Abstract

We present a simple and effective method for denoising phase patterns based on a discrete model. The proposed filtering method transforms the image denoising problem to solving the energy diffusion problem of a system with complex-valued fields. We establish an appropriate cost function that uses the discrete form of complex-valued Markov random fields. The attractiveness of the proposed filtering method includes three points: the first is that the filtering process can be easily implemented using an iterative method, the second is that 2π phase jumps are well preserved, and the third is its little computational effort. The performance of the proposed method is demonstrated by simulated and experimentally obtained phase patterns.

© 2010 Optical Society of America

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase-Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  2. S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
    [CrossRef] [PubMed]
  3. J. A. Langley, R. G. Brice, and Q. Zhao, “Recursive approach to the comment-based phase unwrapping algorithm,” Appl. Opt. 49, 3096–3101 (2010).
    [CrossRef] [PubMed]
  4. M. A. Herraéz, D. R. Burton, and M. J. Lalor, “Clustering-based robust three- dimensional phase unwrapping algorithm,” Appl. Opt. 49, 1780–1788 (2010).
    [CrossRef]
  5. K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier filtered and quality guided phase unwrapping algorithm: on locally high-order polynomial phase,” Appl. Opt. 49, 1764(2010).
    [CrossRef] [PubMed]
  6. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  7. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673(1987).
    [CrossRef] [PubMed]
  8. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]
  9. F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
    [CrossRef]
  10. K. M. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  11. K. M. Qian, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
    [CrossRef]
  12. C. Tang, W. P. Wang, H. Q. Yan, and X. H. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46, 2907–2913 (2007).
    [CrossRef] [PubMed]
  13. K. M. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  14. K. M. Qian, H. X. Wang, and W. J. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408–5419 (2008).
    [CrossRef]
  15. K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
    [CrossRef]
  16. J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
    [CrossRef] [PubMed]
  17. C. Tang, T. Gao, S. Yan, L. L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase pattern,” Opt. Express 18, 8942–8947 (2010).
    [CrossRef] [PubMed]
  18. J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
    [CrossRef]
  19. J. Villa, J. A. Quiroga, and I. de la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34, 1741–1743 (2009).
    [CrossRef] [PubMed]
  20. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins, 1990).
  21. P. Perona and J. Malik, “Scale-pace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
    [CrossRef]
  22. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
    [CrossRef]
  23. S. M. Chao and D. M. Tsai, “Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images,” Pattern Recogn. 43, 1917–1931 (2010).
    [CrossRef]
  24. P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. L. Yao, “Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy,” Opt. Lett. 35, 712–714 (2010).
    [CrossRef] [PubMed]

2010 (9)

S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
[CrossRef] [PubMed]

J. A. Langley, R. G. Brice, and Q. Zhao, “Recursive approach to the comment-based phase unwrapping algorithm,” Appl. Opt. 49, 3096–3101 (2010).
[CrossRef] [PubMed]

M. A. Herraéz, D. R. Burton, and M. J. Lalor, “Clustering-based robust three- dimensional phase unwrapping algorithm,” Appl. Opt. 49, 1780–1788 (2010).
[CrossRef]

K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier filtered and quality guided phase unwrapping algorithm: on locally high-order polynomial phase,” Appl. Opt. 49, 1764(2010).
[CrossRef] [PubMed]

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
[CrossRef] [PubMed]

C. Tang, T. Gao, S. Yan, L. L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase pattern,” Opt. Express 18, 8942–8947 (2010).
[CrossRef] [PubMed]

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

S. M. Chao and D. M. Tsai, “Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images,” Pattern Recogn. 43, 1917–1931 (2010).
[CrossRef]

P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. L. Yao, “Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy,” Opt. Lett. 35, 712–714 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

2007 (3)

2004 (2)

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

K. M. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

1999 (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

1997 (1)

1990 (1)

P. Perona and J. Malik, “Scale-pace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

1987 (1)

1982 (1)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Brice, R. G.

Burton, D. R.

Chao, S. M.

S. M. Chao and D. M. Tsai, “Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images,” Pattern Recogn. 43, 1917–1931 (2010).
[CrossRef]

Cheng, J.

de la Rosa, I.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

J. Villa, J. A. Quiroga, and I. de la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34, 1741–1743 (2009).
[CrossRef] [PubMed]

Feng, L.

Fienup, J. R.

Gao, P.

Gao, T.

Gao, W. J.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase-Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Golub, G. H.

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

Gonocalves, E.

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

González, E.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Gu, X. H.

Harder, I.

Herraéz, M. A.

Heshmat, S.

Ina, H.

Jiang, J.

Kobayashi, S.

Lalor, M. J.

Langley, J. A.

Luong, B.

Malik, J.

P. Perona and J. Malik, “Scale-pace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

Mantel, K.

Miyamoto, N.

Nam, L. T. H.

Nercissian, V.

Nishiyama, S.

Palacios, F.

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Perona, P.

P. Perona and J. Malik, “Scale-pace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase-Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Qian, K. M.

Quiroga, J. A.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

J. Villa, J. A. Quiroga, and I. de la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34, 1741–1743 (2009).
[CrossRef] [PubMed]

Ricardo, J.

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Roddier, C.

Roddier, F.

Soon, S. H.

Takeda, M.

Tang, C.

Tomioka, S.

Tsai, D. M.

S. M. Chao and D. M. Tsai, “Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images,” Pattern Recogn. 43, 1917–1931 (2010).
[CrossRef]

Valin, J. L.

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Van Loan, C. F.

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

Vera, R. R.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Villa, J.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

J. Villa, J. A. Quiroga, and I. de la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34, 1741–1743 (2009).
[CrossRef] [PubMed]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Wang, H. X.

Wang, L. L.

Wang, W. P.

Wu, J.

Yan, H. Q.

Yan, S.

Yao, B. L.

Zhao, Q.

Appl. Opt. (12)

S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
[CrossRef] [PubMed]

J. A. Langley, R. G. Brice, and Q. Zhao, “Recursive approach to the comment-based phase unwrapping algorithm,” Appl. Opt. 49, 3096–3101 (2010).
[CrossRef] [PubMed]

M. A. Herraéz, D. R. Burton, and M. J. Lalor, “Clustering-based robust three- dimensional phase unwrapping algorithm,” Appl. Opt. 49, 1780–1788 (2010).
[CrossRef]

K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier filtered and quality guided phase unwrapping algorithm: on locally high-order polynomial phase,” Appl. Opt. 49, 1764(2010).
[CrossRef] [PubMed]

K. M. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

K. M. Qian, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[CrossRef]

C. Tang, W. P. Wang, H. Q. Yan, and X. H. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46, 2907–2913 (2007).
[CrossRef] [PubMed]

K. M. Qian, H. X. Wang, and W. J. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408–5419 (2008).
[CrossRef]

K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
[CrossRef]

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
[CrossRef] [PubMed]

C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673(1987).
[CrossRef] [PubMed]

J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

P. Perona and J. Malik, “Scale-pace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

F. Palacios, E. Gonocalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

K. M. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Opt. Lett. (2)

Pattern Recogn. (1)

S. M. Chao and D. M. Tsai, “Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images,” Pattern Recogn. 43, 1917–1931 (2010).
[CrossRef]

Other (2)

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

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase-Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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Figures (6)

Fig. 1
Fig. 1

Pixel m neighborhood.

Fig. 2
Fig. 2

First experiment with a simulated phase pattern. (a) Noise phase pattern. (b) Result obtained using the proposed method. (c) Result obtained using WFT.

Fig. 3
Fig. 3

Second experiment with a simulated phase pattern. (a) Noisy phase pattern. (b) Result obtained using WFT. (c) Result obtained using the proposed method. (d) Result obtained using the isotropic model reported in [18].

Fig. 4
Fig. 4

The filtered result with β = 0 for the second experiment with a simulated wrapped phase pattern. (a) Result obtained using the isotropic model reported in [18]. (b) Result obtained using our proposed method.

Fig. 5
Fig. 5

(a) The relation curve of ε and β for the same iterative times. (b) The relation curve of ε and iterative numbers ( Δ T ) for the same β.

Fig. 6
Fig. 6

(a) Experimentally obtained wrapped phase pattern. (b) Result obtained using the proposed method. (c) Result obtained using the WFT method.

Tables (1)

Tables Icon

Table 1 Performance of the Proposed Filter, the Isotropic Filter Reported in [18], and the Windowed Fourier Filter, represented by F A , U A and WFT, Respectively

Equations (10)

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v = e i φ ,
E d ( u ) = β m ( u m v m ) 2 + μ m p n m ( u m u p ) 2 ,
E d ( u ) u m = 2 β ( u m v m ) + 4 μ p n m ( u m u p ) .
2 β ( u m v m ) + 4 μ p n m ( u m u p ) = 0.
u t m = u t 1 m λ [ β ( u t 1 m v m ) + 2 μ p n m ( u t 1 m u t 1 p ) ] ,
I i ( x , y ) = 1 + cos [ φ ( x , y ) + ( i 1 ) π / 2 ] + n i ( x , y ) , i = 1 , 2 , 3 , 4.
φ = φ 1 ( x , y ) = { 0.002 × ρ 2 ρ 2 = x 2 + y 2 127 2 0.002 × 127 2 ρ 2 = x 2 + y 2 > 127 2 .
f p s e ( x , y ) = [ I 1 ( x , y ) I 3 ( x , y ) + j I 4 ( x , y ) j I 2 ( x , y ) ] / 2 = exp ( j φ 1 ) ,
φ = φ 2 ( x , y ) = 0.001 [ ( x 256 ) 2 + ( y 256 ) 2 ] + 6 [ peaks ( 512 ) ] .
ε = | α α ^ | 2 / | α | 2 ,

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