Abstract

The applicability of the wavelet-transform profilometry is examined in detail. The wavelet-ridge-based phase demodulation is an integral operation of the fringe signal in the spatial domain. The accuracy of the phase demodulation is related to the local linearity of the phase modulated by the object surface. We present a more robust applicability condition which is based on the evaluation of the local linearity. Since high carrier frequency leads to the phase demodulation integral in a narrow interval and the narrow interval results in the high local linearity of modulated phase, we propose to increase the carrier fringe frequency to improve the applicability of the wavelet-transform profilometry and the measurement accuracy. The numerical simulations and the experiment are presented.

© 2013 OSA

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2012 (1)

L. Watkins, “Review of fringe pattern phase recovery using the 1-D and 2-D continuous wavelet transforms,” Opt. Lasers Eng.50(8), 1015–1022 (2012).
[CrossRef]

2009 (3)

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett.34(8), 1210–1212 (2009).
[CrossRef] [PubMed]

G. Zhou, Z. Li, C. Wang, and Y. Shi, “A novel method for human expression rapid reconstruction,” Tsinghua Sci. Technol.14, 62–65 (2009).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

2007 (4)

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

L. Chen and Y. Chang, “High accuracy confocal full-field 3-D surface profilometry for micro lenses using a digital fringe projection strategy,” Key Eng. Mater.113, 364–366 (2007).

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

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt.46(24), 6120–6126 (2007).
[CrossRef] [PubMed]

2006 (4)

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]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng.45(12), 123601 (2006).
[CrossRef]

K. Genovese and C. Pappalettere, “Whole 3D shape reconstruction of vascular segments under pressure via fringe projection techniques,” Opt. Lasers Eng.44(12), 1311–1323 (2006).
[CrossRef]

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).

2005 (1)

2004 (3)

2003 (1)

2001 (1)

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

2000 (1)

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng.39(1), 10–22 (2000).
[CrossRef]

1999 (2)

J. Villa and M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng.31(4), 279–288 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett.24(13), 905–907 (1999).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

R. A. Carmona, W. L. Hwang, and B. Torrésani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on45(10), 2586–2590 (1997).
[CrossRef]

1996 (1)

W. Chen, Y. Tan, and H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng.25(2-3), 111–120 (1996).
[CrossRef]

1993 (1)

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98(1-3), 141–150 (1993).
[CrossRef]

1992 (1)

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

1990 (1)

M. Takeda, “Spatial–carrier fringe–pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol.1(2), 79–99 (1990).
[CrossRef]

1983 (1)

Abid, A.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

Abid, A. Z.

Asundi, A.

Barnes, T. H.

Bian, X.

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

Brown, G.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng.39(1), 10–22 (2000).
[CrossRef]

Burton, D. R.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt.46(24), 6120–6126 (2007).
[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]

Cai, L. Z.

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torrésani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on45(10), 2586–2590 (1997).
[CrossRef]

Chang, Y.

L. Chen and Y. Chang, “High accuracy confocal full-field 3-D surface profilometry for micro lenses using a digital fringe projection strategy,” Key Eng. Mater.113, 364–366 (2007).

Chen, F.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng.39(1), 10–22 (2000).
[CrossRef]

Chen, L.

L. Chen and Y. Chang, “High accuracy confocal full-field 3-D surface profilometry for micro lenses using a digital fringe projection strategy,” Key Eng. Mater.113, 364–366 (2007).

Chen, W.

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).

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

W. Chen, Y. Tan, and H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng.25(2-3), 111–120 (1996).
[CrossRef]

Gdeisat, M. A.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt.46(24), 6120–6126 (2007).
[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]

Genovese, K.

K. Genovese and C. Pappalettere, “Whole 3D shape reconstruction of vascular segments under pressure via fringe projection techniques,” Opt. Lasers Eng.44(12), 1311–1323 (2006).
[CrossRef]

Guo, J. P.

Huang, P. S.

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng.45(12), 123601 (2006).
[CrossRef]

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torrésani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on45(10), 2586–2590 (1997).
[CrossRef]

Kemao, Q.

Lalor, M. J.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt.46(24), 6120–6126 (2007).
[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]

Leizerson, I.

Li, A. M.

Li, Z.

G. Zhou, Z. Li, C. Wang, and Y. Shi, “A novel method for human expression rapid reconstruction,” Tsinghua Sci. Technol.14, 62–65 (2009).
[CrossRef]

Lilley, F.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt.46(24), 6120–6126 (2007).
[CrossRef] [PubMed]

Lipson, S. G.

Meng, X. F.

Moore, C.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

Mutoh, K.

Pappalettere, C.

K. Genovese and C. Pappalettere, “Whole 3D shape reconstruction of vascular segments under pressure via fringe projection techniques,” Opt. Lasers Eng.44(12), 1311–1323 (2006).
[CrossRef]

Peng, X.

Qudeisat, M.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

Servin, M.

J. Villa and M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng.31(4), 279–288 (1999).
[CrossRef]

Shi, Y.

G. Zhou, Z. Li, C. Wang, and Y. Shi, “A novel method for human expression rapid reconstruction,” Tsinghua Sci. Technol.14, 62–65 (2009).
[CrossRef]

Song, M.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng.39(1), 10–22 (2000).
[CrossRef]

Su, X.

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).

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

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98(1-3), 141–150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

Sun, S.

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

Takeda, M.

M. Takeda, “Spatial–carrier fringe–pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol.1(2), 79–99 (1990).
[CrossRef]

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]

Tan, S. M.

Tan, Y.

W. Chen, Y. Tan, and H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng.25(2-3), 111–120 (1996).
[CrossRef]

Torrésani, B.

R. A. Carmona, W. L. Hwang, and B. Torrésani, “Characterization of signals by the ridges of their wavelet transforms,” Signal Processing, IEEE Transactions on45(10), 2586–2590 (1997).
[CrossRef]

Vander, R.

Villa, J.

J. Villa and M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng.31(4), 279–288 (1999).
[CrossRef]

von Bally, G.

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98(1-3), 141–150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

Vukicevic, D.

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98(1-3), 141–150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

Wang, C.

G. Zhou, Z. Li, C. Wang, and Y. Shi, “A novel method for human expression rapid reconstruction,” Tsinghua Sci. Technol.14, 62–65 (2009).
[CrossRef]

Wang, Y. R.

Watkins, L.

L. Watkins, “Review of fringe pattern phase recovery using the 1-D and 2-D continuous wavelet transforms,” Opt. Lasers Eng.50(8), 1015–1022 (2012).
[CrossRef]

Watkins, L. R.

Weng, J.

Wensen, Z.

Zeng, H.

Zhang, S.

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng.45(12), 123601 (2006).
[CrossRef]

Zhao, H.

W. Chen, Y. Tan, and H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng.25(2-3), 111–120 (1996).
[CrossRef]

Zheng, S.

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).

Zhong, J.

Zhou, G.

G. Zhou, Z. Li, C. Wang, and Y. Shi, “A novel method for human expression rapid reconstruction,” Tsinghua Sci. Technol.14, 62–65 (2009).
[CrossRef]

Zhou, W.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

Appl. Opt. (8)

Ind. Metrol. (1)

M. Takeda, “Spatial–carrier fringe–pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol.1(2), 79–99 (1990).
[CrossRef]

J. Mod. Opt. (1)

W. Chen, S. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt.54(18), 2747–2762 (2007).
[CrossRef]

Key Eng. Mater. (1)

L. Chen and Y. Chang, “High accuracy confocal full-field 3-D surface profilometry for micro lenses using a digital fringe projection strategy,” Key Eng. Mater.113, 364–366 (2007).

Opt. Commun. (2)

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun.94(6), 561–573 (1992).
[CrossRef]

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98(1-3), 141–150 (1993).
[CrossRef]

Opt. Eng. (4)

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]

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng.39(1), 10–22 (2000).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng.45(12), 123601 (2006).
[CrossRef]

Opt. Lasers Eng. (6)

K. Genovese and C. Pappalettere, “Whole 3D shape reconstruction of vascular segments under pressure via fringe projection techniques,” Opt. Lasers Eng.44(12), 1311–1323 (2006).
[CrossRef]

J. Villa and M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng.31(4), 279–288 (1999).
[CrossRef]

W. Chen, Y. Tan, and H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng.25(2-3), 111–120 (1996).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges, and suggested developments,” Opt. Lasers Eng.47(12), 1348–1361 (2009).
[CrossRef]

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

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Figures (25)

Fig. 1
Fig. 1

Optical geometry of fringe projection profilometry.

Fig. 2
Fig. 2

Illustration of the obtained modulated phase. Each point of the modulated phase (c) is obtained by computing an integral of the signal (a) and the analytic Morlet wavelet (b) within the interval that is the support of the scaled wavelet [ LB,UB ] .

Fig. 3
Fig. 3

Illustration of local linearity. The height distribution of object surface (d), resulting phase (b), resulting modulated phase (c) and the Gaussian function (a).

Fig. 4
Fig. 4

Chart of relationship between carrier frequency f 0 and local nonlinearity Δ S h .

Fig. 5
Fig. 5

Fourier spectrums of deformed fringe patterns.

Fig. 6
Fig. 6

Height distribution of object for simulation.

Fig. 7
Fig. 7

Deformed fringes patterns at 256th row.

Fig. 8
Fig. 8

Distributions of ϕ ( x ) (a) and of ϕ ( x ) (b).

Fig. 9
Fig. 9

Distributions of η ϕ (a) and of Δ S h (x) (b).

Fig. 10
Fig. 10

Comparison of the demodulated height with different carrier frequencies.

Fig. 11
Fig. 11

Comparison of demodulated absolute height error.

Fig. 12
Fig. 12

Distributions of modulated phase (a) with its first derivative (b) and second derivative (c).

Fig. 13
Fig. 13

Comparison of demodulated heights (a) and absolute height errors (b).

Fig. 14
Fig. 14

Distributions of η ϕ (a) and Δ S h (x) (b).

Fig. 15
Fig. 15

Distributions of modulated phase (a) with its first derivative (b) and second derivative (c).

Fig. 16
Fig. 16

Comparison of demodulated heights (a) and absolute height errors (b).

Fig. 17
Fig. 17

Distributions of η ϕ (a) and Δ S h (x) (b).

Fig. 18
Fig. 18

Captured deformed fringe pattern for WTP. Carrier frequencies of the fringe pattern are 0.045 cycle/pixel (a), 0.06 cycle/pixel (b) and 0.075 cycle/pixel (c), respectively. Red line in (a) indicates the 518th column of the fringe pattern.

Fig. 19
Fig. 19

Set of deformed fringe pattern for phase demodulation by four-step PSP.

Fig. 20
Fig. 20

Distributions of intensity (a), wavelet coefficients amplitudes (b) and wavelet coefficients phase map (c) corresponding to the 518th-column fringe with the carrier frequency f 0 =0.075 cycle/pixel .

Fig. 21
Fig. 21

Comparison of the demodulated height at the 518th column.

Fig. 22
Fig. 22

Comparison of demodulated height at the 518th column (corresponding to Local 1, y[ 560,610 ] in Fig. 21).

Fig. 23
Fig. 23

Comparison of demodulated height at the 518th column (corresponding to Local 2, y[ 760,830 ] in Fig. 21).

Fig. 24
Fig. 24

2-D distribution of the wrapped phase Δϕ with the carrier frequency f 0 =0.075 cycle/pixel .

Fig. 25
Fig. 25

3-D distribution of demodulated height with the carrier frequency f 0 =0.075 cycle/pixel .

Tables (1)

Tables Icon

Table 1 Comparison of Height Errors Demodulated by WTPa

Equations (37)

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I( x )= I 0 ( x )+A( x )cos[ ϕ( x ) ],
ϕ( x )=2π f 0 x+Δϕ( x ).
Δϕ( x,y )= 2π f 0 h( x,y )d l 0 h( x,y ) ,
Δϕ( x,y )= 2π f 0 h( x,y )d l 0 .
W( a,b )= I( x ) 1 a ψ * ( xb a )dx ,
A w ( a,b )= { real[ W( a,b ) ] } 2 + { imag[ W( a,b ) ] } 2 ,
R( a r (b),b )=max[ A w ( a,b ) ],
ϕ wr ( a r (b),b )=arg[ W( a r (b),b ) ],
ψ( x )= 1 π 4 exp( j2πx )exp( x 2 2 ).
η A ( x )= ( 2π ) 2 | ϕ ( x ) | | A ( x ) | | A( x ) | 1
η ϕ ( x )= ( 2π ) 2 | ϕ ( x ) | | ϕ ( x ) | 2 1,
ϕ( b ) ϕ wr ( a r ( b ),b )=arg[ W( a r ( b ),b ) ].
ϕ ( x )=2π f 0 +Δ ϕ ( x )
ϕ ( x )=Δ ϕ ( x ),
ϕ ( x )=2π f 0 2π f 0 d l 0 h ( x )
ϕ ( x )= 2π f 0 d l 0 h ( x ).
ϕ( b )arg[ W( a r ( b ),b ) ] =arg{ 1 a r ( b ) π 4 + I( x )exp[ j2π( xb a r ( b ) ) ]exp[ 1 2 ( xb a r ( b ) ) 2 ]dx }.
ϕ( b )arg[ W( a r ( b ),b ) ] =arg{ 1 a r ( b ) π 4 LB UB I( x )exp[ j2π( xb a r ( b ) ) ]exp[ 1 2 ( xb a r ( b ) ) 2 ]dx }.
ΔS=| LB UB [ ϕ( x )L( x ) ]dx |, with L( x )=[ ϕ( UB )ϕ( LB ) UBLB ]( xUB )+ϕ( UB ),
Δ S w ( b )=| LB UB g( xb a r ( b ) )[ ϕ( x )L( x ) ]dx |, with g( x )=exp( x 2 2 ),
Δ S w ( b )0.
Δ S w ( b )=| LB UB g( xb a r ( b ) )[ Δϕ( x ) L Δ ( x ) ]dx |, with L Δ ( x )=[ Δϕ( UB )Δϕ( LB ) UBLB ]( xUB )+Δϕ( UB ).
Δ S w ( b )=| 2π f 0 d LB UB g( xb a r ( b ) ){ h( x ) l 0 h( x ) { [ h( UB ) l 0 h( UB ) + h( LB ) l 0 h( LB ) ]( xUB UBLB ) h( UB ) l 0 h( UB ) } }dx |.
Δ S w ( b )=| 2π f 0 d l 0 LB UB g( xb a r ( b ) )[ h( x ) L h ( x ) ]dx |, with L h ( x )=[ h( UB )h( LB ) UBLB ]( xUB )+h( UB ).
Δ S h ( b )= Δ S w ( b ) f 0 =| 2πd l 0 { LB UB g( xb a r ( b ) )[ h( x ) L h ( x ) ]dx } |
Δ S h (b)0.
a r = 1 f inst .
f inst ( x )= 1 2π ϕ ( x ).
f inst ( x )= f 0 + 1 2π Δ ϕ ( x ).
| ϕ ( x ) | 2 = | 2π f 0 [ 1 h ( x )d l 0 ] | 2 f 0 2
| ϕ ( x ) |=| 2π f 0 h ( x )d l 0 | f 0 .
η ϕ 1 f 0 .
f 0 < f sample B 2 ,
h( x,y )= 1 40 peaks( 6x3,6y3 ),
peaks( x,y )=3 ( 1x ) 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 ],
h( x )= 1 3 { exp[ ( 6x3 ) 2 ]1 }exp[ ( 6x3 ) 2 ]sgn( x+0.5 ),
h( x )=| 0.36( x0.5 ) |,

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