following a conventional 2D-CWT analysis. The theories are given in details, and the validity of the proposed technique is verified by computer simulation and real experiments.

© 2011 Optical Society of America

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References

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2010 (4)

2009 (2)

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

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

2007 (1)

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

2006 (2)

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef] [PubMed]

2004 (3)

2003 (1)

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

2001 (2)

1997 (1)

1982 (1)

Ali, S. T.

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

Antoine, J.

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

Basaran, C.

Borghesi, M.

Burton, D.

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Burton, D. R.

Cartwright, A. N.

Chen, W.

Cuevas, F. J.

Galimberti, M.

Gdeisat, M.

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Gdeisat, M. A.

Giulietti, A.

Giulietti, D.

Gizzi, L. A.

Ina, H.

Kadooka, K.

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

Kemao, Q.

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

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

Kobayashi, S.

Kunoo, K.

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

Lalor, M.

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Lalor, M. J.

Li, K.

Li, S.

Lilley, F.

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Liu, H.

Ma, H.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Marroquin, J. L.

Moore, C.

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Murenzi, R.

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

Nagayasu, T.

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

Niu, H.

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

Ono, K.

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

Pan, B.

Qian, K.

Quan, C.

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

Quiroga, J. A.

Servin, M.

Su, X.

Takeda, M.

Tay, C.

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

Tomassini, P.

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moir 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. T. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge University, 2004).
[CrossRef]

Wang, H.

Wang, Z.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Willi, O.

AIP Conf. Proc. (1)

M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010).
[CrossRef]

Appl. Opt. (5)

Exp. Mech. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

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

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

Opt. Lett. (1)

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Traditional 2D-CWT analysis of an interferogram with complex fringes. (a) Original interferogram, (b) extracted phase distribution by using θ [ 0 , 2 π ) .

Fig. 2
Fig. 2

Traditional 2D-CWT analysis of an interferogram with fringe orders monotonically changing. (a) Original interferogram, (b) extracted phase distribution by using θ [ 0 , π ) .

Fig. 3
Fig. 3

Extracted phase distribution of Fig. 1a by using traditional 2D-CWT analysis with θ [ 0 , π ) .

Fig. 4
Fig. 4

Phase determination of the complex interferogram in Fig. 1a. (a) Extracted phase with proposed 2D-CWT, (b) unwrapped phase map.

Fig. 5
Fig. 5

Phase distribution extracted from an interferogram containing a crack. (a) Interferogram in a crack region, (b) detected incorrect phase distribution, (c) boundary mask, (d) detected correct phase distribution.

Fig. 6
Fig. 6

Warpage measurement of an electronic package. (a) Fringe pattern ( 288 × 288 pixels), (b) wrapped phase distribution, (c) refined wrapped phase distribution, (d) 3D warpage.

Fig. 7
Fig. 7

Displacement measurement of a circular disc. (a) Fringe pattern ( 1024 × 1024 pixels), (b) wrapped phase distribution, (c) refined wrapped phase distribution, (d) displacement map.

Equations (15)

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

W ( u , s , θ ) I , ψ u , s , θ = s 2 R 2 I ( x ) ψ * ( s 1 r θ ( x u ) ) d 2 x = s 2 R 2 I ^ ( ω ) ψ ^ * ( s r θ ( ω ) ) e i · ω · u d 2 ω ,
I ( x ) = I b ( x ) + I a ( x ) cos ( ϕ ( x ) ) ,
I ( x ) = I b + I a cos [ 2 π S 1 n · ( x u ) + ϕ ( u ) ] ,
ψ M ( x ) = exp ( i ω 0 · x ) exp ( m | x | 2 ) ,
W ( u , s , θ ) = π m I b exp ( π 2 m ) + π 2 m I a exp ( π 2 m { ( s S 1 ) 2 + 2 s S [ 1 cos ( Θ θ ) ] } ) exp [ i ϕ ( u ) ] + π 2 m I a exp ( π 2 m { ( s S + 1 ) 2 2 s S [ 1 cos ( Θ θ ) ] } ) exp [ i ϕ ( u ) ] .
W ( u , s , θ ) ridge = π m I b exp ( π 2 m ) + π 2 m I a exp [ i ϕ ( u ) ] + π 2 m I a exp ( 4 π 2 m ) exp [ i ϕ ( u ) ] .
W ( u ) ridge = W ( u , s ridge , θ ridge ) ,
( s ridge , θ ridge ) = arg max s R + θ [ 0 , 2 π ) { | W ( u , s , θ ) | } .
ϕ ( u ) = tan 1 { [ W ( u ) ridge ] [ W ( u ) ridge ] } ,
W ( u , s , θ ) = π m I b exp ( π 2 m ) + π 2 m I a exp ( 4 π 2 m ) exp [ i ϕ ( u ) ] + π 2 m I a exp [ i ϕ ( u ) ] .
r θ ( x ) = r π + θ ( x ) .
| W ( u , s , θ ) | = | W ( u , s , π + θ ) | .
| θ ridge ( u i ) θ ridge ( u i 1 ) | Δ θ th ,
[ θ ridge ( u i ) , ϕ ridge ( u i ) ] = { [ θ d t ( u i ) , ϕ d t ( u i ) ] , if     | θ diff ( u i ) π | > θ diff ( u i ) [ θ d t ( u i ) , ϕ d t ( u i ) ] , if     | θ diff ( u i ) π | > | θ diff ( u i ) 2 π | [ θ d t ( u i ) + π , 2 π ϕ d t ( u i ) ] , else
θ diff ( u i ) = | θ d t ( u i ) θ ridge ( u i 1 ) | .

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