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February  2016, 10(1): 27-50. doi: 10.3934/ipi.2016.10.27

## 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

Received  December 2013 Revised  June 2015 Published  February 2016

In this paper, we propose a unified variational framework for noise removal, which uses a combination of different orders of fractional derivatives in the regularization term of the objective function. The principle of the combination is taking the order two or higher derivatives for smoothing the homogeneous regions, and a fractional order less than or equal to one to smooth the locations near the edges. We also introduce a novel edge detector to better detect edges and textures. A main advantage of this framework is the superiority in dealing with textures and repetitive structures as well as eliminating the staircase effect. To effectively solve the proposed model, we extend the first-order primal dual algorithm to minimize a functional involving fractional-order derivatives. A set of experiments demonstrates that the proposed method is able to avoid the staircase effect and preserve accurately edges and structural details of the image while removing the noise.
Citation: Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27
##### References:
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Comput., 22 (2000), 503.  doi: 10.1137/S1064827598344169.  Google Scholar [6] N. Engheta, On the role of fractional calculus in electromagnetic theory,, IEEE Antennas Propagat. Mag., 39 (1997), 35.   Google Scholar [7] P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising,, J. Math. Imaging Vis., 40 (2011), 188.  doi: 10.1007/s10851-010-0256-9.  Google Scholar [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.  doi: 10.1109/ICSAI.2012.6223398.  Google Scholar [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.  doi: 10.1016/j.jvcir.2007.04.005.  Google Scholar [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.   Google Scholar [11] A. Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer,, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, (2004).   Google Scholar [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.  doi: 10.1109/TIP.2003.819229.  Google Scholar [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.  doi: 10.1007/s11263-005-3219-7.  Google Scholar [14] B. Ninness, Estimation of $1/f$ noise,, IEEE Trans. Inf. Theory, 44 (1998), 32.  doi: 10.1109/18.650986.  Google Scholar [15] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629.  doi: 10.1109/34.56205.  Google Scholar [16] I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).   Google Scholar [17] W. Ring, Structural properties of solutions to total variation regularization problems,, Math. Model. Numer. Anal., 34 (2000), 799.  doi: 10.1051/m2an:2000104.  Google Scholar [18] B. Ross, A brief history and exposition of the fundamental theory of fractional calculus,, in Fractional Calculus and Its Applications, (2006), 1.  doi: 10.1007/BFb0067096.  Google Scholar [19] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica. D., 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [20] M. Unser and Splines, A perfect fit for signal and image processing,, IEEE Signal Process. Mag., 16 (1999), 22.  doi: 10.1109/79.799930.  Google Scholar [21] M. Unser and T. Blu, Fractional splines and wavelets,, SIAM Rev., 42 (2000), 43.  doi: 10.1137/S0036144598349435.  Google Scholar [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.  doi: 10.1109/TIP.2003.819861.  Google Scholar [23] J. Weickert, Anisotropic Diffusion in Image Processing,, Stutgart, (1998).   Google Scholar [24] Y. You and M. Kaveh, Fourth order partial differential equations for noise removal,, IEEE Trans. Image Process., 9 (2000), 1723.  doi: 10.1109/83.869184.  Google Scholar [25] J. Zhang and Z. Wei, Fractional variational model and algorithm for image denoising,, IEEE Fourth International Conference on Natural Computation, 5 (2008), 524.  doi: 10.1109/ICNC.2008.172.  Google Scholar [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.  doi: 10.1016/j.apm.2010.11.049.  Google Scholar [27] J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising,, J. Math. Imaging Vis., 43 (2012), 39.  doi: 10.1007/s10851-011-0285-z.  Google Scholar

show all references

##### References:
 [1] J. Bai and X. Feng, Fractional-order anisotropic diffusion for image denoising,, IEEE Trans. Image Process., 16 (2007), 2492.  doi: 10.1109/TIP.2007.904971.  Google Scholar [2] A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems,, Numer. Math., 76 (1997), 167.  doi: 10.1007/s002110050258.  Google Scholar [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.  doi: 10.1016/j.jvcir.2006.12.004.  Google Scholar [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), 4137.  doi: 10.1109/ICIP.2010.5653199.  Google Scholar [5] T. Chan, A. Marquina and P. and Mulet, High-order total variation-based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503.  doi: 10.1137/S1064827598344169.  Google Scholar [6] N. Engheta, On the role of fractional calculus in electromagnetic theory,, IEEE Antennas Propagat. Mag., 39 (1997), 35.   Google Scholar [7] P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising,, J. Math. Imaging Vis., 40 (2011), 188.  doi: 10.1007/s10851-010-0256-9.  Google Scholar [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.  doi: 10.1109/ICSAI.2012.6223398.  Google Scholar [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.  doi: 10.1016/j.jvcir.2007.04.005.  Google Scholar [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.   Google Scholar [11] A. Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer,, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, (2004).   Google Scholar [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.  doi: 10.1109/TIP.2003.819229.  Google Scholar [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.  doi: 10.1007/s11263-005-3219-7.  Google Scholar [14] B. Ninness, Estimation of $1/f$ noise,, IEEE Trans. Inf. Theory, 44 (1998), 32.  doi: 10.1109/18.650986.  Google Scholar [15] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629.  doi: 10.1109/34.56205.  Google Scholar [16] I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).   Google Scholar [17] W. Ring, Structural properties of solutions to total variation regularization problems,, Math. Model. Numer. Anal., 34 (2000), 799.  doi: 10.1051/m2an:2000104.  Google Scholar [18] B. Ross, A brief history and exposition of the fundamental theory of fractional calculus,, in Fractional Calculus and Its Applications, (2006), 1.  doi: 10.1007/BFb0067096.  Google Scholar [19] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica. D., 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [20] M. Unser and Splines, A perfect fit for signal and image processing,, IEEE Signal Process. Mag., 16 (1999), 22.  doi: 10.1109/79.799930.  Google Scholar [21] M. Unser and T. Blu, Fractional splines and wavelets,, SIAM Rev., 42 (2000), 43.  doi: 10.1137/S0036144598349435.  Google Scholar [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.  doi: 10.1109/TIP.2003.819861.  Google Scholar [23] J. Weickert, Anisotropic Diffusion in Image Processing,, Stutgart, (1998).   Google Scholar [24] Y. You and M. Kaveh, Fourth order partial differential equations for noise removal,, IEEE Trans. Image Process., 9 (2000), 1723.  doi: 10.1109/83.869184.  Google Scholar [25] J. Zhang and Z. Wei, Fractional variational model and algorithm for image denoising,, IEEE Fourth International Conference on Natural Computation, 5 (2008), 524.  doi: 10.1109/ICNC.2008.172.  Google Scholar [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.  doi: 10.1016/j.apm.2010.11.049.  Google Scholar [27] J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising,, J. Math. Imaging Vis., 43 (2012), 39.  doi: 10.1007/s10851-011-0285-z.  Google Scholar
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