Abstract

The use of an optical fringe projection method with two-step phase shifting for three-dimensional (3-D) shape measurement of small objects is described. In this method, sinusoidal linear fringes are projected onto an object’s surface by a programmable liquid-crystal display (LCD) projector and a long-working-distance microscope (LWDM). The image of the fringe pattern is captured by another LWDM and a CCD camera and processed by a phase-shifting technique. Usually a minimum of three phase-shifted fringe patterns is necessary for extraction of the object shape. In this method, a new algorithm based on a two-step phase-shifting technique produces the 3-D object shape. Unlike in the conventional method, phase unwrapping is performed directly by use of an arccosine function without the need for a wrapped phase map. Hence, shape measurement can be speeded up greatly with this approach. A small coin is evaluated to demonstrate the validity of the proposed measurement method, and the experimental results are compared with those of the four-step phase-shifting method and the conventional mechanical stylus method.

© 2003 Optical Society of America

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References

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    [CrossRef]
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2001 (3)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

2000 (1)

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

1998 (2)

Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37, 1005–1010 (1998).
[CrossRef]

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

1997 (2)

1995 (2)

1992 (1)

1990 (1)

1989 (1)

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

1984 (1)

1982 (1)

Barrett, H. H.

Bo, T.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Brown, G. M.

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

Burton, D. R.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Chen, F.

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

Choi, Y. B.

Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37, 1005–1010 (1998).
[CrossRef]

de Groot, P.

Freischlad, K.

Guo, Y. F.

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Halioua, M.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef]

He, X. Y.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Hong, C. K.

Kim, S. W.

Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37, 1005–1010 (1998).
[CrossRef]

Kolipoulos, C. L.

Kujawinska, M.

M. Kujawinska, J. Wojciak, “Spatial phase-shifting technique of fringe pattern analysis in photomechanics,” in Second International Conference on Photomechanics and Speckle Metrology: Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics, F.-P. Chiang, ed., Proc. SPIE1554B, 503–513 (1991).

Lalor, M. J.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Larkin, K. G.

Lim, H. C.

Liu, H. C.

Liu, H.-C.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Liu, S.

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Morgan, C. J.

Ngoi, B. K. A.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Oreb, B. F.

Quan, C.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Rogala, E. W.

Ryu, H. S.

Shang, H. M.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Sivakumar, N. R.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Song, M.

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

Srinivasan, V.

Tay, C. J.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Venkatakrishnan, K.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Wang, C. F.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Wojciak, J.

M. Kujawinska, J. Wojciak, “Spatial phase-shifting technique of fringe pattern analysis in photomechanics,” in Second International Conference on Photomechanics and Speckle Metrology: Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics, F.-P. Chiang, ed., Proc. SPIE1554B, 503–513 (1991).

Wu, F.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Zhang, H.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Zou, D. Q.

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Appl. Opt. (3)

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

Opt. Commun. (3)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surface,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Opt. Eng. (3)

X. Y. He, D. Q. Zou, S. Liu, Y. F. Guo, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37, 1005–1010 (1998).
[CrossRef]

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

Opt. Lasers Eng. (1)

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Opt. Lett. (2)

Other (1)

M. Kujawinska, J. Wojciak, “Spatial phase-shifting technique of fringe pattern analysis in photomechanics,” in Second International Conference on Photomechanics and Speckle Metrology: Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics, F.-P. Chiang, ed., Proc. SPIE1554B, 503–513 (1991).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the method for obtaining a continuous phase value.

Fig. 2
Fig. 2

Flow chart of the proposed algorithm.

Fig. 3
Fig. 3

Schematic diagram of the experimental setup.

Fig. 4
Fig. 4

Two phase-shifted sinusoidal fringe patterns projected onto the number 20 in the coin with a phase shift of π.

Fig. 5
Fig. 5

(a) Surface of the 20-cent coin; (b) background intensity R(x, y)A(x, y) of the number 20 on the coin obtained from Eq. (9).

Fig. 6
Fig. 6

(a) Gray-scale distribution of a cross section marked by AA in Fig. 4. (b) Distribution of I″(x, y) for the corresponding cross section.

Fig. 7
Fig. 7

Continuous phase map extracted by the proposed two-step phase-shifting method.

Fig. 8
Fig. 8

(a) Four phase-shifted sinusoidal fringe patterns projected on the number 20 in the coin with a phase shift of π/2. (b) Continuous phase map obtained by four-step phase shifting.

Fig. 9
Fig. 9

Contour maps of the surface of the number 20 on the coin: (a) the two-step phase-shifting method and (b) by the four-step phase-shifting method.

Fig. 10
Fig. 10

Corresponding reconstructed 3-D plots of the number 20 on the coin: (a) by the two-step phase-shifting method and (b) by four-step phase-shifting method.

Fig. 11
Fig. 11

Comparison of cross-section profiles of the number 20 marked by BB in Fig. 7 for the two- and the four-step phase-shifting methods.

Fig. 12
Fig. 12

Cross-section profiles of the number 20 marked by CC in Fig. 7 (a) by the two-step phase-shifting method and (b) by the mechanical stylus method.

Equations (19)

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

Ix, y=Rx, yAx, y1+Bx, yAx, ycos2πfx+ϕx, y,
Ix, y=Rx, yBx, ycos2πfx+ϕx, y,
Ix, y=Bx, yAx, ycos2πfx+ϕx, y,
Ix, y=Ix, yRx, yAx, y.
Φx, y=ΦxK, y+π2-arccos|Ix, y|Bx, y/Ax, yKxK<xxP,π2+arccos|Ix, y|Bx, y/Ax, yKxP<xxK+1,
ϕx, y=Φx, y-2πfx,
Iix, y=Rx, yAx, y+Bx, ycos2πfx+ϕx, y+αi=ux, y+vx, ycos2πfx+ϕx, y+αii=1, 2N,
E=i=1N(Iix, y-ux, y+vx, ycos2πfx+ϕx, y+αi)2.
i=1NIix, y-i=1Nux, y-i=1Nvx, ycos2πfx+ϕx, y+αi=0,
Nux, y=i=1NIix, y-vx, yi=1Ncos2πfx+ϕx, y+αi.
02πFx, y, θdθ=02πcos2πfx+ϕx, y+θdθ=0.
i=1Ncos2πfx+ϕx, y+αiΔθ=0,
Rx, yAx, y=1Ni=1N Iix, yN2.
Rx, yAx, y=½ I1x, y+I2x, y.
Ix, y=I1x, y-Rx, yAx, yRx, yAx, y,
Ix, y=I1x, y-I2x, yI1x, y+I2x, y,
=1/ni=1n|Δhi|hmax×100%,
Δ=hmin,1-hmin,2,
=1/ni=1n|Δhi|-|Δ|hmax+Δ×100%,

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