A fractional-order derivative based variational framework for image denoising

Previous Article
On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy
IPI Home
This Issue
Next Article
The topological gradient method for semi-linear problems and application to edge detection and noise removal

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

Abstract
Related Papers

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.

Keywords: fractional-order derivative, first-order primal dual algorithm., Image denoising.

Mathematics Subject Classification: Primary: 49M29, 65K10, 68U10; Secondary: 49K2.

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:

[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.
[2]	A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems,, Numer. Math., 76 (1997), 167. 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. 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), 4137. 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. doi: 10.1137/S1064827598344169.
[6]	N. Engheta, On the role of fractional calculus in electromagnetic theory,, IEEE Antennas Propagat. Mag., 39 (1997), 35.
[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.
[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.
[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.
[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.
[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. 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. doi: 10.1007/s11263-005-3219-7.
[14]	B. Ninness, Estimation of $1/f$ noise,, IEEE Trans. Inf. Theory, 44 (1998), 32. 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. 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. 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, (2006), 1. doi: 10.1007/BFb0067096.
[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.
[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.
[21]	M. Unser and T. Blu, Fractional splines and wavelets,, SIAM Rev., 42 (2000), 43. 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. doi: 10.1109/TIP.2003.819861.
[23]	J. Weickert, Anisotropic Diffusion in Image Processing,, Stutgart, (1998).
[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.
[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.
[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.
[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.

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.
[2]	A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems,, Numer. Math., 76 (1997), 167. 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. 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), 4137. 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. doi: 10.1137/S1064827598344169.
[6]	N. Engheta, On the role of fractional calculus in electromagnetic theory,, IEEE Antennas Propagat. Mag., 39 (1997), 35.
[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.
[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.
[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.
[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.
[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. 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. doi: 10.1007/s11263-005-3219-7.
[14]	B. Ninness, Estimation of $1/f$ noise,, IEEE Trans. Inf. Theory, 44 (1998), 32. 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. 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. 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, (2006), 1. doi: 10.1007/BFb0067096.
[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.
[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.
[21]	M. Unser and T. Blu, Fractional splines and wavelets,, SIAM Rev., 42 (2000), 43. 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. doi: 10.1109/TIP.2003.819861.
[23]	J. Weickert, Anisotropic Diffusion in Image Processing,, Stutgart, (1998).
[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.
[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.
[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.
[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.

[1]	Huan Han. A variational model with fractional-order regularization term arising in registration of diffusion tensor image. Inverse Problems & Imaging, 2018, 12 (6) : 1263-1291. doi: 10.3934/ipi.2018053
[2]	Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations & Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355
[3]	Zeng-bao Wu, Yun-zhi Zou, Nan-jing Huang. A new class of global fractional-order projective dynamical system with an application. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2018139
[4]	Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065
[5]	Ansgar Jüngel, Ingrid Violet. First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 861-877. doi: 10.3934/dcdsb.2007.8.861
[6]	Xiaoling Guo, Zhibin Deng, Shu-Cherng Fang, Wenxun Xing. Quadratic optimization over one first-order cone. Journal of Industrial & Management Optimization, 2014, 10 (3) : 945-963. doi: 10.3934/jimo.2014.10.945
[7]	Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225
[8]	Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353
[9]	Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281
[10]	Simone Fiori. Synchronization of first-order autonomous oscillators on Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1725-1741. doi: 10.3934/dcdsb.2018233
[11]	Sylvia Anicic. Existence theorem for a first-order Koiter nonlinear shell model. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1535-1545. doi: 10.3934/dcdss.2019106
[12]	Gábor Kiss, Bernd Krauskopf. Stability implications of delay distribution for first-order and second-order systems. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 327-345. doi: 10.3934/dcdsb.2010.13.327
[13]	Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[14]	Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 683-693. doi: 10.3934/dcdss.2020037
[15]	Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 823-834. doi: 10.3934/dcdss.2020047
[16]	Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
[17]	Noboru Okazawa, Tomomi Yokota. Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1081-1090. doi: 10.3934/dcds.2008.22.1081
[18]	Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851
[19]	Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064
[20]	Emiliano Cristiani, Smita Sahu. On the micro-to-macro limit for first-order traffic flow models on networks. Networks & Heterogeneous Media, 2016, 11 (3) : 395-413. doi: 10.3934/nhm.2016002

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences