Abstract

The theory of generalized grating imaging for a one-dimensional grating is applied to a pattern projection system in pattern projection profilometry. Contrast of the projected grating image is calculated under various conditions. The results help to determine the conditions suitable for obtaining high contrast grating images in a large space. Although the gratings required for the profilometry are hexagonal, the theory for two-dimensional gratings is prohibitively complex. Therefore, the projection system was designed using the one-dimensional theory. The projection system using two-dimensional hexagonal gratings was constructed and experiments were done with it. The result agrees approximately with the theoretical calculations for one-dimensional gratings. This suggests that the one-dimensional theory may be used for estimating the approximated behavior for hexagonal gratings for use in pattern projection profilometry. Some discussions are given for the application of the projection system for profiling the mannequin or human body.

© 2011 Optical Society of America

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References

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2009 (2)

2008 (3)

2006 (1)

N. D’Apuzzo, “Overview of 3D surface digitization technologies in Europe,” Proc. SPIE 6056, 605605 (2006).
[CrossRef]

2004 (1)

L. M. Sanchez-Brea, J. Aloso, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusodial grating imaging using an extended light source,” Opt. Commun. 236, 53–58(2004).
[CrossRef]

2002 (1)

2000 (1)

1999 (1)

1985 (1)

1983 (1)

Alonso, J.

Aloso, J.

L. M. Sanchez-Brea, J. Aloso, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusodial grating imaging using an extended light source,” Opt. Commun. 236, 53–58(2004).
[CrossRef]

Bernabeu, E.

Crespo, D.

D’Apuzzo, N.

N. D’Apuzzo, “Overview of 3D surface digitization technologies in Europe,” Proc. SPIE 6056, 605605 (2006).
[CrossRef]

Fukuda, H.

Halioua, M.

Iwata, K.

Kakunai, S.

Kusunoki, F.

Liu, H. C.

Minchev, G.

Moriwaki, K.

Mutoh, K.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North Holland, 1989) Vol.  27, pp. 3–108.
[CrossRef]

Sainov, V.

Sakamoto, T.

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, J. Aloso, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusodial grating imaging using an extended light source,” Opt. Commun. 236, 53–58(2004).
[CrossRef]

Srinivasan, V.

Stoykova, E.

Takeda, M.

Tomii, T.

Appl. Opt. (7)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

L. M. Sanchez-Brea, J. Aloso, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusodial grating imaging using an extended light source,” Opt. Commun. 236, 53–58(2004).
[CrossRef]

Proc. SPIE (1)

N. D’Apuzzo, “Overview of 3D surface digitization technologies in Europe,” Proc. SPIE 6056, 605605 (2006).
[CrossRef]

Other (1)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North Holland, 1989) Vol.  27, pp. 3–108.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a pattern projection profilometer.

Fig. 2
Fig. 2

Optical system for generalized grating imaging.

Fig. 3
Fig. 3

Contrast of fringes as a function of L 1 and L 2 . p 1 = 100 μm , p 2 = 96 : 8 μm , λ = 617 nm , L 0 = 5 mm , S = 0.5 mm .

Fig. 4
Fig. 4

Contrast of fringes for L 1 = 15.8 mm as a function of S and L 2 . p 1 = 100 μm , p 2 = 96 : 8 μm , λ = 617 nm , L 0 = 5 mm .

Fig. 5
Fig. 5

Contrast of fringes for L 1 = 15.8 mm as a function of L 0 and L 2 . p 1 = 100 μm , p 2 = 96 : 8 μm , λ = 617 nm , S = 0.5 mm .

Fig. 6
Fig. 6

Contrast of fringes as a function of δ and L 2 . L 1 = 15.8 mm , p 1 = 100 μm , p 2 = 96 : 8 μm , λ = 617 nm , L 0 = 5 mm , S = 0.5 mm .

Fig. 7
Fig. 7

Hexagonal binary gratings obtained from Eq. (15): (a)  ϕ = 0 (b)  ϕ = π / 2 .

Fig. 8
Fig. 8

Projected grating image. (a) Intensity image. (b) Intensity variation along the solid line in (a).

Fig. 9
Fig. 9

Variation of contrast and pitch in relation to L T . Filled circles are the experimental results of the contrast. Solid curve is calculated by the theory for a one-dimensional grating with S = 1.1 mm and L 1 = 15 mm , L 0 = 5 mm . The point E corresponds to Fig. 8. Broken line is calculated by Eq. (14), and the open circles are the experimental result of the pitch.

Fig. 10
Fig. 10

Hexagonal grating projected on a mannequin body.

Equations (18)

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a 0 = 1 , a n = sin ( π n γ a ) / ( π n γ a ) n = ± 1 , ± 2 , b 0 = 1 , b m = sin ( π m γ b ) / ( π m γ b ) m = ± 1 , ± 2 , ,
I ( X ) = j = amp ( j ) cos [ 2 π X j μ 1 + j Ψ 1 ] ,
amp ( j ) = W ( j μ 2 ) A ( j ) B ( j ) , A ( j ) = n = a n a n + j cos [ ( 2 n + j ) j E ] , B ( j ) = m = b m b m j cos [ ( 2 m j ) j F ] ,
W ( j μ 2 ) = S S exp [ i 2 π j μ 2 x ] d x = sin ( 2 π j S μ 2 ) / ( 2 π j S μ 2 ) ,
μ 1 = [ L 0 / p 1 ( L 0 + L 1 ) / p 2 ] / L T ,
μ 2 = [ ( L 1 + L 2 ) / p 1 L 2 / p 2 ] / L T ,
E = ( π λ L 0 / p 1 ) μ 2 , F = ( π λ L 2 / p 2 ) μ 1 ,
Ψ 1 = 2 π { ε 1 / p 1 ε 2 / p 2 } .
P = 1 / μ 1 .
C ( j ) = 2 W ( j μ 2 ) C A ( j ) C B ( j ) , where     C A ( j ) = A ( j ) / A ( 0 ) , C B ( j ) = B ( j ) / B ( 0 ) .
L 20 = L 1 p 2 / ( p 1 p 2 ) ,
1 / P 0 = 1 / p 1 + 1 / p 2 .
F 0 = π λ L 1 / ( p 1 p 2 ) .
L 10 = ν p 1 p 2 / λ .
P = 1 / μ 1 = P 0 ( L 0 + L 1 + L 2 ) / ( L 0 + L 1 + L 20 ) ,
I δ ( X ) = λ 0 δ λ 0 + δ I ( X ) d λ .
f ( x , y ) = cos [ 2 π x / p + ϕ ] + cos [ 2 π ( x / 2 + 3 y / 2 ) / p ] + cos [ 2 π ( x / 2 + 3 y / 2 ) / p ] .
C = I max I min I max + I min ,

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