Abstract

We use the regularization theory in a Bayesian framework to derive a quadratic cost function for denoising fringe patterns. As prior constraints for the regularization problem, we propose a Markov random field model that includes information about the fringe orientation. In our cost function the regularization term imposes constraints to the solution (i.e., the filtered image) to be smooth only along the fringe’s tangent direction. In this way as the fringe information and noise are conveniently separated in the frequency space, our technique avoids blurring the fringes. The attractiveness of the proposed filtering method is that the minimization of the cost function can be easily implemented using iterative methods. To show the performance of the proposed technique we present some results obtained by processing simulated and real fringe patterns.

© 2009 Optical Society of America

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References

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  1. S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
    [CrossRef]
  2. M. Rivera and J. L. Marroquin, Image Vis. Comput. 21, 345 (2003).
    [CrossRef]
  3. J. E. Besag, J. R. Stat. Soc. Ser. B (Methodol.) 36, 192 (1974).
  4. J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
    [CrossRef]
  5. M. Rivera and J. L. Marroquin, Opt. Lett. 29, 504 (2004).
    [CrossRef] [PubMed]
  6. C. Tang, L. Han, H. Ren, D. Zhou, Y. Chang, X. Wang, and X. Cui, Opt. Lett. 33, 2179 (2008).
    [CrossRef] [PubMed]
  7. H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
    [CrossRef]
  8. C. Bouman and K. Sauer, IEEE Trans. Nucl. Sci. 99, 1144 (1992).
  9. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Press, 1990).
  10. X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
    [CrossRef]

2008 (1)

2007 (1)

X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
[CrossRef]

2004 (1)

2003 (1)

M. Rivera and J. L. Marroquin, Image Vis. Comput. 21, 345 (2003).
[CrossRef]

1999 (1)

J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
[CrossRef]

1992 (1)

C. Bouman and K. Sauer, IEEE Trans. Nucl. Sci. 99, 1144 (1992).

1984 (2)

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

1974 (1)

J. E. Besag, J. R. Stat. Soc. Ser. B (Methodol.) 36, 192 (1974).

Besag, J. E.

J. E. Besag, J. R. Stat. Soc. Ser. B (Methodol.) 36, 192 (1974).

Bouman, C.

C. Bouman and K. Sauer, IEEE Trans. Nucl. Sci. 99, 1144 (1992).

Castillo, L.

J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
[CrossRef]

Chang, Y.

Cristi, R.

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

Cui, X.

Derin, H.

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

Elliot, H.

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

Fu, S.

X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
[CrossRef]

Geman, D.

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Press, 1990).

Han, L.

Marroquin, J. L.

M. Rivera and J. L. Marroquin, Opt. Lett. 29, 504 (2004).
[CrossRef] [PubMed]

M. Rivera and J. L. Marroquin, Image Vis. Comput. 21, 345 (2003).
[CrossRef]

Ren, H.

Rivera, M.

M. Rivera and J. L. Marroquin, Opt. Lett. 29, 504 (2004).
[CrossRef] [PubMed]

M. Rivera and J. L. Marroquin, Image Vis. Comput. 21, 345 (2003).
[CrossRef]

Sauer, K.

C. Bouman and K. Sauer, IEEE Trans. Nucl. Sci. 99, 1144 (1992).

Servín, M.

J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
[CrossRef]

Tang, C.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Press, 1990).

Villa, J.

J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
[CrossRef]

Wang, X.

Yang, X.

X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
[CrossRef]

Yu, Q.

X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
[CrossRef]

Zhou, D.

IEEE Trans. Nucl. Sci. (1)

C. Bouman and K. Sauer, IEEE Trans. Nucl. Sci. 99, 1144 (1992).

IEEE Trans. Pattern Anal. Mach. Intell. (2)

H. Derin, H. Elliot, R. Cristi, and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 707 (1984).
[CrossRef]

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

Image Vis. Comput. (1)

M. Rivera and J. L. Marroquin, Image Vis. Comput. 21, 345 (2003).
[CrossRef]

J. R. Stat. Soc. Ser. B (Methodol.) (1)

J. E. Besag, J. R. Stat. Soc. Ser. B (Methodol.) 36, 192 (1974).

Opt. Commun. (2)

J. Villa, M. Servín, and L. Castillo, Opt. Commun. 161, 13 (1999).
[CrossRef]

X. Yang, Q. Yu, and S. Fu, Opt. Commun. 274, 286 (2007).
[CrossRef]

Opt. Lett. (2)

Other (1)

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Press, 1990).

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Figures (3)

Fig. 1
Fig. 1

(a) Simulated noisy fringe pattern. (b) Result obtained by minimizing cost function (7). (c) Result obtained by minimizing cost function (10).

Fig. 2
Fig. 2

(a) Real moiré fringe pattern. (b) Result obtained by minimizing cost function (7). (c) Result obtained by minimizing cost function (10).

Fig. 3
Fig. 3

(a) Real ESPI fringe pattern. (b) Result obtained by minimizing cost function (7). (c) Result obtained by minimizing cost function (10).

Equations (18)

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y = H ( x ) + n ,
P x / y ( x ) = K P y / x ( x ) P x ( x ) ,
P y / x ( x ) = K 1   exp [ m L Φ ( H ( x m ) y m ) ]
P x ( x ) = K 2   exp [ C V c ( x m ) ] ,
U ( x ) = m L Φ ( H ( x m ) y m ) + μ C V c ( x m ) ,
x ̂ = argmin x [ U ( x ) ] .
U ( x m ) = m L [ x m y m ] 2 + μ m , l L [ x m x l ] 2 .
x ρ = ( x m x h ) cos   θ m + ( x m x v ) sin   θ m ,
U ( x m ) = m L [ x m y m ] 2 + μ m , h , v L [ ( x m x h ) cos   θ m + ( x m x v ) sin   θ m ] 2 .
U ( x i , j ) = ( i , j ) L { [ x i , j y i , j ] 2 + μ [ ( x ρ ) i , j 2 + ( x ρ ) i + 1 , j 2 + ( x ρ ) i , j + 1 2 ] } ,
( x ρ ) i , j = ( x i , j x i 1 , j ) c i , j + ( x i , j x i , j 1 ) s i , j ,
( x ρ ) i + 1 , j = ( x i + 1 , j x i , j ) c i + 1 , j + ( x i + 1 , j x i + 1 , j 1 ) s i + 1 , j ,
( x ρ ) i , j + 1 = ( x i , j + 1 x i 1 , j + 1 ) c i , j + 1 + ( x i , j + 1 x i , j ) s i , j + 1 ,
U x i , j = 2 [ x i , j y i , j ] + 2 μ [ ( x i , j x i 1 , j ) c i , j + ( x i , j x i , j 1 ) s i , j ] ( c i , j + s i , j ) 2 μ [ ( x i + 1 , j x i , j ) c i + 1 , j + ( x i + 1 , j x i + 1 , j 1 ) s i + 1 , j ] c i + 1 , j 2 μ [ ( x i , j + 1 x i 1 , j + 1 ) c i , j + 1 + ( x i , j + 1 x i , j ) s i , j + 1 ] s i , j + 1 = 0.
x k + 1 = x k λ U k x ,
d i , j 0 = | y i 1 , j y i + 1 , j | 2 ,     d i , j 45 = | y i 1 , j + 1 y i + 1 , j 1 | ,
d i , j 90 = | y i , j 1 y i , j + 1 | 2 ,     d i , j 135 = | y i 1 , j 1 y i + 1 , j + 1 | .
β i , j = 1 2 arctan ( ( k , l ) Γ d k , l 45 ( k , l ) Γ d k , l 135 , ( k , l ) Γ d k , l 0 ( k , l ) Γ d k , l 90 ) ,

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