Abstract

The two-dimensional continuous wavelet transform (2D-CWT) technique provides robust processing for digital fringe pattern analysis. To cope with the problem of long computation time, a concept called the cover map is introduced to speed up the 2D-CWT analysis. The cover map is constructed by discretizing the continuous dilation and rotation parameters. The discretized parameters help substantially reduce the processing time without affecting the analysis accuracy. The theories are presented and the validity and effectiveness of the proposed concept are demonstrated by computer simulation and real experiment.

© 2011 Optical Society of America

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References

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  1. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [CrossRef]
  2. R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
    [CrossRef]
  3. P. Tomassini, A. Giulietti, L. A. Gizzi, M. Galimberti, D. Giulietti, 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]
  4. H. Liu, A. N. Cartwright, and C. Basaran, “Moiré interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
    [CrossRef] [PubMed]
  5. A. Federico and G. H. 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]
  6. 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]
  7. 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]
  8. J. Weng, J. Zhong, and C. Hu, “Phase reconstruction of digital holography with the peak of the two-dimensional Gabor wavelet transform,” Appl. Opt. 48, 3308–3316 (2009).
    [CrossRef] [PubMed]
  9. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
    [CrossRef]
  10. S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27, 1245–1254(2010).
    [CrossRef]
  11. C. K. Chui, An Introduction to Wavelets (Academic, 1992), Vol.  1.
  12. D. W. Robinson, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).
  13. C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
    [CrossRef]
  14. J. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge University, 2004).
    [CrossRef]
  15. J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011).
    [CrossRef] [PubMed]
  16. D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
    [CrossRef]

2011

2010

2009

J. Weng, J. Zhong, and C. Hu, “Phase reconstruction of digital holography with the peak of the two-dimensional Gabor wavelet transform,” Appl. Opt. 48, 3308–3316 (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]

2006

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

2004

2003

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

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

A. Federico and G. H. 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

1999

1997

R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

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]

Barnes, T. H.

Basaran, C.

Borghesi, M.

Carmona, R.

R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

Cartwright, A. N.

Chen, W.

Chui, C. K.

C. K. Chui, An Introduction to Wavelets (Academic, 1992), Vol.  1.

Federico, A.

A. Federico and G. H. 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]

Galimberti, M.

Giulietti, A.

Giulietti, D.

Gizzi, L. A.

Han, B.

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
[CrossRef]

Hoang, T.

Hu, C.

Hwang, W.

R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

Ifju, P.

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
[CrossRef]

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. H.

A. Federico and G. H. 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]

Kim, T.

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[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]

Li, S.

Liu, H.

Luu, L.

Ma, H.

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

Ma, J.

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 moiré 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 moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Pan, B.

Post, D.

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
[CrossRef]

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]

Robinson, D. W.

D. W. Robinson, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Su, X.

Tan, S. 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.

Torresani, B.

R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

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. T. Ali, Two-Dimensional Wavelets and Their Relatives (Cambridge University, 2004).
[CrossRef]

Vo, M.

Wang, Z.

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011).
[CrossRef] [PubMed]

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

Watkins, L. R.

Weng, J.

Willi, O.

Zhong, J.

Appl. Opt.

Exp. Mech.

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]

IEEE Trans. Signal Process.

R. Carmona, W. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

A. Federico and G. H. 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]

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

C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Opt. Lasers Eng.

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.

Other

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

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
[CrossRef]

C. K. Chui, An Introduction to Wavelets (Academic, 1992), Vol.  1.

D. W. Robinson, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

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Figures (10)

Fig. 1
Fig. 1

Wavelet modulated window in the frequency domain.

Fig. 2
Fig. 2

Examples of cover maps. (a)  σ = 1.0 and q = 1.2536 , (b)  σ = 0.5 and q = 1.5809 , (c)  σ = 0.3376 and q = 2.0 , and (d)  σ = 0.5 and q = 2.0 .

Fig. 3
Fig. 3

Schematic of the calculation of rotation angle.

Fig. 4
Fig. 4

Complete dyadic cover map for σ = 0.3376 .

Fig. 5
Fig. 5

Frequency response.

Fig. 6
Fig. 6

Frequency response for σ = 0.5 . (a)  s = 3.2162 , (b)  s = 8.0380 , (c)  s = 20.0892 , and (d)  s = 125.4847 .

Fig. 7
Fig. 7

Simulation results: (a) fringe pattern, (b) fringe pattern with noise, (c) extracted phase, (d) phase gradient, (e) phase gain, and (f) frequency gain.

Fig. 8
Fig. 8

Moiré fringe pattern of a PBGA package and the analysis results obtained by the conventional 2D-CWT scheme: (a) fringe pattern, (b) boundary mask, (c) spectrum, and (d) phase map.

Fig. 9
Fig. 9

Analysis results with different cover map parameters: (a) cover map for σ = 1.0 , (b) phase map from σ = 1.0 , (c) cover map for σ = 0.5 , (d) phase map from σ = 0.5 , (e) cover map for σ = 0.3376 , and (f) phase map from σ = 0.3376 .

Fig. 10
Fig. 10

Comparison of analysis speed associated with different cover maps.

Equations (29)

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Δ f Δ f ^ 1 2 or Δ f Δ f ^ 1 4 π ,
Δ f : = 1 f 2 ( ( t t 0 ) 2 | f ( t ) | 2 d t ) 1 / 2 ,
t 0 : = 1 f 2 2 t | f ( t ) | 2 d t .
Δ f Δ f ^ = 1 2 or Δ f Δ f ^ = 1 4 π ,
f ( t ) = c exp ( i k t ) exp ( ( x μ ) 2 2 σ 2 ) ,
g ( t ) = 1 2 π σ exp ( ( t μ ) 2 2 σ 2 ) ,
t 0 = 1 g 2 2 t | g ( t ) | 2 d t = μ ,
Δ g = 1 g 2 ( ( t t 0 ) 2 | g ( t ) | 2 d t ) 1 / 2 = 2 2 σ .
Δ g ^ = 2 2 σ .
ψ u , s , θ ( x ) = 1 s ψ ( r θ ( x u s ) ) ,
ψ M ( x ) = exp ( i ω 0 · x ) exp ( 1 2 σ 2 | x | 2 ) ,
ψ ^ M ( ω ) = exp ( σ 2 2 | ω ω 0 | 2 ) .
[ | u 0 | s Δ ψ , | u 0 | + s Δ ψ ] × [ | ω 0 | s 1 s Δ ψ ^ , | ω 0 | s + 1 s Δ ψ ^ ] .
Δ ψ Δ ψ ^ = 1 2 .
Δ ψ = 2 2 σ = 2 4 , Δ ψ ^ = 2 .
Δ ψ = 2 2 , Δ ψ ^ = 2 2 .
I ( x ) = I b ( x ) + I a ( x ) cos ( ϕ ( x ) ) ,
U ( x ) = m ϕ ( x ) ,
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 ,
W ( u ) ridge = W ( u , s ridge , θ ridge ) ,
( s ridge , θ ridge ) = argmax s R + θ [ 0 , 2 π ) { | W ( u , s , θ ) | } .
ϕ ( u ) = tan 1 { [ W ( u ) ridge ] [ W ( u ) ridge ] } ,
Δ ψ ^ s i + 1 + Δ ψ ^ s i = | ω 0 | s i | ω 0 | s i + 1 .
q = s i + 1 s i = | ω 0 | + Δ ψ ^ | ω 0 | Δ ψ ^ .
ω c = ω 0 s .
q = s i + 1 s i = 2 π + 2 2 σ 2 π 2 2 σ .
Δ θ = 2 c arcsin ( Δ ψ ^ / s | ω 0 | / s ) = 2 c arcsin ( Δ ψ ^ | ω 0 | ) ,
I ( x ) = A cos ( 2 π · f ( x ) ) + N ( x ) .
f ( x ) = f 0 + p 2 x , x [ 0 , 511 ] and x Z , k R

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