Abstract

For many phase extraction algorithms, a priori knowledge of a fringe-pattern density distribution is beneficial for later processing. A fringe-density estimation method based on a continuous wavelet transform (CWT) is proposed. For a one-dimensional signal the instantaneous frequency detected at the CWT ridge is directly adopted as a measure of the local fringe density. For a two-dimensional signal the instantaneous frequency components in both the x and the y directions are detected. Their reliability is evaluated by the CWT coefficient magnitude, based on which an approximate density value is given. The capability for noise reduction and the accuracy of the method are discussed.

© 2005 Optical Society of America

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References

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    [CrossRef]
  14. H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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2004

2003

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

2002

2001

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

O. Marklund, “Robust fringe density and direction estimation in noisy phase maps,” J. Opt. Soc. Am. A 18, 2717–2727 (2001).
[CrossRef]

1998

1997

1996

A. Davila, G. H. Kaufmann, D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35, 3549–3554 (1996).
[CrossRef]

B. Strobel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
[CrossRef] [PubMed]

1994

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

1986

1983

1966

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Asundi, A.

Basaran, C.

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

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Burton, D. R.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Carré, P.

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Cartwright, A. N.

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

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Colonna de Lega, X.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds.(Institute of Physics, Bristol, UK, 1993), pp. 94–140.

Cuevas, F. J.

Davila, A.

A. Davila, G. H. Kaufmann, D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35, 3549–3554 (1996).
[CrossRef]

Franze, B.

Haible, P.

Jacquot, P.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Joenathan, C.

Kaufmann, G. H.

A. Davila, G. H. Kaufmann, D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35, 3549–3554 (1996).
[CrossRef]

Kerr, D.

A. Davila, G. H. Kaufmann, D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35, 3549–3554 (1996).
[CrossRef]

Kreis, T.

Lalor, M. I.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Le, X.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Liu, H.

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

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1999).

Marklund, O.

Marroquin, J. L.

Mutoh, K.

Rodriguez-Vera, R.

Servin, M.

Stephenson, P.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Strobel, B.

Takeda, M.

Tao, G.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Tiziani, H. J.

Wang, J.

Yang, Y.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Metrologia

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Opt. Eng.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35, 3549–3554 (1996).
[CrossRef]

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Opt. Laser Technol.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Other

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds.(Institute of Physics, Bristol, UK, 1993), pp. 94–140.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1999).

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Figures (5)

Fig. 1
Fig. 1

(a) Sinusoidal signal with high frequency at the center. (b) CWT magnitude map. Density curves obtained for scale increment steps set at (c) 1.0, (d) 0.2, and (e) 1.0 with a mean filter.

Fig. 2
Fig. 2

(a) Sinusoidal signal with additive noise. (b) CWT magnitude map. Density curves obtained for scale increment steps set at (c) 1.0, (d) 0.2, and (e) 1.0, and a mean filter.

Fig. 3
Fig. 3

(a) Vertical fringe pattern. (b) Density map obtained by use of a weight.

Fig. 4
Fig. 4

Cross sections A–A and B–B in the vertical fringe pattern. (a) Intensities of A–A (dashed curve) and B–B (solid curve). Density curves (b) without weight and (c) with weight.

Fig. 5
Fig. 5

(a) Circular fringe pattern with parabola density distribution. (b) Density map.

Equations (9)

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I = I 0 [ 1 + γ cos ϕ ( x , y ) ] ,
W f ( a , b ) = 1 a + f ( x ) M ( x b a ) d x ,
M ( x ) = exp ( x 2 2 ) exp ( j ω 0 x ) ,
W f ( a r , b ) = 2 π 2 I 0 γ exp ( ± j ω 0 b a r ) ,
a r ( b ) = max { | W f ( a , b ) | , a [ s min , s max ] } ,
d ( b ) = 1 N i = b ( N / 2 ) b + ( N / 2 ) ω 0 a r ( i ) ,
d x ( x , y ) = 1 N 2 i = x ( N / 2 ) x + ( N / 2 ) j = y ( N / 2 ) y + ( N / 2 ) ω 0 a r , x ( i , j ) | W f [ a r , x ( i , j ) , b ] | { | W f [ a r , x ( i , j ) , b ] 2 + | W f [ a r , y ( i , j ) , b ] | 2 } 1 / 2 ,
d y ( x , y ) = 1 N 2 i = x ( N / 2 ) x + ( N / 2 ) j = y ( N / 2 ) y + ( N / 2 ) ω 0 a r , y ( i , j ) | W f [ a r , y ( i , j ) , b ] | { | W f [ a r , x ( i , j ) , b ] | 2 + | W f [ a r , y ( i , j ) , b ] | 2 } 1 / 2 ,
d ( x , y ) = [ d x 2 ( x , y ) + d y 2 ( x , y ) ] 1 / 2 .

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