Abstract

A novel method for windowed Fourier transform (WFT) profilometry is presented. This method is based on improved S-transform. The impact of the second order derivative of the phase (φ(b)) to the ridge of S-transform is derived, and how to estimate this deviation is discussed. An important conclusion that more accurate instantaneous frequency can be obtained after removing this deviation is shown. Thus, an accurate phase map of the fringe pattern is obtained by using the WFT based on the window size map, and this map is related to the instantaneous frequency. The method is compared with the WFT based on the wavelet transform. A numerical simulation and experimental example have shown its validity in practical applications.

© 2012 Optical Society of America

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References

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  1. M. Takeda and K. Mutoh, Appl. Opt. 22, 3977 (1983).
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    [CrossRef]

2012 (2)

2008 (1)

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

2007 (1)

2005 (1)

2004 (1)

1996 (1)

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process 44, 998 (1996).
[CrossRef]

1983 (1)

Chen, W.

Coskun, E.

Da, F.

F. Da and H. Huang, Opt. Commun. 285, 421 (2012).
[CrossRef]

Dursun, A.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

Ecevit, F. N.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

Göktas, H.

Huang, H.

F. Da and H. Huang, Opt. Commun. 285, 421 (2012).
[CrossRef]

Jiang, M.

Kocahan, Ö.

Lowe, R. P.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process 44, 998 (1996).
[CrossRef]

Mansinha, L.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process 44, 998 (1996).
[CrossRef]

Mutoh, K.

Özder, S.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

S. Özder, Ö. Kocahan, E. Coskun, and H. Göktas, Opt. Lett. 32, 591 (2007).
[CrossRef]

Saraç, Z.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

Stockwell, R. G.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process 44, 998 (1996).
[CrossRef]

Takeda, M.

Topkara, H. S.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

Wen, J.

Weng, J.

Zhong, J.

Zhong, M.

Appl. Opt. (3)

IEEE Trans. Signal Process (1)

R. G. Stockwell, L. Mansinha, and R. P. Lowe, IEEE Trans. Signal Process 44, 998 (1996).
[CrossRef]

J. Int. Meas. Confed. (1)

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, J. Int. Meas. Confed. 41, 403 (2008).

Opt. Commun. (1)

F. Da and H. Huang, Opt. Commun. 285, 421 (2012).
[CrossRef]

Opt. Lett. (2)

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Figures (4)

Fig. 1.
Fig. 1.

Ridge of S-transform before and after reducing the influence of φ(b). (a) Ridge of S-transform before reducing the influence of φ(b). (b) Ridge of S-transform after reducing the influence of φ(b).

Fig. 2.
Fig. 2.

Comparison of wrapped peak fringe pattern between two different methods. (a) Fringe pattern of “peak.” (b) 3-D shape of “peak.” (c) Unwrapped phase map of “peak” with deviation. (d) Unwrapped phase map of “peak” by using our method.

Fig. 3.
Fig. 3.

Comparison of unwrapped phase between different methods. (a) Error of unwrapped phase between WWFT and two kinds of SWFT in row 450. (b) Standard deviation of unwrapped phase between WWFT and two kinds of SWFT from row 1 to 900.

Fig. 4.
Fig. 4.

Experiment of the Venus model. (a) Fringe pattern of the Venus model. (b) Right part of the 3-D reconstruction face. (c) Left part of the 3-D reconstruction face.

Equations (14)

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S(b,f)=+g(x)ω(bx,f)exp(i2πfx)dx,
ω(bx,f)=|f|2πexp[f2(bx)22].
ω(bx,f)=+W(α,f)exp[i2πα(bx)]dα,
S(b,f)=+g(x)exp(i2πfx){+W(α,f)exp[i2πα(bx)]dα}dx=F1{G(α+f)·W(α,f)}.
g(x)=I0(x)+I1(x)cos[2πf0x+φ(x)],
φ(x)=φ(b)+φ(b)(xb)+ϕ(b)(xb)22+o(φ(b)).
S(b,f)=S0(b,f)+S1(b,f)+S2(b,f).
S0(b,f)=I0(b)exp(2π2)exp(i2πfb),S1(b,f)=12I1(b)exp{2π2f2[f+f0+φ(b)2π]2}exp{i[2πfb2πf0bφ(b)]},S2(b,f)=12I1(b)exp{2π2f2[ff0φ(b)2π]2}exp{i[2πfb+2πf0b]+φ(b)}.
fb=f0+φ(b)2π.
ε1(f,b)=f2+I1exp{j[12φ(b)(xb)2]}exp[f2(xb)22]exp(j2πfx)dx=I1exp(j2πfb)2+cos[12φ(b)t2f2]exp(t22)cos(2πt)dt+jI1exp(j2πfb)2+sin[12φ(b)t2f2]exp(t22)cos(2πt)dt,ε2(f,b)=f2+I1exp{j[12φ(b)(xb)2]}exp[f2(xb)22]exp(j2πfx)dx=I1exp(j2πfb)2+cos[12φ(b)t2f2]exp(t22)cos(2πt)dtjI1exp(j2πfb)2+sin[12φ(b)t2f2]exp(t22)cos(2πt)dt.
ε0=ε1(f,b)+ε2(f,b)=I1exp(j2πfb)+cos[12φ(b)t2f2]exp(t22)cos(2πt)dt.
fb=f0+φ(b)2π+K(φ(b)).
fb=φ(b)2π+K(o(φ(b)))φ(b)2π.
φ(x,y)=2{3(1x)2exp[x2(y+1)2]}10(x5x3y5)exp(x2y2)13exp{(x+1)2y2}.

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