Abstract

A generalized Fourier analysis, by use of an adaptive multiscale windowed Fourier transform (AWFT), has been presented for the phase retrieval of fringe patterns. The Fourier transform method can be considered as a special case of AWFT method with a maximum window. The instantaneous frequency of the local signal is introduced to estimate whether the condition for separating the first spectrum component is satisfied for the phase retrieval of fringe patterns. The adaptive window width for this algorithm is determined by the length of the local stationary fringe pattern in order to balance the frequency and space resolution. The local stationary length of fringe pattern is defined as the signal satisfying the condition that whose first spectrum component is separated from all the other spectra within the local spatial area. In comparison with Fourier transform, fixed windowed Fourier transform and wavelet transform in numerical simulation and experiment, the adaptive multiscale windowed Fourier transform can present more accurate results of phase retrieval.

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2010 (1)

M. Takeda, “Measurements of extreme physical phenomena by Fourier fringe analysis,” International conference on advanced phase measurement methods in optics and imaging,” AIP Conf. Proc. 1236, 445–448 (2010).
[CrossRef]

2009 (1)

2008 (1)

2007 (2)

2006 (1)

2005 (1)

2004 (4)

2001 (3)

2000 (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11(10), 1480–1492 (2000).
[CrossRef]

1999 (1)

1996 (1)

1992 (1)

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

1988 (1)

R. J. Green, J. G. Walker, and D. W. Robinson, “Investigation of the Fourier-transform method of fringe pattern analysis,” Opt. Lasers Eng. 8(1), 29–44 (1988).
[CrossRef]

1985 (1)

1983 (1)

1982 (1)

Barnes, T. H.

Bernini, M. B.

Bone, D. J.

Bruno, L.

Burton, D. R.

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(34), 8722–8732 (2006).
[CrossRef] [PubMed]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11(10), 1480–1492 (2000).
[CrossRef]

Chen, W.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

Delprat, N.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Escudie, B.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Federico, A.

Gao, W.

Gdeisat, M. A.

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(34), 8722–8732 (2006).
[CrossRef] [PubMed]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11(10), 1480–1492 (2000).
[CrossRef]

Green, R. J.

R. J. Green, J. G. Walker, and D. W. Robinson, “Investigation of the Fourier-transform method of fringe pattern analysis,” Opt. Lasers Eng. 8(1), 29–44 (1988).
[CrossRef]

Guillemain, P.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Ina, H.

Kaufmann, G. H.

Kemao, Q.

Kobayashi, S.

Kronland-Martinet, R.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Lalor, M. J.

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(34), 8722–8732 (2006).
[CrossRef] [PubMed]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11(10), 1480–1492 (2000).
[CrossRef]

Larkin, K. G.

Mutoh, K.

Nugent, K. A.

Oldfield, M. A.

Robin, E.

Robinson, D. W.

R. J. Green, J. G. Walker, and D. W. Robinson, “Investigation of the Fourier-transform method of fringe pattern analysis,” Opt. Lasers Eng. 8(1), 29–44 (1988).
[CrossRef]

Su, X.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

Surrel, Y.

Takeda, M.

Tan, S. M.

Tchamitchian, P.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Torresani, B.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

Valle, V.

Walker, J. G.

R. J. Green, J. G. Walker, and D. W. Robinson, “Investigation of the Fourier-transform method of fringe pattern analysis,” Opt. Lasers Eng. 8(1), 29–44 (1988).
[CrossRef]

Wang, H.

Watkins, L. R.

Weng, J.

Zeng, H.

Zhong, J.

AIP Conf. Proc. (1)

M. Takeda, “Measurements of extreme physical phenomena by Fourier fringe analysis,” International conference on advanced phase measurement methods in optics and imaging,” AIP Conf. Proc. 1236, 445–448 (2010).
[CrossRef]

Appl. Opt. (10)

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
[CrossRef] [PubMed]

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985).
[CrossRef] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[CrossRef] [PubMed]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[CrossRef] [PubMed]

E. Robin and V. Valle, “Phase demodulation from a single fringe pattern based on a correlation technique,” Appl. Opt. 43(22), 4355–4361 (2004).
[CrossRef] [PubMed]

J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43(26), 4993–4998 (2004).
[CrossRef] [PubMed]

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(34), 8722–8732 (2006).
[CrossRef] [PubMed]

J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46(14), 2670–2675 (2007).
[CrossRef] [PubMed]

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47(29), 5408–5419 (2008).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38(2), 644–664 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11(10), 1480–1492 (2000).
[CrossRef]

Opt. Eng. (1)

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43(4), 895–899 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

R. J. Green, J. G. Walker, and D. W. Robinson, “Investigation of the Fourier-transform method of fringe pattern analysis,” Opt. Lasers Eng. 8(1), 29–44 (1988).
[CrossRef]

Opt. Lett. (2)

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Figures (23)

Fig. 1
Fig. 1

The 1D modulated (a) sinusoidal and (b) non-sinusoidal fringe patterns of 512 pixels.

Fig. 2
Fig. 2

The FT spectra of (a) sinusoidal and (b) non-sinusoidal fringe patterns without the zero component.

Fig. 3
Fig. 3

The spectra of the two local (a) sinusoidal and (b) non-sinusoidal fringe pattern center at 128th and 375th pixel.

Fig. 4
Fig. 4

Condition for separating the first spectrum component.

Fig. 5
Fig. 5

The spectrum of the fringe pattern with a = 30 and a = 10 .

Fig. 6
Fig. 6

(a) The noisy original and (b) modulated fringe pattern

Fig. 7
Fig. 7

The spectrum..

Fig. 8
Fig. 8

The instantaneous frequency.

Fig. 9
Fig. 9

The local stationary length.

Fig. 10
Fig. 10

The scale factor of the Gaussian window.

Fig. 11
Fig. 11

The local fringe pattern center at 50th pixel and 380th pixel selected by Gaussian window.

Fig. 12
Fig. 12

The spectra of the local fringe pattern center at (a) 50th pixel and (b) 380th pixel.

Fig. 13
Fig. 13

The unwrapped demodulated phase by different methods.

Fig. 14
Fig. 14

The errors by different methods. (a) the AWFT and FT; (b) the AWFT and WFT with a = 30 ; (c) the AWFT and MWFT with threshold 0.005; (d) the AWFT and WT.

Fig. 15
Fig. 15

The modulated fringe pattern.

Fig. 16
Fig. 16

The spectrum of the 100th row.

Fig. 17
Fig. 17

The instantaneous frequency of the 100th row.

Fig. 18
Fig. 18

The scale factor of the Gaussian window of the 100th row

Fig. 19
Fig. 19

The local fringe pattern of the 100th row center at 100th pixel and 350th pixel selected by Gaussian window.

Fig. 20
Fig. 20

The spectra of the local fringe pattern of the 100th row center at 100th pixel and 350th pixel.

Fig. 21
Fig. 21

The demodulated wrapphase by the (a) FT, (b) WT and (c) AWFT methods.

Fig. 22
Fig. 22

The wrapphase of the 100th row by different methods.

Fig. 23
Fig. 23

The unwrapped demodulated phase by the (a) FT, (b) WT, (c) AWFT and (d) phase shifting methods

Equations (31)

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I ( x ) = A ( x ) + B ( x ) cos [ 2 π f 0 x + φ ( x ) ]
I ( x ) = r ( x ) n = + A n exp { j [ 2 π n f 0 x + n φ ( x ) ] }
F ( f ) = I ( x ) exp ( j 2 π f x ) d x
I 1 ( x ) = F ( f 0 ) exp ( j 2 π f x ) d f
φ ( x ) = arctan Im [ I 1 ( x ) ] Re [ I 1 ( x ) ] 2 π f 0 x
{ f 0 + 1 2 π φ ( x ) max < n f 0 + n 2 π φ ( x ) min         ( n = 2 , 3 , ) f b < f 0 + 1 2 π φ ( x ) min
f 0 + 1 2 π φ ( x ) max < 2 f 0 + 2 2 π φ ( x ) min         ( n = 2 )
f i n s t ( x ) = f 0 + 1 2 π φ ( x )
{ ( f i n s t ) max = f 0 + 1 2 π φ ( x ) max ( f i n s t ) min = f 0 + 1 2 π φ ( x ) min
( f i n s t ) max < 2 ( f i n s t ) min
W F T a 0 b ( f ) = [ I ( x ) exp ( j 2 π f x ) ] G a 0 b ( x ) d x
G a 0 b ( x ) = | a 0 | 1 g ( x b a 0 ) ( a 0 > 0 )
g ( x ) = 1 2 π exp ( x 2 2 )
δ x δ f 1 / 4 π
A W F T a b ( f ) = [ I ( x ) exp ( j 2 π f x ) ] G a b ( x ) d x
G a b ( x ) d x = G a b ( x ) d b = 1
A W F T a b ( f ) d b = [ I ( x ) exp ( j 2 π f x ) ] G a b ( x ) d x d b                                               = { I ( x ) exp ( j 2 π f x ) G a b ( x ) d b } d x                                               = I ( x ) exp ( j 2 π f x ) d x                                               = F ( f )
F ( f 0 ) = A W F T a b ( f 0 ) d b
L s i g n a l ( x ) = 2 ( Δ x ) max
L w i n d o w ( x ) = 4 a
a = L s i g n a l ( x ) 4
f b < ( f i n s t ) min
A W F T a b ( f ) = [ I ( x ) exp ( j 2 π f x ) ] G a b ( x ) d x                             = 1 2 π a r ( x ) A 0 exp [ ( x b ) 2 2 a 2 ] exp ( j 2 π f x ) d x                                 + 1 2 π a r ( x ) { n = n 0 + A n exp [ j 2 π n f 0 x + j n φ ( x ) ] } exp [ ( x b ) 2 2 a 2 ] exp ( j 2 π f x ) d x
1 2 π a r ( x ) A 0 exp [ ( x b ) 2 2 a 2 ] exp ( j 2 π f x ) d x = r ( x ) A 0 exp ( j 2 π f b ) exp ( 2 π 2 a 2 f 2 )
| F ( f ) | n = 0 2 = [ r ( x ) A 0 ] 2 exp ( 4 π 2 a 2 f 2 )
f b = 1 2 π a < ( f i n s t ) min
a > 1 2 π ( f i n s t ) min
{ a = L s i g n a l ( x ) 4 ,                   i f L s i g n a l ( x ) > 2 2 π ( f i n s t ) min a = 1 2 π ( f i n s t ) min         i f L s i g n a l ( x ) 2 2 π ( f i n s t ) min
I 0 ( x ) = 255 × n = 0 50 A n exp ( j 2 π n f 0 x ) + n o i s e
I ( x ) = 255 × n = 0 50 A n exp { j [ 2 π n f 0 x + n φ ( x ) ] } + n o i s e
φ ( x ) = 100 π 16 × [ 1 cos ( 2 π x 512 ) ]

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