Abstract

Phase unwrapping plays a very important role in noncontact optical profilometries and interforometric techniques. This phase unwrapping process is often hindered by the presence of noise, spots of low-intensity modulation, and the instability of the solutions. We present a systematic approach to the problem by formulating the phase solving and unwrapping as an ill-posed problem. By using the regularization method and taking intensity modulation into consideration, we will be able to provide a unified view to address various challenges in the problem. A new fully automated algorithm for phase unwrapping based on this approach is also presented.

© 2007 Optical Society of America

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References

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  1. X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
    [CrossRef]
  2. X. Y. Su, W. J. Chen, Opt. Lasers Eng. 42, 245 (2004).
    [CrossRef]
  3. J. L. Marroquin, M. Rivera, J. Opt. Soc. Am. A 12, 2393 (1995).
    [CrossRef]
  4. A. N. Tikhonov, Solutions of Ill-Posed Problems (V. H. Winston, 1977).
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    [CrossRef]

2004

1995

1994

1993

X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
[CrossRef]

Chen, W. J.

X. Y. Su, W. J. Chen, Opt. Lasers Eng. 42, 245 (2004).
[CrossRef]

Ghiglia, D. C.

Marroquin, J. L.

Rivera, J. M.

Rivera, M.

Romero, L. A.

Su, X. Y.

X. Y. Su, W. J. Chen, Opt. Lasers Eng. 42, 245 (2004).
[CrossRef]

X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, Solutions of Ill-Posed Problems (V. H. Winston, 1977).

von Bally, G.

X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
[CrossRef]

Vukicevic, D.

X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

X. Y. Su, G. von Bally, D. Vukicevic, Opt. Commun. 98, 141 (1993).
[CrossRef]

Opt. Lasers Eng.

X. Y. Su, W. J. Chen, Opt. Lasers Eng. 42, 245 (2004).
[CrossRef]

Opt. Lett.

Other

A. N. Tikhonov, Solutions of Ill-Posed Problems (V. H. Winston, 1977).

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Figures (3)

Fig. 1
Fig. 1

A graph model for images.

Fig. 2
Fig. 2

(a) Original five phase shifted images and (b) the wrapped phase.

Fig. 3
Fig. 3

Unwrapped phase obtained by (a) a path-dependent method and (b) our unwrapping algorithm followed by five iterations of optimization.

Equations (10)

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n = 0 N 1 I n ( x , y ) cos ( 2 π n N ) = M ( x , y ) cos [ ϕ ( x , y ) ] ,
n = 0 N 1 I n ( x , y ) sin ( 2 π n N ) = M ( x , y ) sin [ ϕ ( x , y ) ] ,
ϕ ( x , y ) = tan 1 [ n = 0 N 1 I n ( x , y ) sin ( 2 π n N ) n = 0 N 1 I n ( x , y ) cos ( 2 π n N ) ] .
[ M ( x , y ) cos ϕ ( x , y ) M ( x , y ) sin ϕ ( x , y ) ] = [ n = 0 N 1 I n ( x , y ) cos ( 2 π n N ) n = 0 N 1 I n ( x , y ) sin ( 2 π n N ) ] = [ C ( x , y ) S ( x , y ) ] .
F ( ϕ ) = ( x , y ) [ M ( x , y ) cos ϕ ( x , y ) C ( x , y ) ] 2 + [ M ( x , y ) sin ϕ ( x , y ) S ( x , y ) ] 2 .
Ω ( ϕ ) = ( x i , y i ) ( x j , y j ) [ ϕ ( x i , y i ) ϕ ( x j , y j ) ] 2 .
R ( ϕ ) = F ( ϕ ) + λ Ω ( ϕ ) .
R ϕ x , y = M ( x , y ) [ C ( x , y ) sin ϕ x , y S ( x , y ) cos ϕ x , y ] + λ [ ϕ x , y 1 N ( x , y ) ( u , v ) N ( x , y ) ϕ u , v ] .
[ C ( x 1 , y 1 ) C ( x 2 , y 2 ) ] 2 + [ S ( x 1 , y 1 ) S ( x 2 , y 2 ) ] 2 1 + [ C ( x 1 , y 1 ) 2 + S ( x 1 , y 1 ) 2 ] [ C ( x 2 , y 2 ) 2 + S ( x 2 , y 2 ) 2 ] .
ϕ x , y = 1 N ( x , y ) ( u , v ) N ( x , y ) ϕ u , v .

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