Abstract

We present a novel ridge extraction algorithm for use with the two-dimensional continuous wavelet transform to extract the phase information from a fringe pattern. A cost function is employed for the detection of the ridge. The results of the proposed algorithm on simulated and real fringe patterns are illustrated. Moreover, the proposed algorithm outperforms the maximum ridge extraction algorithm and it is found to be robust and reliable.

© 2007 Optical Society of America

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References

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  1. 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]
  2. H. Zeng and J. Zhong, "Local frequency and phase analysis of interferogram," Proc. SPIE 6150, 61503S (2006).
    [CrossRef]
  3. 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]
  4. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform," Opt. Commun. 266, 482-489 (2006).
    [CrossRef]
  5. R. A. Carmona, W. L. Hwang, and B. Torresani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
    [CrossRef]
  6. L. R. Watkins, "Phase recovery from fringe patterns using the continuous wavelet transform," Opt. Lasers Eng. 45, 298-303 (2007).
    [CrossRef]
  7. B. E. N. Delprat, P. Gillemani, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequency," IEEE Trans. Inf. Theory 38, 644-664 (1992).
    [CrossRef]
  8. H. Liu, A. N. Cartwright, and C. Basaran, "Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms," Appl. Opt. 43, 850-857 (2004).
    [CrossRef] [PubMed]
  9. A. Dursun, S. Ozder, and F. N. Ecevit, "Continuous wavelet transform analysis of projected fringe patterns," Meas. Sci. Technol. 15, 1768-1772 (2004).
    [CrossRef]
  10. F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
    [CrossRef]
  11. Yet Another Wavelet Toolbox (YAWTb) home page (accessed in April 2007). http://www.fyma.ucl.ac.be/projects/yawtb/.
  12. K. Itoh, "Analysis of the phase unwrapping algorithm," Appl. Opt. 21, 2470-2471 (1982).
    [CrossRef] [PubMed]

2007 (1)

L. R. Watkins, "Phase recovery from fringe patterns using the continuous wavelet transform," Opt. Lasers Eng. 45, 298-303 (2007).
[CrossRef]

2006 (3)

H. Zeng and J. Zhong, "Local frequency and phase analysis of interferogram," Proc. SPIE 6150, 61503S (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]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform," Opt. Commun. 266, 482-489 (2006).
[CrossRef]

2004 (3)

2000 (1)

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

1997 (1)

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

1992 (1)

B. E. N. Delprat, P. Gillemani, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequency," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

1982 (1)

Appl. Opt. (4)

IEEE Trans. Inf. Theory (1)

B. E. N. Delprat, P. Gillemani, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequency," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

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

Meas. Sci. Technol. (1)

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

Opt. Commun. (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform," Opt. Commun. 266, 482-489 (2006).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lasers Eng. (1)

L. R. Watkins, "Phase recovery from fringe patterns using the continuous wavelet transform," Opt. Lasers Eng. 45, 298-303 (2007).
[CrossRef]

Proc. SPIE (1)

H. Zeng and J. Zhong, "Local frequency and phase analysis of interferogram," Proc. SPIE 6150, 61503S (2006).
[CrossRef]

Other (1)

Yet Another Wavelet Toolbox (YAWTb) home page (accessed in April 2007). http://www.fyma.ucl.ac.be/projects/yawtb/.

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Figures (18)

Fig. 1
Fig. 1

Fan mother wavelet (a) real part, (b) imaginary part, and (c) their Fourier transform.

Fig. 2
Fig. 2

Illustration of a four-dimensional array produced by 2D-CWT.

Fig. 3
Fig. 3

(a) Noise-free simulated object and (b) its fringe pattern.

Fig. 4
Fig. 4

Modulus of 1D-CWT for row 120 of the fringe pattern shown in Fig. 3(b) in the presence of noise.

Fig. 5
Fig. 5

(Color online) (a) Column 50 of the modulus array shown in Fig. 4 and (b) candidate ridge points.

Fig. 6
Fig. 6

Illustrative example of candidate ridge points with the optimal ridge of the CWT shown as black arrows.

Fig. 7
Fig. 7

(Color online) Illustration of constructing three ridges matrices of 2D-CWT.

Fig. 8
Fig. 8

(Color online) Example of three extracted ridges from three matrices.

Fig. 9
Fig. 9

Noisy simulated fringe pattern.

Fig. 10
Fig. 10

Unwrapped phase of the noisy fringe pattern shown in Fig. 9 using maximum 2D method.

Fig. 11
Fig. 11

Unwrapped phase of the noisy fringe pattern shown in Fig. 9 using cost 2D method.

Fig. 12
Fig. 12

Unwrapped phase of the noisy fringe pattern shown in Fig. 9 using cost 1D method.

Fig. 13
Fig. 13

Real fringe pattern.

Fig. 14
Fig. 14

Unwrapped phase of the real fringe pattern shown in Fig. 13 using maximum 2D method.

Fig. 15
Fig. 15

Unwrapped phase, shown inverted, of the real fringe pattern shown in Fig. 13 using the cost 2D method.

Fig. 16
Fig. 16

Unwrapped phase of the real fringe pattern shown in Fig. 13 using cost 1D method.

Fig. 17
Fig. 17

Unwrapped phase of the noisy fringe pattern shown in Fig. 9 using FT.

Fig. 18
Fig. 18

Unwrapped phase of the real fringe pattern shown in Fig. 13 using FT.

Equations (6)

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ψ F ( x , y ) = Σ j = 0 N θ 1 exp ( i k o ( x  cos   θ j + y   sin   θ j ) ) × exp ( 1 2 x 2 + y 2 ) ,
S ( a , b , s , θ ) = s 1 ψ F ( x a s , y b s , r θ ) × f ( x , y ) d x d y .
ϕ ( x , y ) = 3 * ( 1 x ) 2 *  exp ( x 2 ( y + 1 ) 2 ) 10 * ( x 5 x 3 y 5 ) * exp ( x 2 y 2 ) 1 3 *  exp ( ( x + 1 ) 2 y 2 ) ,
I ( x , y ) = 0.3 ϕ ( x , y ) + cos ( 2 π f o x + ϕ ( x , y ) ) + N O I S E ,
C o s t [ ϕ ( b ) , b ] = C o b | S [ ϕ ( b ) , b ] | 2 d b + C l b | ϕ ( b ) b | 2 d b ,
C o s t = Σ b = 2 W { C o | [ S ( ϕ ( b ) , b ) ] | 2 + C l ϕ ( b ) ϕ ( b 1 ) 2 } ,

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