Abstract

Moiré topography is a well-established optical technique to measure the shape of three-dimensional surfaces, based on the geometric interference between an optical grid and its image deformed by an object surface. The technique produces fringes that represent contours of equal height, and from the recordings of several phase-shifted topograms surface height coordinates can be calculated. To perform these calculations, it is assumed that object height variation is small in comparison with the measurement setup dimensions, and this approximation leads to systematic errors in measurement accuracy. We present the mathematical description of the fringe formation process in projection moiré topography, and on the basis of these equations we establish the relation between setup geometry and upper limits of the systematic measurement errors. We derive the equations that determine design specifications needed to reduce the effects of approximations to be below the measurement resolution of the setup. It is shown that setup geometry should be adapted to the gray-scale measurement resolution of the imaging system. We show that, using an iterative correction from one fringe order to the next, measurement accuracy can be maintained over the entire object depth.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. H. Takasaki, "Moiré topography," Appl. Opt. 9, 1467-1472 (1970).
    [CrossRef] [PubMed]
  3. K. Patorski, The Moiré Fringe Technique (Elsevier, 1993).
  4. C. A. Walker, Handbook of Moiré Measurement (Institute of Physics, 2004).
    [CrossRef]
  5. J. J. J. Dirckx and W. F. Decraemer, "Optoelectronic moiré projector for real-time shape and deformation studies of the tympanic membrane," J. Biomed. Opt. 2, 176-185 (1997).
    [CrossRef]
  6. P. Hopstone, A. Katz, and J. Politch, "Infrastructure of time-averaged projection moiré fringes in vibration analysis," Appl. Opt. 28, 5305-5311 (1989).
    [CrossRef] [PubMed]
  7. J. Politch and S. Gryc, "Second derivative of displacement on cylindrical shells by projection moiré," Appl. Opt. 28, 111-118 (1989).
    [CrossRef] [PubMed]
  8. M. Halioua, R. S. Krishnamurthy, H. Liu, and F.-P. Chiang, "Projection moiré with moving gratings for automated 3-D topography," Appl. Opt. 22, 850-855 (1983).
    [CrossRef] [PubMed]
  9. J. J. J. Dirckx and W. F. Decraemer, "Grating noise removal in moiré topography," Optik (Stuttgart) 86, 107-110 (1990).
  10. H. Takasaki, "Moiré topography from its birth to practical applications," Opt. Lasers Eng. 3, 3-13 (1982).
    [CrossRef]
  11. B. Drerup, "Some problems in analytical reconstruction of biological shapes from moiré topograms," in Optics in Biomedical Sciences, G.V.Bally and P.Greguss, eds. (Springer, 1982), pp. 258-261.
  12. K. Andersen, "Das Phasenshift Verfahren zum Moiré-Bildauswehrtung," Optik (Stuttgart) 72, 115-119 (1986).
  13. G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, "Temporal phase unwrapping of digital hologram sequences," Appl. Opt. 42, 5846-5854 (2003).
    [CrossRef] [PubMed]
  14. M.-S. Jeong and S.-W. Kim, "Phase-shifting projection moiré for out-of-plane displacement measurement," Proc. SPIE 4317, 170-179 (2001).
    [CrossRef]
  15. J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, "Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moiré topograms," Appl. Opt. 27, 1164-1169 (1988).
    [CrossRef] [PubMed]
  16. C. Han and B. Han, "Error analysis of the phase-shifting technique when applied to shadow moiré," Appl. Opt. 45, 1124-1133 (2006).
    [CrossRef] [PubMed]
  17. J. J. J. Dirckx and W. F. Decraemer, "Automatic calibration method for phase shift shadow moiré interferometry," Appl. Opt. 29, 1474-1476 (1990).
    [CrossRef] [PubMed]
  18. K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
    [CrossRef]
  19. K. Patorski and A. Styk, "Tilt-shift error detection in phase-shifting interferometry," Opt. Express 14, 5232-5249 (2006).
    [CrossRef] [PubMed]

2006 (2)

2005 (1)

K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
[CrossRef]

2004 (1)

C. A. Walker, Handbook of Moiré Measurement (Institute of Physics, 2004).
[CrossRef]

2003 (1)

2001 (1)

M.-S. Jeong and S.-W. Kim, "Phase-shifting projection moiré for out-of-plane displacement measurement," Proc. SPIE 4317, 170-179 (2001).
[CrossRef]

1997 (1)

J. J. J. Dirckx and W. F. Decraemer, "Optoelectronic moiré projector for real-time shape and deformation studies of the tympanic membrane," J. Biomed. Opt. 2, 176-185 (1997).
[CrossRef]

1993 (1)

K. Patorski, The Moiré Fringe Technique (Elsevier, 1993).

1990 (2)

J. J. J. Dirckx and W. F. Decraemer, "Automatic calibration method for phase shift shadow moiré interferometry," Appl. Opt. 29, 1474-1476 (1990).
[CrossRef] [PubMed]

J. J. J. Dirckx and W. F. Decraemer, "Grating noise removal in moiré topography," Optik (Stuttgart) 86, 107-110 (1990).

1989 (2)

1988 (1)

1986 (1)

K. Andersen, "Das Phasenshift Verfahren zum Moiré-Bildauswehrtung," Optik (Stuttgart) 72, 115-119 (1986).

1983 (1)

1982 (2)

H. Takasaki, "Moiré topography from its birth to practical applications," Opt. Lasers Eng. 3, 3-13 (1982).
[CrossRef]

B. Drerup, "Some problems in analytical reconstruction of biological shapes from moiré topograms," in Optics in Biomedical Sciences, G.V.Bally and P.Greguss, eds. (Springer, 1982), pp. 258-261.

1970 (2)

Alexeenko, I.

Allen, J. B.

Andersen, K.

K. Andersen, "Das Phasenshift Verfahren zum Moiré-Bildauswehrtung," Optik (Stuttgart) 72, 115-119 (1986).

Chiang, F.-P.

Decraemer, W. F.

J. J. J. Dirckx and W. F. Decraemer, "Optoelectronic moiré projector for real-time shape and deformation studies of the tympanic membrane," J. Biomed. Opt. 2, 176-185 (1997).
[CrossRef]

J. J. J. Dirckx and W. F. Decraemer, "Grating noise removal in moiré topography," Optik (Stuttgart) 86, 107-110 (1990).

J. J. J. Dirckx and W. F. Decraemer, "Automatic calibration method for phase shift shadow moiré interferometry," Appl. Opt. 29, 1474-1476 (1990).
[CrossRef] [PubMed]

J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, "Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moiré topograms," Appl. Opt. 27, 1164-1169 (1988).
[CrossRef] [PubMed]

Dielis, G.

Dirckx, J. J. J.

J. J. J. Dirckx and W. F. Decraemer, "Optoelectronic moiré projector for real-time shape and deformation studies of the tympanic membrane," J. Biomed. Opt. 2, 176-185 (1997).
[CrossRef]

J. J. J. Dirckx and W. F. Decraemer, "Grating noise removal in moiré topography," Optik (Stuttgart) 86, 107-110 (1990).

J. J. J. Dirckx and W. F. Decraemer, "Automatic calibration method for phase shift shadow moiré interferometry," Appl. Opt. 29, 1474-1476 (1990).
[CrossRef] [PubMed]

J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, "Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moiré topograms," Appl. Opt. 27, 1164-1169 (1988).
[CrossRef] [PubMed]

Drerup, B.

B. Drerup, "Some problems in analytical reconstruction of biological shapes from moiré topograms," in Optics in Biomedical Sciences, G.V.Bally and P.Greguss, eds. (Springer, 1982), pp. 258-261.

Gryc, S.

Halioua, M.

Han, B.

Han, C.

Hopstone, P.

Jeong, M.-S.

M.-S. Jeong and S.-W. Kim, "Phase-shifting projection moiré for out-of-plane displacement measurement," Proc. SPIE 4317, 170-179 (2001).
[CrossRef]

Johnson, W. O.

Katz, A.

Kim, S.-W.

M.-S. Jeong and S.-W. Kim, "Phase-shifting projection moiré for out-of-plane displacement measurement," Proc. SPIE 4317, 170-179 (2001).
[CrossRef]

Krishnamurthy, R. S.

Liu, H.

Meadows, D. M.

Osten, W.

Patorski, K.

K. Patorski and A. Styk, "Tilt-shift error detection in phase-shifting interferometry," Opt. Express 14, 5232-5249 (2006).
[CrossRef] [PubMed]

K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
[CrossRef]

K. Patorski, The Moiré Fringe Technique (Elsevier, 1993).

Pedrini, G.

Politch, J.

Sienicki, Z.

K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
[CrossRef]

Styk, A.

K. Patorski and A. Styk, "Tilt-shift error detection in phase-shifting interferometry," Opt. Express 14, 5232-5249 (2006).
[CrossRef] [PubMed]

K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
[CrossRef]

Takasaki, H.

H. Takasaki, "Moiré topography from its birth to practical applications," Opt. Lasers Eng. 3, 3-13 (1982).
[CrossRef]

H. Takasaki, "Moiré topography," Appl. Opt. 9, 1467-1472 (1970).
[CrossRef] [PubMed]

Tiziani, H. J.

Walker, C. A.

C. A. Walker, Handbook of Moiré Measurement (Institute of Physics, 2004).
[CrossRef]

Appl. Opt. (9)

J. Biomed. Opt. (1)

J. J. J. Dirckx and W. F. Decraemer, "Optoelectronic moiré projector for real-time shape and deformation studies of the tympanic membrane," J. Biomed. Opt. 2, 176-185 (1997).
[CrossRef]

Opt. Eng. (1)

K. Patorski, Z. Sienicki, and A. Styk, "Phase-shifting method contrast calculations in time-averaged interferometry: error analysis," Opt. Eng. 44, 1124-1133 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

H. Takasaki, "Moiré topography from its birth to practical applications," Opt. Lasers Eng. 3, 3-13 (1982).
[CrossRef]

Optik (Stuttgart) (2)

K. Andersen, "Das Phasenshift Verfahren zum Moiré-Bildauswehrtung," Optik (Stuttgart) 72, 115-119 (1986).

J. J. J. Dirckx and W. F. Decraemer, "Grating noise removal in moiré topography," Optik (Stuttgart) 86, 107-110 (1990).

Proc. SPIE (1)

M.-S. Jeong and S.-W. Kim, "Phase-shifting projection moiré for out-of-plane displacement measurement," Proc. SPIE 4317, 170-179 (2001).
[CrossRef]

Other (3)

K. Patorski, The Moiré Fringe Technique (Elsevier, 1993).

C. A. Walker, Handbook of Moiré Measurement (Institute of Physics, 2004).
[CrossRef]

B. Drerup, "Some problems in analytical reconstruction of biological shapes from moiré topograms," in Optics in Biomedical Sciences, G.V.Bally and P.Greguss, eds. (Springer, 1982), pp. 258-261.

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Figures (3)

Fig. 1
Fig. 1

Schematic drawing of a telecentric projection moiré setup. Light is modulated by the grating G 1 and projected by a lens L 1 onto the surface of an object positioned near the focal plane of the lenses. The reflected light is imaged by lens L 2 onto a second grating G 2 . The geometrical construction shows that an object point with coordinates ( x , y , z ) is illuminated by a ray of light coming from a point ( 2 D x + d x 1 , y d y , 2 L ) on the first grating and that the light from that point is imaged on the point ( 2 D x d x 2 , y d y , 2 L ) on the second grating. The distance d x 1 is determined by the similar triangles with sides d x 1 and z and with sides L z and D x . The distance d x 2 is determined by the similar triangles d x 2 and z and L z and D + x .

Fig. 2
Fig. 2

Relative intensity error Δ I ( I max I min ) as a function of object depth z. The horizontal line indicates the intensity difference resolving power 2 3 θ = 0.59 % for an imaging system with dynamic range θ = 256 . The thin solid curve and the dashed curve, respectively, indicate the error and its amplitude for a setup with dimensions 2 D = 54 mm and L = 268 mm . The figure shows that for these dimensions, the amplitude of the error remains smaller than 0.59% for z values in the zeroth-order fringe plane and increases dramatically for higher z values. The thick solid curve and the dotted curve indicate the error and its amplitude for a setup with dimensions 2 D = 5.39 m and L = 26.8 m . The figure shows that for these (very large) dimensions the amplitude of the error remains smaller than 0.59% for all z values up to 5 mm .

Fig. 3
Fig. 3

Amplitude of the relative intensity error Δ I ( I max I min ) as a function of object depth z n + n λ 0 , for setup dimensions 2 D = 55 mm and L = 277 mm and with compensation for the change of fringe plane distance with fringe plane order. The distance z n is the distance of the z coordinate within the n th -order fringe plane, and n λ 0 is n times the zeroth-order fringe plane distance. The thick (upper) horizontal line indicates the imaging system intensity difference resolving power of 0.59%. The figure shows that for all height values up to z = 5 mm the intensity error remains smaller than the error that can be detected by the imaging system. For higher values, the error increases slightly.

Equations (77)

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I pr = I 0 T pr ,
I pr = I 0 [ 1 2 + 1 2 sin ( 2 π x 1 p + ϕ 1 ) ] ,
I obs ( x , y , z ) = ρ 2 T obs R ( x , y , z ) r 2 ( x , y , z ) ρ 1 I pr
= I 0 ρ 1 ρ 2 R ( x , y , z ) r 2 ( x , y , z ) T pr T obs
= I 0 ρ 1 ρ 2 R ( x , y ) r 2 ( x , y , z ) [ 1 2 + 1 2 sin ( 2 π x 1 p + ϕ 1 ) ] [ 1 2 + 1 2 sin ( 2 π x 2 p + ϕ 2 ) ] ,
x 1 = 2 D x + d x 1 ,
y 1 = y d y ,
z 1 = 2 L ,
tan α 1 = d x 1 z = D x + d x 1 L = 2 D x + d x 1 2 L z ,
d x 1 = z D z L x ,
d y = z y L z .
x 2 = 2 D x d x 2 ,
y 2 = y 1 ,
z 2 = 2 L ,
tan α 2 = d x 2 z = D + x + d x 2 L = 2 D + x + d x 2 2 L z ,
d x 2 = z D + x L z .
I obs ( x , y , z ) = A [ 1 2 + 1 2 sin ( 2 π ( 2 D x + d x 1 ) p + ϕ 1 ) ] [ 1 2 + 1 2 sin ( 2 π ( 2 D x d x 2 ) p + ϕ 2 ) ] ,
A ( x , y , z ) = I 0 ρ 1 ρ 2 R ( x , y ) r 2 ( x , y , z ) .
I obs = A 4 { 1 + sin ( 2 π ( 2 D x z D + x L z ) p + ϕ 1 ) + sin ( 2 π ( 2 D x + z D x L z ) p + ϕ 2 ) + 1 2 [ cos ( 2 π ( 4 D z 2 D L z ) p + ϕ 1 ϕ 2 ) cos ( 2 π ( 2 x z 2 x L z ) p + ϕ 1 + ϕ 2 ) ] } .
A 4 { sin ( 2 π ( 2 D z D + x L z ) p + ϕ 1 ) cos ( 2 π x p ) cos ( 2 π ( 2 D z D + x L z ) p + ϕ 1 ) sin ( 2 π x p ) } .
Φ 1 = ϕ 1 + 2 π v p t ,
Φ 2 = ϕ 2 + 2 π v p t .
I n 1 = A 4 t = 0 T [ sin ( 2 π ( 2 D x d x 1 ) p + Φ 1 ) ] d t
= A 4 t = 0 T [ sin ( 2 π ( 2 D x d x 1 + v t ) p + ϕ 1 ) ] d t
= p A 8 π v [ cos ( 2 π ( 2 D x d x 1 ) p + ϕ 1 ) + cos ( 2 π ( 2 D x d x 1 + v T ) p + ϕ 1 ) ] .
I m = A 8 t = 0 T [ cos ( 2 π ( 4 D z 2 D L z ) p + Φ 1 Φ 2 ) ] d t
= A 8 t = 0 T [ cos ( 2 π ( 4 D z 2 D L z + v t v t ) p + ϕ 1 ϕ 2 ) ] d t
= T A 8 cos ( 2 π ( 4 D z 2 D L z ) p + ϕ 1 ϕ 2 )
= T A 8 cos ( 2 π ( z 2 D L z ) p ) ,
θ = I sat δ I det .
max ( I m ) max ( I n 1 ) + max ( I n 2 ) + max ( I n 3 ) = n v T 5 p θ .
I T = 0 T I obs ( t ) d t
[ T A 4 + I m ] .
I = T A ( x , y ) 4 [ 1 + 1 2 cos ( 2 π ( 2 z D L z ) p ) ] + B ( x , y ) .
T s = 1 2 + 2 π n = 1 , odd 1 n sin ( 2 π n x p ) .
I m = 2 T A π 2 [ n = 1 , odd 1 n 2 cos ( n z 2 π p 2 D L z ) ]
= 2 T A π 2 [ cos ( 2 π ( z 2 D L z ) p ) + n = 3 , odd 1 n 2 cos ( n z 2 π p 2 D L z ) ] .
arctan ( I 3 I 1 I 4 I 2 ) = 4 π D p z L z .
arctan ( I 3 I 1 I 4 I 2 ) = 4 π D p z L .
λ 0 = p L 2 D ,
λ 1 = p ( L λ 0 ) 2 D .
λ n = p ( L i = 0 n 1 λ i ) 2 D .
λ n p ( L n λ 0 ) 2 D = p L ( 2 D n ) 4 D 2 .
L z L z = L ( z L z 2 L 2 + z 3 L 3 )
ϵ = 4 π D z p ( L z ) 4 π D z p L = 4 π D p z 2 L ( L z ) .
I max = T A 4 3 2 .
I max = I sat ,
I min = T A 4 1 2 .
Δ I I max I min = Δ I 2 3 I sat δ I det 2 3 I sat = 3 2 θ .
Δ I = T A ( x , y ) 8 cos ( 2 π z 2 D L z p ) T A ( x , y ) 8 cos ( 2 π z 2 D L p ) .
Δ I I max I min = 4 T A ( x , y ) T A ( x , y ) 8 cos ( 2 π z 2 D L z p ) T A ( x , y ) 8 cos ( 2 π z 2 D L p )
= sin ( 2 π D z 2 p L ( L z ) ) sin ( 2 π D ( 2 z L z 2 ) p L ( L z ) )
= sin ( 2 π D z 2 p L ( L z ) ) sin ( 4 π D z p L )
3 2 θ .
2 π D z 2 p L ( L z ) 3 2 θ ,
2 D 2 θ 3 ( π p + 3 p 2 θ ) = 2 π θ 3 p + p 2 θ p ,
L = 2 λ D p λ ( 2 θ 3 ( π p + 3 p 2 θ ) ) p = 2 π 3 λ θ + λ 2 λ θ .
2 D 2 n 2 π 3 p θ + n p 2 n 2 θ p
L 2 n 2 π 3 λ θ + n λ 2 n 2 θ λ .
2 D 2 π 3 × 0.1 × 256 + 0.1 54 mm ,
L 2 π 3 × 0.5 × 256 + 0.5 268 mm .
2 D 10 2 × 2 π 3 × 0.1 × 256 + 10 × 0.1 5390 mm ,
L 10 2 × 2 π 3 × 0.5 × 256 + 10 × 0.5 26 , 810 mm .
z = z n + i = 0 n 1 λ i ,
L n = L i = 0 n 1 λ i .
Δ I = T A ( x , y ) 8 cos ( 4 π z n D p ( L i = 0 n 1 λ i z n ) ) T A ( x , y ) 8 cos ( 4 π D p ( L i = 0 n 1 λ i ) ) .
2 π D p ( L i = 0 n 1 λ i ) z n ( L i = 0 n 1 λ i z n ) z n ( L i = 0 n 1 λ i z n ) ( L i = 0 n 1 λ i ) 3 2 θ .
2 π D p λ n 2 ( L i = 0 n λ i ) ( L i = 0 n 1 λ i ) 3 2 θ .
λ n = p ( L i = 0 n 1 λ i ) 2 D p L 2 D = λ 0 .
2 π D p λ 0 2 ( L ( n + 1 ) λ n ) ( L n λ 0 ) 3 2 θ ,
π p 2 D ( 1 p ( 2 n + 1 ) 2 D + p 2 n ( n + 1 ) 4 D 2 ) 3 2 θ .
2 D 2 π 3 p θ + ( 2 n + 1 ) p ,
L 2 π 3 λ 0 θ + ( 2 n + 1 ) λ 0 .
2 D 2 π 3 × 0.1 × 256 + 19 × 0.1 55 mm ,
L 2 π 3 × 0.5 × 256 + 19 × 0.5 277 mm .
1 2 [ cos ( ϕ + δ ϕ ) cos ϕ ] 1 2 ( δ ϕ sin ϕ ) 3 2 θ .
δ x 3 p 2 π θ .

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