Abstract

We describe a technique for the measurement of non-full-field reflective surfaces by using phase-stepping profilometry. We explain the principles of phase demodulation and discuss three-dimensional (3-D) height reconstruction in the case of measuring surfaces with reflective properties such as plain glass and mirrored glass. A number of required calibration algorithms are described to obtain surface slopes and reconstructed 3-D heights of the whole surface. Masking for non-full-field objects and the surface reconstruction procedure are demonstrated mathematically and algorithmically. Several experimental results are given for glass with different shapes and defects. Measurement of a spherical mirror with a known radius has also allowed us to show the performance of the proposed technique. This allows for the possibility to compare 3-D data from the known object with data taken from the measurement system.

© 2005 Optical Society of America

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References

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2002

2000

R. Hofling, P. Aswendt, R. Neugebauer, “Phase reflection—a new solution for the detection on shape defects on car body sheets,” Opt. Eng. 39, 175–182 (2000).
[CrossRef]

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

1999

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

1996

1987

1986

1983

1982

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

1974

Aswendt, P.

R. Hofling, P. Aswendt, R. Neugebauer, “Phase reflection—a new solution for the detection on shape defects on car body sheets,” Opt. Eng. 39, 175–182 (2000).
[CrossRef]

Bachor, H. A.

Beyerer, J.

D. Perard, J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, K. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 74–80 (1997).
[CrossRef]

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970), Chap. 7.

Brangaccio, D. J.

Bruning, J. H.

Burton, D. R.

O. A. Skydan, M. J. Lalor, D. R. Burton, “Technique for phase measurement and surface reconstruction by use of colored structured light,” Appl. Opt. 41, 6104–6117 (2002).
[CrossRef] [PubMed]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Chau, S. C.

R. S. Sirohi, S. C. Chau, Optical Methods of Measurement (Marcel Dekker, 1999).

Creath, K.

K. Creath, Interferogram Analysis Digital Fringe Pattern Measurement Techniques, W. R. Robinson, G. T. Reid (Institute of Physics, 1993), Chap. 4.

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).

Eiju, T.

Gallagher, J. E.

Hariharan, P.

Herriot, D. R.

Hofling, R.

R. Hofling, P. Aswendt, R. Neugebauer, “Phase reflection—a new solution for the detection on shape defects on car body sheets,” Opt. Eng. 39, 175–182 (2000).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Ina, H.

Juptner, W. P.

T. M. Kreis, W. P. Juptner, “Fourier-transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Prypatniewicz, ed., Proc. SPIE1162, 116–125 (1989).
[CrossRef]

Kobayashi, S.

Kreis, T. M.

T. M. Kreis, W. P. Juptner, “Fourier-transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Prypatniewicz, ed., Proc. SPIE1162, 116–125 (1989).
[CrossRef]

Lalor, M. J.

O. A. Skydan, M. J. Lalor, D. R. Burton, “Technique for phase measurement and surface reconstruction by use of colored structured light,” Appl. Opt. 41, 6104–6117 (2002).
[CrossRef] [PubMed]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Lin, L.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Macy, W.

Neugebauer, R.

R. Hofling, P. Aswendt, R. Neugebauer, “Phase reflection—a new solution for the detection on shape defects on car body sheets,” Opt. Eng. 39, 175–182 (2000).
[CrossRef]

Oreb, B. F.

Park, B. G.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Perard, D.

D. Perard, J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, K. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 74–80 (1997).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Optical Measurement Techniques and Applications (Artech House, 1997).

Rosenfeld, D. P.

Sandeman, R. J.

Schmit, J.

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).

Shang, H. M.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Sirohi, R. S.

R. S. Sirohi, S. C. Chau, Optical Methods of Measurement (Marcel Dekker, 1999).

Skydan, O. A.

Surrel, Y.

Takeda, M.

White, A. D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970), Chap. 7.

Wyant, J. C.

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Zhang, H.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Laser Focus

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Opt. Eng.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

R. Hofling, P. Aswendt, R. Neugebauer, “Phase reflection—a new solution for the detection on shape defects on car body sheets,” Opt. Eng. 39, 175–182 (2000).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970), Chap. 7.

P. K. Rastogi, Optical Measurement Techniques and Applications (Artech House, 1997).

R. S. Sirohi, S. C. Chau, Optical Methods of Measurement (Marcel Dekker, 1999).

D. Perard, J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, K. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 74–80 (1997).
[CrossRef]

K. Creath, Interferogram Analysis Digital Fringe Pattern Measurement Techniques, W. R. Robinson, G. T. Reid (Institute of Physics, 1993), Chap. 4.

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).

T. M. Kreis, W. P. Juptner, “Fourier-transform evaluation of interference patterns: the role of filtering in the spatial frequency domain,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Prypatniewicz, ed., Proc. SPIE1162, 116–125 (1989).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Fringe reflection.

Fig. 2
Fig. 2

Calibration chessboard with recognized cross points.

Fig. 3
Fig. 3

Profile of the slope calibration mirror.

Fig. 4
Fig. 4

Camera view of the slope calibration mirror.

Fig. 5
Fig. 5

Kpx parameter distribution along the x coordinate axis.

Fig. 6
Fig. 6

Results of integration of (a) the full-field surface slope and (b) the same non-full-field surface slope with some missing points.

Fig. 7
Fig. 7

Measurement of the spherical calibration mirror: (a) and (b) the wrapped phase in the x and y directions, respectively; (c) and (d) the respective unwrapped phase distributions; (e) a gray-scale map of the surface height; (f) a 3-D view of the reconstructed surface.

Fig. 8
Fig. 8

Measurement of the automotive side glass: (a) and (b) the wrapped phase in the x and y directions, respectively; (c) and (d) the respective unwrapped phase distributions; (e) a gray-scale map of the surface height; (f) a 3-D view of the reconstructed glass surface.

Equations (22)

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i x 1 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ x ( x , y ) ] ,
i x 2 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ x ( x , y ) + π / 2 ] ,
i x 3 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ x ( x , y ) + π ] ,
i x 4 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ x ( x , y ) + 3 π / 2 ] ,
ϕ x ( x , y ) = tan - 1 ( i x 4 - i x 2 i x 1 - i x 3 ) = 2 π f x x + k x ( x , y , Δ ϕ x ) z x ,
ϕ r x ( x , y ) = 2 π f x x .
ϕ x ( x , y ) - ϕ r x ( x , y ) = k x ( x , y , Δ ϕ x ) z x ,
s x ( x , y ) = z x = ϕ x ( x , y ) - ϕ r x ( x , y ) k x ( x , y , Δ ϕ x ) = Δ ϕ x k x ( x , y , Δ ϕ x ) ,
s y ( x , y ) = z y = Δ ϕ y k y ( x , y , Δ ϕ y ) .
z x = 0 x s x ( x , y ) d x ,
z y = 0 y s y ( x , y ) d y ,
z = R { z x , z y } ,
B x = [ x 11 x 21 x n 1 x 12 x 22 x n 2 x 1 m x 2 m x n m ] ,     B y = [ y 11 y 21 y n 1 y 12 y 22 y n 2 y 1 m y 2 m y n m ] ,
p i = x i , j - x i + 1 , j d x ,     x i , j x x i + 1 , j ,
q j = y i , j - y i , j + 1 d y ,     y i , j y i , j + 1 ,
α = 2 arcsin ( x 1 + x 2 2 r ) .
γ = arcsin ( x 1 + x 2 2 r ) .
k x = ϕ x ( x ,     y ) - ϕ r x ( x ,     y ) tan ( γ ) .
m ( x ,     y ) = i x 1 - i x 2 + i x 2 - i x 3 + i x 3 - i x 4 + i x 4 - i x 1 .
M ( x , y ) = { 0 , m ( x , y ) < γ 1 , m ( x , y ) γ .
d ( i ) = z y ( i , j ) - z x ( i , j ) ,             for             i l i i r ,
z ( i , j ) = z y ( i , j ) - d ( i ) ,             for             j t j j b ,

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