Abstract

We derive the fourth-order oriented partial-differential equations (PDEs) for noise removal of two-photon fluorescence images. We consider it from two aspects: one is based on a variational method; the other is based on controlling the diffusion direction. Our filtering model makes the diffusion along only the special orientation—decided via all the information in the established filtering window, so the edges are protected during filtering. Compared with related PDEs models, our model shows superior performance in terms of both objective criteria and subjective human vision via processing simulated and experimental noisy images.

© 2010 Optical Society of America

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    [CrossRef]

2009 (2)

2007 (2)

H. M. Zhang, Q. M. Luo, and S. Q. Zeng, Proc. SPIE 6534, 65343H (2007).
[CrossRef]

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

2006 (1)

2004 (1)

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

2001 (1)

S. T. Li, T. K. James, and Y. N. Wang, Inf. Fusion 2, 169 (2001).
[CrossRef]

2000 (1)

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[CrossRef]

1992 (2)

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

L. Alvarez, P. L. Lions, and J. M. Morel, SIAM J. Numer. Anal. 29, 845 (1992).
[CrossRef]

1990 (1)

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Machine Intell. 12, 629 (1990).
[CrossRef]

Alan, C. B.

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

Alvarez, L.

L. Alvarez, P. L. Lions, and J. M. Morel, SIAM J. Numer. Anal. 29, 845 (1992).
[CrossRef]

Catté, F.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

Coll, T.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

Fu, S. H.

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

Gao, W. J.

Ismael, R.

James, T. K.

S. T. Li, T. K. James, and Y. N. Wang, Inf. Fusion 2, 169 (2001).
[CrossRef]

Jesús, V.

Juan, A. Q.

Kaveh, M.

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[CrossRef]

Li, B. T.

Li, S. T.

S. T. Li, T. K. James, and Y. N. Wang, Inf. Fusion 2, 169 (2001).
[CrossRef]

Lin, F.

Lions, P. L.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

L. Alvarez, P. L. Lions, and J. M. Morel, SIAM J. Numer. Anal. 29, 845 (1992).
[CrossRef]

Luo, Q. M.

H. M. Zhang, Q. M. Luo, and S. Q. Zeng, Proc. SPIE 6534, 65343H (2007).
[CrossRef]

Malik, J.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Machine Intell. 12, 629 (1990).
[CrossRef]

Morel, J. M.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

L. Alvarez, P. L. Lions, and J. M. Morel, SIAM J. Numer. Anal. 29, 845 (1992).
[CrossRef]

Perona, P.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Machine Intell. 12, 629 (1990).
[CrossRef]

Qian, K. M.

Seah, H. S.

Sheikh, H. R.

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

Simoncelli, E. P.

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

Tang, C.

Wang, H. X.

Wang, Y. N.

S. T. Li, T. K. James, and Y. N. Wang, Inf. Fusion 2, 169 (2001).
[CrossRef]

Wang, Z.

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

Witkin, A. P.

A. P. Witkin, in Proceedings of the International Joint Conferences on Artificial Intelligence (Morgan Kaufmann, 1983), pp. 1019–1021.

Yan, H. Q.

Yang, X.

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

You, Y. L.

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[CrossRef]

Yu, Q. F.

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

Zeng, S. Q.

H. M. Zhang, Q. M. Luo, and S. Q. Zeng, Proc. SPIE 6534, 65343H (2007).
[CrossRef]

Zhang, F.

Zhang, H. M.

H. M. Zhang, Q. M. Luo, and S. Q. Zeng, Proc. SPIE 6534, 65343H (2007).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Imag. Proc. (1)

Z. Wang, C. B. Alan, H. R. Sheikh, and E. P. Simoncelli, IEEE Trans. Imag. Proc. 13, 600 (2004).
[CrossRef]

IEEE Trans. Image Process. (1)

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[CrossRef]

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

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Machine Intell. 12, 629 (1990).
[CrossRef]

Inf. Fusion (1)

S. T. Li, T. K. James, and Y. N. Wang, Inf. Fusion 2, 169 (2001).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (2)

Proc. SPIE (1)

H. M. Zhang, Q. M. Luo, and S. Q. Zeng, Proc. SPIE 6534, 65343H (2007).
[CrossRef]

SIAM J. Numer. Anal. (2)

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, SIAM J. Numer. Anal. 29, 182 (1992).
[CrossRef]

L. Alvarez, P. L. Lions, and J. M. Morel, SIAM J. Numer. Anal. 29, 845 (1992).
[CrossRef]

Other (1)

A. P. Witkin, in Proceedings of the International Joint Conferences on Artificial Intelligence (Morgan Kaufmann, 1983), pp. 1019–1021.

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Figures (2)

Fig. 1
Fig. 1

(a) Original “Lena” image, (b) its Gaussian noisy image, and (c) and (d) the denoised images with n = 4 in Eqs. (1, 8), respectively.

Fig. 2
Fig. 2

Experimentally obtained noisy microscopic image and its denoised images: (a) noisy microscopic image, (b) and (c) denoised images with n = 15 in Eqs. (1, 8), respectively, and (d) and (e) the denoised images with n = 20 in Eqs. (1, 8), respectively.

Equations (9)

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u t = g ( | ( G σ * u ) | ) | u | div ( u | u | ) .
E ( u ) = Ω 1 2 | 2 u ρ 2 | 2 d x d y .
f u x ( f u x ) y ( f u y ) + x x ( f u x x ) + y y ( f u y y ) = 0 .
f = 1 2 | 2 u ρ 2 | 2 = 1 2 ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) 2 ,
u t = ( u x x x x cos 4 θ + u y y x x sin 2 θ cos 2 θ + 2 u x y x x sin θ cos 3 θ + u y y y y sin 4 θ + u x x y y sin 2 θ cos 2 θ + 2 u x y y y sin 3 θ cos θ ) ,
u t = g ( · ) ( u x x x x cos 4 θ + u y y x x sin 2 θ cos 2 θ + 2 u x y x x sin θ cos 3 θ + u y y y y sin 4 θ + u x x y y sin 2 θ cos 2 θ + 2 u x y y y sin 3 θ cos θ ) .
θ ( i , j ) = 1 2 arctan 1 ( ( 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 ) .
u i , j n + 1 = u i , j n Δ t g i , j n ( s ) ( ( u x x x x ) i , j n cos 4 θ + ( u y y x x ) i , j n sin 2 θ cos 2 θ + 2 ( u x y x x ) i , j n sin θ cos 3 θ + ( u y y y y ) i , j n sin 4 θ + ( u x x y y ) i , j n sin 2 θ cos 2 θ + 2 ( u x y y y ) i , j n sin 3 θ cos θ ) ,
g i , j n = 1 1 + k ( ( ( G σ * u ) x x + ( G σ * u ) y y ) 2 ) i j n ,

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