Abstract

Phase-shift shadow-moiré topography is a noncontact optical technique for measuring the shapes of surfaces. Artifactual bands resembling isoheight surface contours are observed during measurement of small changes in shape by use of this technique. The shape-reconstruction algorithm used in shadow-moiré topography is based on a mathematical model of the fringe patterns generated on the surface to be measured. We hypothesize that the observed bands reflect systematic errors caused by ignoring height-dependent terms in the mathematical model of the fringe patterns. We test the assumption by simulating the fringe patterns for a virtual test surface by using a model that contains height-dependent terms and one term that is idealized by ignoring these terms. Small systematic errors in shape are observed only when the surface is reconstructed from fringe patterns simulated with a model containing the height-dependent terms. Shape-error curves are computed as a function of the surface height by the subtraction of the reconstructed shape from the known shape. Simulated shape-error curves agree with experimental measurements in that they show an increase in error with surface height, and both the experimental and the simulated shape-error curves contain ripples. Although the errors are small in comparison with the dimensions of the surface and are negligible in shape measurements and in most deformation measurements, they may show up as noticeable bands in images of small deformations.

© 2000 Optical Society of America

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References

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  1. F. Chen, G. M. Brown, M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
    [Crossref]
  2. H. Ladak, “Finite-element and experimental analyses of the response of the cat eardrum to large static pressures,” Ph.D. dissertation (McGill University, Montréal, Québec, Canada, 1998).
  3. M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
    [Crossref]
  4. S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
    [Crossref]
  5. D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
    [Crossref]
  6. A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 120–129 (1992).
  7. K. E. Perry, J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
    [Crossref]
  8. A. Asundi, C. S. Chan, “Phase shifting applied to nonsinusoidal intensity distribution—an error simulation,” Opt. Lasers Eng. 21, 3–30 (1994).
    [Crossref]
  9. Y. Arai, S. Yokozeki, “Improvement of measurement accuracy in shadow moiré by considering the influence of harmonics in the moiré profile,” Appl. Opt. 38, 3503–3507 (1999).
    [Crossref]
  10. C. Wykes, R. Morshedizadeh, “Surface topography measurement using digital moiré contouring—errors and limitations,” J. Eng. Manuf. B 209, 317–325 (1995).
    [Crossref]
  11. D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [Crossref] [PubMed]
  12. J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).
  13. H. Takasaki, “Moiré topography from its birth to practical applications,” Opt. Lasers Eng. 3, 3–13 (1982).
    [Crossref]
  14. J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).
  15. J. J. J. Dirckx, W. F. Decraemer, “Phase shift moiré apparatus for automatic 3-D surface measurement,” Rev. Sci. Instrum. 60, 3698–3701 (1989).
    [Crossref]

2000 (1)

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

1999 (2)

M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
[Crossref]

Y. Arai, S. Yokozeki, “Improvement of measurement accuracy in shadow moiré by considering the influence of harmonics in the moiré profile,” Appl. Opt. 38, 3503–3507 (1999).
[Crossref]

1997 (1)

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

1996 (1)

S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
[Crossref]

1995 (2)

K. E. Perry, J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[Crossref]

C. Wykes, R. Morshedizadeh, “Surface topography measurement using digital moiré contouring—errors and limitations,” J. Eng. Manuf. B 209, 317–325 (1995).
[Crossref]

1994 (1)

A. Asundi, C. S. Chan, “Phase shifting applied to nonsinusoidal intensity distribution—an error simulation,” Opt. Lasers Eng. 21, 3–30 (1994).
[Crossref]

1990 (1)

J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).

1989 (1)

J. J. J. Dirckx, W. F. Decraemer, “Phase shift moiré apparatus for automatic 3-D surface measurement,” Rev. Sci. Instrum. 60, 3698–3701 (1989).
[Crossref]

1985 (1)

J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).

1982 (1)

H. Takasaki, “Moiré topography from its birth to practical applications,” Opt. Lasers Eng. 3, 3–13 (1982).
[Crossref]

1970 (1)

Allen, J. B.

Arai, Y.

Asundi, A.

A. Asundi, C. S. Chan, “Phase shifting applied to nonsinusoidal intensity distribution—an error simulation,” Opt. Lasers Eng. 21, 3–30 (1994).
[Crossref]

Brown, G. M.

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

Chan, C. S.

A. Asundi, C. S. Chan, “Phase shifting applied to nonsinusoidal intensity distribution—an error simulation,” Opt. Lasers Eng. 21, 3–30 (1994).
[Crossref]

Chen, F.

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

Chiang, F. P.

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

Chou, S. C.

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

Decraemer, W. F.

J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).

J. J. J. Dirckx, W. F. Decraemer, “Phase shift moiré apparatus for automatic 3-D surface measurement,” Rev. Sci. Instrum. 60, 3698–3701 (1989).
[Crossref]

J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).

Dirckx, J. J. J.

J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).

J. J. J. Dirckx, W. F. Decraemer, “Phase shift moiré apparatus for automatic 3-D surface measurement,” Rev. Sci. Instrum. 60, 3698–3701 (1989).
[Crossref]

Eyckmans, M. M. K.

J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).

Fagg, A. H.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 120–129 (1992).

Gee, L.

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

Guralnick, S. A.

S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
[Crossref]

Hales, B.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 120–129 (1992).

Hinds, G.

M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
[Crossref]

Janssens, J. L.

J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).

Johnson, W. O.

Kokidko, D.

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

Ladak, H.

H. Ladak, “Finite-element and experimental analyses of the response of the cat eardrum to large static pressures,” Ph.D. dissertation (McGill University, Montréal, Québec, Canada, 1998).

McKelvie, J.

K. E. Perry, J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[Crossref]

Meadows, D. M.

Morshedizadeh, R.

C. Wykes, R. Morshedizadeh, “Surface topography measurement using digital moiré contouring—errors and limitations,” J. Eng. Manuf. B 209, 317–325 (1995).
[Crossref]

Perry, K. E.

K. E. Perry, J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[Crossref]

Singhatanadgid, P.

M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
[Crossref]

Song, M. M.

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

Stahl, H. P.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 120–129 (1992).

Suen, E. S.

S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
[Crossref]

Takasaki, H.

H. Takasaki, “Moiré topography from its birth to practical applications,” Opt. Lasers Eng. 3, 3–13 (1982).
[Crossref]

Tuttle, M.

M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
[Crossref]

Vanhuyse, V. J.

J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).

Wykes, C.

C. Wykes, R. Morshedizadeh, “Surface topography measurement using digital moiré contouring—errors and limitations,” J. Eng. Manuf. B 209, 317–325 (1995).
[Crossref]

Yokozeki, S.

Zoruba, S.

S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
[Crossref]

Appl. Opt. (2)

Exp. Mech. (1)

M. Tuttle, P. Singhatanadgid, G. Hinds, “Buckling of composite panels subjected to biaxial loading,” Exp. Mech. 39, 191–201 (1999).
[Crossref]

Int. J. Impact Eng. (1)

D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang, “Method for measuring transient out-of-plane deformation during impact,” Int. J. Impact Eng. 19, 127–133 (1997).
[Crossref]

J. Eng. Manuf. B (1)

C. Wykes, R. Morshedizadeh, “Surface topography measurement using digital moiré contouring—errors and limitations,” J. Eng. Manuf. B 209, 317–325 (1995).
[Crossref]

Opt. Eng. (1)

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

Opt. Lasers Eng. (3)

K. E. Perry, J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[Crossref]

A. Asundi, C. S. Chan, “Phase shifting applied to nonsinusoidal intensity distribution—an error simulation,” Opt. Lasers Eng. 21, 3–30 (1994).
[Crossref]

H. Takasaki, “Moiré topography from its birth to practical applications,” Opt. Lasers Eng. 3, 3–13 (1982).
[Crossref]

Optik (2)

J. J. J. Dirckx, W. F. Decraemer, M. M. K. Eyckmans, “Grating noise removal in moiré topography,” Optik 86, 107–110 (1990).

J. L. Janssens, W. F. Decraemer, V. J. Vanhuyse, “Visibility depth of shadow-moiré fringes in function of extending of light source and aperture of recording system,” Optik 71, 45–51 (1985).

Rev. Sci. Instrum. (1)

J. J. J. Dirckx, W. F. Decraemer, “Phase shift moiré apparatus for automatic 3-D surface measurement,” Rev. Sci. Instrum. 60, 3698–3701 (1989).
[Crossref]

Transp. Res. Rec. (1)

S. A. Guralnick, E. S. Suen, S. Zoruba, “Development of automated road inspection vehicle for nondestructive evaluation of road surface condition,” Transp. Res. Rec. 1536, 125–129 (1996).
[Crossref]

Other (2)

H. Ladak, “Finite-element and experimental analyses of the response of the cat eardrum to large static pressures,” Ph.D. dissertation (McGill University, Montréal, Québec, Canada, 1998).

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 120–129 (1992).

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Figures (8)

Fig. 1
Fig. 1

Phase-shift shadow-moiré apparatus: h is the distance between the camera and the grating, and d is the distance between each light guide and the camera. For clarity the origin of the xyz coordinate system is shown on the left-hand side of the figure. In reality the origin coincides with the center of the grating. O, object; G, grating; TS1, translation stage for moving the object; SM1, stepper motor for moving stage TS1; TS2, translation stage for moving the grating in its own plane; SM2, stepper motor for moving stage TS2; L, light source; L1, light guide 1; L2, light guide 2; C, camera; CI, computer interface.

Fig. 2
Fig. 2

Gray-level images of the shape and the displacement patterns of a cat’s eardrum: (a) The shape of the eardrum in which the gray levels vary from black (the deepest point) to white (the closest point). (b) Displacement pattern for a high pressure of 2.2 kPa. The gray levels vary from black (zero displacement) to white (maximum displacement). (c) Displacement pattern for a low pressure of 0.4 kPa. The gray levels vary as for (b). Adjacent tick marks on the axes are 1 mm apart.

Fig. 3
Fig. 3

Simulated data of the reconstruction of a virtual inclined-plane surface in moiré topography when a lens setting of f/8 is used: (a) Horizontal profile through the object. (b) Approximate (dashed–dotted curve) and realistic (solid curve) fringe patterns. (c) Shapes reconstructed from the approximate fringe pattern (dashed–dotted curve) and the realistic pattern (solid curve).

Fig. 4
Fig. 4

Shape-reconstruction error plotted versus the distance z from the grating for different f-numbers: (a) f/2.8, (b) f/5.6, (c) f/8, (d) f/11. The curves in black are for simulated errors, and the curves in gray are for measured errors.

Fig. 5
Fig. 5

Reconstructed displacement field for a spherical surface that is translated 100 µm away from the grating: (a) The gray-level displacement image. The gray levels vary from black (the smallest displacement magnitude) to white (the largest displacement magnitude). (b) Horizontal profile through the center of the gray-level image.

Fig. 6
Fig. 6

Simulated displacement error plotted versus the distance z from the grating for translations away from the grating of 100 µm (solid curve) and 500 µm (dashed curve) for different f-numbers: (a) f/2.8, (b) f/5.6, (c) f/8, (d) f/11.

Fig. 7
Fig. 7

Sensitivity of the simulated shape-reconstruction errors to variations in the grating period and the geometry of the apparatus: (a) Variation of the grating period p. (b) Variation of the distance d between the camera and the light sources as shown in Fig. 1. (c) Variation of the distance h between the camera and the grating. The solid black curve represents the 100% value used; the gray curve represents a value that is 50% less than that for the solid black curve; the dashed–dotted curve represents a value that is 50% larger than that for the solid black curve.

Fig. 8
Fig. 8

Simulated shape-reconstruction error plotted versus the distance z from the grating for three fringe-plane spacing values. The camera was set at f/8.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

Ix, y=Ax, yMzsin2πdzph+z+Bx, y,
Mz=2ph+zπDsz J1πDszph+z×2ph+zπDcz J1πDczph+z,
Ix, y=Ax, ysinSzx, y+Bx, y,
Inx, y=Ax, y(sinSzx, y+δn+Bx, y),
δ1=0,  δ2=λ4,  δ3=λ2,  δ4=3λ4.
zx, y=1S arctanI3x, y-I1x, yI4x, y-I2x, y.
ex, y=zx, y-zrx, y.
d=114 mm,  h=333 mm,  p=0.254 mm,  Ds=10 mm.
MT=ff-h-z.
dx, y=zdx, y-zix, y.
edx, y=dax, y-dx, y.

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