Abstract

In fringe projection profilometry, an object shape is evaluated through phase distribution extracted from a projected fringe pattern. For parallel illumination geometry, the carrier phase component introduced by the fringes is spatially linear, whereas nonparallel illumination would lead to a nonlinear carrier. In this study, a general approach for the removal of a nonlinear-carrier phase component is proposed. A series expansion technique is used to approximate the carrier phase function, and a least-squares method is developed to estimate the unknown coefficients of the series. The theoretical analysis is given on the basis of a divergent illumination geometry with carrier fringes in the x direction. The method is also extended to include a curved surface-fitting approach, which is applicable to various measurement system geometries.

© 2005 Optical Society of America

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Errata

Lujie Chen and Cho Jui Tay, "Carrier phase component removal: a generalized least-squares approach: errata," J. Opt. Soc. Am. A 23, 2053-2053 (2006)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-23-8-2053

References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2004

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

2003

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

1998

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

1994

W. S. Zhou and X. Y. Su, "A direct mapping algorithm for phase-measuring profilometry," J. Mod. Opt. 41, 89-94 (1994).
[CrossRef]

1985

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry: a phase mapping approach," J. Opt. Soc. Am. A 24, 185-188 (1985).

1984

V. Srinivasan, H. C. Liu, and M. Halious, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
[CrossRef] [PubMed]

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

1983

1982

Bosch, T.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

Cha, S. S.

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

Dallier, R.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

Flannery, B. P.

W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes Example Book (C++), 2nd ed. (Cambridge U. Press, 2002).

Garcia, V.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

Halioua, M.

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry: a phase mapping approach," J. Opt. Soc. Am. A 24, 185-188 (1985).

Halious, M.

Ina, H.

Kobayashi, S.

Li, J. L.

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

Liu, H. C.

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry: a phase mapping approach," J. Opt. Soc. Am. A 24, 185-188 (1985).

V. Srinivasan, H. C. Liu, and M. Halious, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
[CrossRef] [PubMed]

Luna, E.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

Morgan, C. J.

Mutoh, K.

Pavageau, S.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

Press, W. H.

W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes Example Book (C++), 2nd ed. (Cambridge U. Press, 2002).

Salas, L.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

Salinas, J.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

Servagent, N.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

Servin, M.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

Srinivasan, V.

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry: a phase mapping approach," J. Opt. Soc. Am. A 24, 185-188 (1985).

V. Srinivasan, H. C. Liu, and M. Halious, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
[CrossRef] [PubMed]

Su, H. J.

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

Su, X. Y.

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

W. S. Zhou and X. Y. Su, "A direct mapping algorithm for phase-measuring profilometry," J. Mod. Opt. 41, 89-94 (1994).
[CrossRef]

Takeda, M.

Teukolsky, S. A.

W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes Example Book (C++), 2nd ed. (Cambridge U. Press, 2002).

Vetterling, W. T.

W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes Example Book (C++), 2nd ed. (Cambridge U. Press, 2002).

Zhou, W. S.

W. S. Zhou and X. Y. Su, "A direct mapping algorithm for phase-measuring profilometry," J. Mod. Opt. 41, 89-94 (1994).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

W. S. Zhou and X. Y. Su, "A direct mapping algorithm for phase-measuring profilometry," J. Mod. Opt. 41, 89-94 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry: a phase mapping approach," J. Opt. Soc. Am. A 24, 185-188 (1985).

Opt. Eng. (Bellingham)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, "Profilometry by fringe projection," Opt. Eng. (Bellingham) 42, 3307-3314 (2003).
[CrossRef]

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

Opt. Lasers Eng.

J. L. Li, X. Y. Su, H. J. Su, and S. S. Cha, "Removal of carrier frequency in phase-shifting techniques," Opt. Lasers Eng. 30, 107-115 (1998).
[CrossRef]

Opt. Lett.

Sens. Actuators, A

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, "A new algorithm for large surfaces profiling by fringe projection," Sens. Actuators, A 115, 178-184 (2004).
[CrossRef]

Other

W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes Example Book (C++), 2nd ed. (Cambridge U. Press, 2002).

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Figures (6)

Fig. 1
Fig. 1

(a) Geometry of the measurement system, (b) vicinity of E.

Fig. 2
Fig. 2

(a) Partial-sphere specimen with carrier fringes in the x direction, (b) unwrapped phase map.

Fig. 3
Fig. 3

Phase distribution after the removal of (a) a linear carrier, (b) a carrier obtained by second-order line fitting.

Fig. 4
Fig. 4

(a) Partial-sphere specimen with carrier fringes in an arbitrary direction, (b) phase distribution after the removal of a carrier obtained by independent line fitting in the x and y directions.

Fig. 5
Fig. 5

Coin specimen with carrier fringes in an arbitrary direction.

Fig. 6
Fig. 6

Phase distribution after the removal of (a) a linear carrier, (b) a carrier obtained by second-order surface fitting.

Equations (19)

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I ( x , y ) = I 0 ( x , y ) + I M ( x , y ) cos [ 2 π f x + ϕ ( x , y ) ] ,
cos α = h h 2 + ( d + x ) 2 ,
f E F = l 1 l 2 + x sin β f A B ,
f ( x ) = f E F cos α .
f ( x ) = f A B l 1 h ( l 2 + x sin β ) h 2 + ( d + x ) 2 .
ϕ r ( x ) = 0 x 2 π f ( u ) d u + ϕ r ( 0 ) = 2 π f A B l 1 h 0 x ( l 2 + u sin β ) 1 [ h 2 + ( d + u ) 2 ] 1 2 d u + ϕ r ( 0 ) ,
( l 2 + u sin β ) 1 = n = 0 p n ( u sin β l 2 ) n , converges when u sin β l 2 < 1 ,
[ h 2 + ( d + u ) 2 ] 1 2 = n = 0 q n ( d + u h ) 2 n , converges when ( d + u h ) 2 < 1 ,
n = 0 p n ( u sin β l 2 ) n = n = 0 b n u n ,
n = 0 q n ( d + u h ) 2 n = n = 0 c n u n .
ϕ r ( x ) = n = 0 a n x n ,
E r ( a 0 , a 1 , a N ) = ( x , y ) U [ a 0 + a 1 x + + a N x N ϕ r , exp ( x , y ) ] 2 ,
X ( N + 1 ) × ( N + 1 ) A ( N + 1 ) × 1 = B ( N + 1 ) × 1 ,
A ( N + 1 ) × 1 = [ a 0 a 1 a N ] T ,
B ( N + 1 ) × 1 = ( x , y ) U [ ϕ r , exp ( x , y ) ϕ r , exp ( x , y ) x ϕ r , exp ( x , y ) x N ] T ,
X ( N + 1 ) × ( N + 1 ) = ( x , y ) U [ 1 x x N x x 2 x N + 1 x N x N + 1 x 2 N ]
ϕ r ( x , y ) = a 0 , 0 + a 0 , 1 x + + a 0 , N 1 x N 1 + a 0 , N x N + a 1 , 0 y + a 1 , 1 x y + + + + + a N 1 , 0 y N 1 + a N 1 , 1 x y N 1 + a N , 0 y N ,
E r ( a 0 , 0 , , a 0 , N , , a N , 0 ) = ( x , y ) U [ ϕ r ( x , y ) ϕ r , exp ( x , y ) ] 2 ,
X [ ( N + 1 ) ( N + 2 ) 2 ] × [ ( N + 1 ) ( N + 2 ) 2 ] A [ ( N + 1 ) ( N + 2 ) 2 ] × 1 = B [ ( N + 1 ) ( N + 2 ) 2 ] × 1 ,

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