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On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy
A fractional-order derivative based variational framework for image denoising
1. | School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China |
2. | Department of Mathematics, 358 Little Hall, PO Box 118105, Gainesville, FL 32611 |
References:
[1] |
J. Bai and X. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.
doi: 10.1109/TIP.2007.904971. |
[2] |
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[3] |
T. F. Chan, S. Esedoglu and F. E. Park, Image decomposition combining staircase reduction and texture extraction, Journal of Visual Communication and Image Representation, 18 (2007), 464-486.
doi: 10.1016/j.jvcir.2006.12.004. |
[4] |
T. F. Chan, S. Esedoglu and F. E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, in Image Processing (ICIP), 2010 17th IEEE International Conference on, (2010), 4137-4140.
doi: 10.1109/ICIP.2010.5653199. |
[5] |
T. Chan, A. Marquina and P. and Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[6] |
N. Engheta, On the role of fractional calculus in electromagnetic theory, IEEE Antennas Propagat. Mag., 39 (1997), 35-46. |
[7] |
P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising, J. Math. Imaging Vis., 40 (2011), 188-198.
doi: 10.1007/s10851-010-0256-9. |
[8] |
X. Hu and Y. Li, A new variational model for image denoising based on fractional-order derivative, in IEEE 2012 International Conference on Systems and Informatics, (2012), 1820-1824.
doi: 10.1109/ICSAI.2012.6223398. |
[9] |
F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image R., 18 (2007), 322-330.
doi: 10.1016/j.jvcir.2007.04.005. |
[10] |
S. C. Liu and S. Chang, Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification, IEEE Trans. Image Process., 6 (1997), 1176-1184. |
[11] |
A. Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 2004. |
[12] |
M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.
doi: 10.1109/TIP.2003.819229. |
[13] |
M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vision, 66 (2006), 5-18.
doi: 10.1007/s11263-005-3219-7. |
[14] |
B. Ninness, Estimation of $1/f$ noise, IEEE Trans. Inf. Theory, 44 (1998), 32-46.
doi: 10.1109/18.650986. |
[15] |
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[16] |
I. Podlubny, Fractional Differential Equations, Academic Press, 1999. |
[17] |
W. Ring, Structural properties of solutions to total variation regularization problems, Math. Model. Numer. Anal., 34 (2000), 799-810.
doi: 10.1051/m2an:2000104. |
[18] |
B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, in Fractional Calculus and Its Applications, Lecture Notes in Math., 457, Springer-Verlag, New York, 2006, 1-36.
doi: 10.1007/BFb0067096. |
[19] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica. D., 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[20] |
M. Unser and Splines, A perfect fit for signal and image processing, IEEE Signal Process. Mag., 16 (1999), 22-38.
doi: 10.1109/79.799930. |
[21] |
M. Unser and T. Blu, Fractional splines and wavelets, SIAM Rev., 42 (2000), 43-67.
doi: 10.1137/S0036144598349435. |
[22] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[23] |
J. Weickert, Anisotropic Diffusion in Image Processing, Stutgart, B.G. Teubner, 1998. |
[24] |
Y. You and M. Kaveh, Fourth order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.
doi: 10.1109/83.869184. |
[25] |
J. Zhang and Z. Wei, Fractional variational model and algorithm for image denoising, IEEE Fourth International Conference on Natural Computation, 5 (2008), 524-528.
doi: 10.1109/ICNC.2008.172. |
[26] |
J. Zhang and Z. Wei, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, Applied Mathematical Modelling, 35 (2011), 2516-2528.
doi: 10.1016/j.apm.2010.11.049. |
[27] |
J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.
doi: 10.1007/s10851-011-0285-z. |
show all references
References:
[1] |
J. Bai and X. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.
doi: 10.1109/TIP.2007.904971. |
[2] |
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[3] |
T. F. Chan, S. Esedoglu and F. E. Park, Image decomposition combining staircase reduction and texture extraction, Journal of Visual Communication and Image Representation, 18 (2007), 464-486.
doi: 10.1016/j.jvcir.2006.12.004. |
[4] |
T. F. Chan, S. Esedoglu and F. E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, in Image Processing (ICIP), 2010 17th IEEE International Conference on, (2010), 4137-4140.
doi: 10.1109/ICIP.2010.5653199. |
[5] |
T. Chan, A. Marquina and P. and Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[6] |
N. Engheta, On the role of fractional calculus in electromagnetic theory, IEEE Antennas Propagat. Mag., 39 (1997), 35-46. |
[7] |
P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising, J. Math. Imaging Vis., 40 (2011), 188-198.
doi: 10.1007/s10851-010-0256-9. |
[8] |
X. Hu and Y. Li, A new variational model for image denoising based on fractional-order derivative, in IEEE 2012 International Conference on Systems and Informatics, (2012), 1820-1824.
doi: 10.1109/ICSAI.2012.6223398. |
[9] |
F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image R., 18 (2007), 322-330.
doi: 10.1016/j.jvcir.2007.04.005. |
[10] |
S. C. Liu and S. Chang, Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification, IEEE Trans. Image Process., 6 (1997), 1176-1184. |
[11] |
A. Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 2004. |
[12] |
M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.
doi: 10.1109/TIP.2003.819229. |
[13] |
M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vision, 66 (2006), 5-18.
doi: 10.1007/s11263-005-3219-7. |
[14] |
B. Ninness, Estimation of $1/f$ noise, IEEE Trans. Inf. Theory, 44 (1998), 32-46.
doi: 10.1109/18.650986. |
[15] |
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[16] |
I. Podlubny, Fractional Differential Equations, Academic Press, 1999. |
[17] |
W. Ring, Structural properties of solutions to total variation regularization problems, Math. Model. Numer. Anal., 34 (2000), 799-810.
doi: 10.1051/m2an:2000104. |
[18] |
B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, in Fractional Calculus and Its Applications, Lecture Notes in Math., 457, Springer-Verlag, New York, 2006, 1-36.
doi: 10.1007/BFb0067096. |
[19] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica. D., 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[20] |
M. Unser and Splines, A perfect fit for signal and image processing, IEEE Signal Process. Mag., 16 (1999), 22-38.
doi: 10.1109/79.799930. |
[21] |
M. Unser and T. Blu, Fractional splines and wavelets, SIAM Rev., 42 (2000), 43-67.
doi: 10.1137/S0036144598349435. |
[22] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[23] |
J. Weickert, Anisotropic Diffusion in Image Processing, Stutgart, B.G. Teubner, 1998. |
[24] |
Y. You and M. Kaveh, Fourth order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.
doi: 10.1109/83.869184. |
[25] |
J. Zhang and Z. Wei, Fractional variational model and algorithm for image denoising, IEEE Fourth International Conference on Natural Computation, 5 (2008), 524-528.
doi: 10.1109/ICNC.2008.172. |
[26] |
J. Zhang and Z. Wei, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, Applied Mathematical Modelling, 35 (2011), 2516-2528.
doi: 10.1016/j.apm.2010.11.049. |
[27] |
J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.
doi: 10.1007/s10851-011-0285-z. |
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