Abstract

We introduce an adaptive window Wigner–Ville-distribution-based method to directly estimate the phase derivative from a single fringe pattern. In the proposed method, the phase derivative is estimated by using the peak detection of the pseudo-Wigner–Ville distribution for a set of different window lengths. Then the optimal window length is selected from the set by resolving the estimator’s bias variance trade-off, using the intersection of confidence intervals rule. Finally, the phase derivative estimate corresponding to the optimum window is selected. Simulation and experimental results are presented to demonstrate the method’s applicability for the phase derivative estimation.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  2. Y. Surrel, Appl. Opt. 35, 51 (1996).
    [CrossRef] [PubMed]
  3. Q. Kemao, S. H. Soon, and A. Asundi, Appl. Opt. 42, 6504 (2003).
    [CrossRef] [PubMed]
  4. K. Qian, S. H. Soon, and A. Asundi, Opt. Lett. 28, 1657 (2003).
    [CrossRef] [PubMed]
  5. C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
    [CrossRef]
  6. C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
    [CrossRef]
  7. A. Federico and G. H. Kaufmann, Appl. Opt. 42, 7066 (2003).
    [CrossRef] [PubMed]
  8. L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).
  9. V. Katkovnik and L. Stankovic, IEEE Trans. Signal Process. 45, 2147 (1997).
  10. S. C. Sekhar and T. V. Sreenivas, Signal Process. 86, 716 (2006).
    [CrossRef]
  11. J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
    [CrossRef]

2007

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

2006

S. C. Sekhar and T. V. Sreenivas, Signal Process. 86, 716 (2006).
[CrossRef]

2005

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

2003

2001

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

1997

V. Katkovnik and L. Stankovic, IEEE Trans. Signal Process. 45, 2147 (1997).

1996

1982

Antonio Gomez-Pedrero, J.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Asundi, A.

Cohen, L.

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

Federico, A.

Garcia-Botella, A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

He, X. Y.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Ina, H.

Katkovnik, V.

V. Katkovnik and L. Stankovic, IEEE Trans. Signal Process. 45, 2147 (1997).

Kaufmann, G. H.

Kemao, Q.

Kim, T.

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

Kobayashi, S.

Qian, K.

Quan, C.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Quiroga, J. A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

Sekhar, S. C.

S. C. Sekhar and T. V. Sreenivas, Signal Process. 86, 716 (2006).
[CrossRef]

Soon, S. H.

Sreenivas, T. V.

S. C. Sekhar and T. V. Sreenivas, Signal Process. 86, 716 (2006).
[CrossRef]

Stankovic, L.

V. Katkovnik and L. Stankovic, IEEE Trans. Signal Process. 45, 2147 (1997).

Sun, W.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Surrel, Y.

Takeda, M.

Tay, C. J.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Appl. Opt.

Exp. Mech.

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

IEEE Trans. Signal Process.

V. Katkovnik and L. Stankovic, IEEE Trans. Signal Process. 45, 2147 (1997).

J. Opt. Soc. Am.

Opt. Commun.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Opt. Lett.

Signal Process.

S. C. Sekhar and T. V. Sreenivas, Signal Process. 86, 716 (2006).
[CrossRef]

Other

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) Probability density function for phase derivative estimate. Original phase derivative ω and estimated phase derivative ω ̂ in radians/pixel for (b) window length h s = 2 , (c) window length h s = 256 , (d) adaptive window.

Fig. 2
Fig. 2

(a) Simulated fringe pattern ( 256 × 256 ) at SNR of 30 dB . (b) Phase derivative ω x of original fringe pattern along x direction. (c) Original phase derivative ω x and estimated phase derivative ω ̂ x along row y = 129 . (d) Estimated phase derivative ω ̂ x for entire fringe pattern. (e) Cosine fringes of estimated phase derivative.

Fig. 3
Fig. 3

(a) Experimentally recorded fringe pattern in DHI ( 256 × 256 ) . (b) Estimated phase derivative in radians/pixel. (c) Cosine fringes of estimated phase derivative.

Equations (17)

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

I ( x , y ) = A ( x , y ) exp [ j ϕ ( x , y ) ] + η ( x , y ) ,
I ( x ) = A exp [ j ϕ ( x ) ] + η ( x ) ,
ω ( x ) = ϕ ( x ) x .
W ( x , ω ) = τ = w h ( τ ) I ( x + τ ) I * ( x τ ) exp ( j 2 ω τ ) ,
ω ̂ h ( x ) = arg max ω W ( x , ω ) .
B ( h ) = s = 1 h 2 s b s [ ω ( x ) ] 2 s ,
σ 2 ( h ) = σ ϵ 2 2 | A | 2 ( 1 + σ ϵ 2 2 | A | 2 ) E 1 h 3 F 1 2 ,
MSE ( h ) = B 2 ( h ) + σ 2 ( h ) ,
E 1 = 1 2 1 2 w 2 ( x ) x 2 d x ,
F 1 = 1 2 1 2 w ( x ) x 2 d x ,
b s = 1 ( 2 s + 1 ) ! F 1 1 2 1 2 w ( x ) x 2 s + 2 d x ,
A ̂ 2 = { | 2 [ 1 N x = 1 N | I ( x ) | 2 ] 2 1 N x = 1 N | I ( x ) | 4 | } 1 2 ,
σ ̂ ϵ 2 = | x = 1 N | I ( x ) | 2 N { | 2 [ x = 1 N | I ( x ) | 2 N ] 2 x = 1 N | I ( x ) | 4 N | } 1 2 | .
| ω ̂ h ( x ) ω ( x ) | | bias | k σ ( h ) .
| bias | k σ ( h ) .
| ω ̂ h ( x ) ω ( x ) | 2 k σ ( h )
ω ̂ h ( x ) 2 k σ ( h ) ω ( x ) ω ̂ h ( x ) + 2 k σ ( h ) .

Metrics