Abstract

Under a nonparallel illumination condition, fringe patterns projected on an object have unequal fringe spacing that would introduce a nonlinear carrier phase component. This Letter describes a nonlinear carrier removal technique based on a least-squares approach. In contrast with conventional methods, the proposed algorithm would not magnify phase measurement uncertainty, nor does it require direct estimation of system geometrical parameters. The theoretical expression of the carrier phase function on the reference is derived and expanded in a power series. The unknown coefficients in the series are determined by a least-squares method. By subtracting the calculated carrier phase function from the unwrapped phase map, the phase distribution of the object profile is obtained.

© 2005 Optical Society of America

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References

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2004 (1)

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

2003 (1)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

2000 (1)

F. Chen, G. M. Brown, and M. Song, Opt. Eng. 39, 10 (2000).
[CrossRef]

1985 (1)

1984 (1)

1983 (1)

Bosch, T.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, Opt. Eng. 39, 10 (2000).
[CrossRef]

Chen, F.

F. Chen, G. M. Brown, and M. Song, Opt. Eng. 39, 10 (2000).
[CrossRef]

Dallier, R.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

Garcia, V.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

Halioua, M.

Liu, H. C.

Luna, E.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

Mutoh, K.

Pavageau, S.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

Salas, L.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

Salinas, J.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

Servagent, N.

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

Servin, M.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

Song, M.

F. Chen, G. M. Brown, and M. Song, Opt. Eng. 39, 10 (2000).
[CrossRef]

Srinivasan, V.

Takeda, M.

Appl. Opt. (3)

Opt. Eng. (2)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, Opt. Eng. 42, 3307 (2003).
[CrossRef]

F. Chen, G. M. Brown, and M. Song, Opt. Eng. 39, 10 (2000).
[CrossRef]

Sens. Actuators, A (1)

S. Pavageau, R. Dallier, N. Servagent, and T. Bosch, Sens. Actuators, A 115, 178 (2004).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Geometry of the measurement system.

Fig. 2
Fig. 2

(a) Fringe pattern projected on a partial sphere, (b) unwrapped phase map.

Fig. 3
Fig. 3

Phase distribution after removal of (a) linear and (b) nonlinear carriers.

Equations (13)

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cos θ = h [ h 2 + ( d + x ) 2 ] 1 2 ,
P EF = l 2 + x sin α l 1 P AB ,
P ( x ) = P EF cos θ .
P ( x ) = l 2 + x sin α l 1 P AB [ h 2 + ( d + x ) 2 ] 1 2 h .
ϕ r ( x ) = 0 x 2 π P ( u ) d u = 2 l 1 π l 2 P AB 0 x ( 1 + u sin α l 2 ) 1 [ 1 + ( d + u h ) 2 ] 1 2 d u ,
( 1 + u sin α l 2 ) 1 = n = 0 p n ( u sin α l 2 ) n , converges when u sin α l 2 < 1 ,
[ 1 + ( d + u h ) 2 ] 1 2 = n = 0 q n ( d + u h ) 2 n , converges when ( d + u h ) 2 < 1 ,
ϕ r ( x ) = n = 0 a n x n ,
Er ( 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 ,
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 ] ,
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 ,

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