Abstract

We propose three-dimensional (3D) profilometry based on a Fourier transform in which a two- dimensional (2D) Dammann grating and a cylindrical lens are used to generate structured light. The Dammann grating splits most of the illumination power into a 2D diffractive spot matrix. The cylindrical lens transforms these 2D diffractive spots into one-dimensional fringe lines that are projected on an object. The produced projection fringes have the advantages of high brightness and high contrast and compression ratios. The experiments have verified the proposed 3D profilometry. The 3D profilometry using Dammann grating should be of high interest for practical applications.

© 2009 Optical Society of America

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References

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2009 (1)

2008 (3)

2007 (1)

2006 (1)

2005 (1)

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645-658 (2005).
[CrossRef]

2003 (1)

2001 (1)

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

1999 (1)

1997 (1)

1995 (1)

1992 (1)

R. L. Morrison, “Symmetries that simplify the design of spot array phase gratings,” J. Opt. Soc. Am. 9, 464-471 (1992).
[CrossRef]

1983 (1)

1982 (1)

1971 (1)

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Cao, H.

Chen, W.

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

Chung, P. S.

Dai, and E.

X. Wang, C. Zhou, and E. Dai, “Scheme of real-time Dammann grating based miniaturized three-dimensional endoscopic imaging system,” Proc. SPIE 6832, R8320 (2008).

Dammann, H.

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Di, C.

Dobson, K.

Doh, K. B.

Feng, J.

Fukuda, H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

Görtler, K.

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Gu, Q.

Guillaume, P.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645-658 (2005).
[CrossRef]

Hao, Y.

Ina, H.

Iwata, K.

Jia, J.

Jia, W.

Kinoshita, M.

Koboyashi, S.

Kusunoki, F.

Li, D.

Liu, L.

Liu, L. R.

Lv, P.

Moriwaki, K.

Morrison, R. L.

R. L. Morrison, “Symmetries that simplify the design of spot array phase gratings,” J. Opt. Soc. Am. 9, 464-471 (1992).
[CrossRef]

Mutoh, K.

Poon, T.-C.

Su, X.

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

Takahashi, Y.

Takai, H.

Takeda, M.

Tomii, T.

Vanherzeele, J.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645-658 (2005).
[CrossRef]

Vanlanduit, S.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645-658 (2005).
[CrossRef]

Wang, B.

Wang, S.

Wang, X.

X. Wang, C. Zhou, and E. Dai, “Scheme of real-time Dammann grating based miniaturized three-dimensional endoscopic imaging system,” Proc. SPIE 6832, R8320 (2008).

Zhao, Y.

Zheng, J.

Zhou, C.

Zhou, C. H.

Appl. Opt. (8)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Opt. Lasers Eng. (2)

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

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645-658 (2005).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

X. Wang, C. Zhou, and E. Dai, “Scheme of real-time Dammann grating based miniaturized three-dimensional endoscopic imaging system,” Proc. SPIE 6832, R8320 (2008).

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

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Figures (5)

Fig. 1
Fig. 1

Schematic of 3D Fourier transform profilometry using a Dammann grating.

Fig. 2
Fig. 2

Generation of an array line pattern with a 2D Dammann grating and a cylindrical lens. P 1 is the convergent plane of the illuminator consisting of L 1 , the Dammann grating, and L 2 , and P 2 is the reference plane. S n is the extended fringe line formed by the nth diffractive rays.

Fig. 3
Fig. 3

(a)  21 × 21 spot matrix pattern produced by a 2D Dammann grating and (b) the transformed 1D fringe lines from the diffractive pattern in (a) by using a cylindrical lens.

Fig. 4
Fig. 4

(a) Reference image used in the experiment, (b) the deformed image of an object with slope height distribution with a folding line on the surface, and (c) the reconstructed 3D unwrapped phase distribution of the object with a maximum phase difference of 23.99 rad , which corresponds to a maximum height difference of 1 cm and is used to calculate the phase-to-height conversion parameter of the experimental setup.

Fig. 5
Fig. 5

(a) Deformed image of a foot model whose substrate has a small angle with the reference plane, and (b) the top view of the reconstructed 3D surface height distribution of the object. The constructed height differences Δ Z are 0.94 mm , 1.02 mm , and 0.83 mm for these three point pairs: ( X = 117 , Y = 37 ) and ( X = 117 , Y = 77 ) , ( X = 165 , Y = 149 ) and ( X = 189 , Y = 149 ) , and ( X = 37 , Y = 109 ) and ( X = 61 , Y = 133 ) , respectively.

Equations (15)

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g ( x , y ) = r ( x , y ) n = A n exp { i [ 2 π n f 0 x + n φ ( x , y ) ] } .
g 0 ( x , y ) = r 0 ( x , y ) n = A n exp { i [ 2 π n f 0 x + n φ 0 ( x , y ) ] } ,
g ( x , y ) = A 1 r ( x , y ) exp [ i 2 π f 0 x + φ ( x , y ) ] ,
g 0 ( x , y ) = A 1 r 0 ( x , y ) exp [ i 2 π f 0 x + φ 0 ( x , y ) ] .
Δ φ ( x , y ) = Im { log [ g ( x , y ) g 0 * ( x , y ) ] } .
h ( x , y ) = L 0 Δ φ ( x , y ) Δ φ ( x , y ) 2 π f 0 d 0 .
h ( x , y ) L 0 Δ φ ( x , y ) 2 π f 0 d 0 .
p k ( y ) = rect ( y ( y k + 1 + y k ) / 2 y k + 1 y k ) ,
I { p k ( y ) } = 1 2 n π [ ( sin α k + 1 sin α k ) + i ( cos α k + 1 cos α k ) ] .
I n = ( 1 2 n π ) [ ( P n ) R 2 + ( P n ) I 2 ] ,
( P n ) R = k = 0 K ( 1 ) k ( sin α k + 1 sin α k ) = 2 k = 1 K ( 1 ) k + 1 sin α k sin 2 n π ,
( P n ) I = 2 k = 1 K ( 1 ) k + 1 ( cos α k + 1 cos α k ) = 2 k = 1 K ( 1 ) k + 1 cos α k cos 2 n π 1.
I 0 = [ 1 + 2 k = 1 K ( 1 ) k y k ] 2 ,
I n = ( 1 n π ) 2 { [ k = 1 K ( 1 ) k sin α k ] 2 + [ 1 + k = 1 K ( 1 ) k cos α k ] 2 } .
s n ( z 0 + z 1 ) * λ d ,

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