A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising. (English) Zbl 1217.94024

Summary: The total variation model proposed by Rudin, Osher and Fatemi performs very well for removing noise while preserving edges. However, it favors a piecewise constant solution in BV space which often leads to the staircase effect, and small details such as textures are often filtered out with noise in the process of denoising. To preserve the textures and eliminate the staircase effect, we improve the total variation model in this paper. This is accomplished by the following steps: (1) we define a new space of functions of fractional-order bounded variation called the \(BV_{\alpha }\) space by using the Grünwald-Letnikov definition of fractional-order derivative; (2) we model the structure of the image as a function belonging to the \(BV_{\alpha }\) space, and the textures in different scales as functions belonging to different negative Sobolev spaces. Thus, we propose a class of fractional-order multi-scale variational models for image denoising. (3) We analyze some properties of the fraction-order total variation operator and its conjugate operator. By using these properties, we develop an alternation projection algorithm for the new model and propose an efficient condition of the convergence of the algorithm. The numerical results show that the fractional-order multi-scale variational model can improve the peak signal to noise ratio of image, preserve textures and eliminate the staircase effect efficiently in the process of denoising.


94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
49K10 Optimality conditions for free problems in two or more independent variables
26A33 Fractional derivatives and integrals
Full Text: DOI


[1] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 1-4, 259-268 (1992) · Zbl 0780.49028
[2] Gilles, J.; Meyer, Y., Properties of BV-G structures+textures decomposition models. Application to road detection in satellite images, IEEE Trans. Image Process., 9, 11, 2793-2800 (2010) · Zbl 1371.94141
[3] Meyer, Y., Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The 15th Dean Jacqueline B. Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The 15th Dean Jacqueline B, Lewis Memorial Lectures (2001), American Mathematical Society: American Mathematical Society Boston, MA · Zbl 0987.35003
[4] Vese, L.; Osher, S., Image denoising and decomposition with total variation minimization and oscillatory functions, J. Math. Imaging Vision, 20, 7-18 (2004) · Zbl 1366.94072
[5] Osher, S.; Solé, A.; Vese, L., Image decomposition and restoration using total variation minimization and \(H^{−1}\) norm, Multi. Model. Simul., 1, 3, 349-370 (2003) · Zbl 1051.49026
[7] Aubert, G.; Aujol, J. F., Modeling very oscillating signals. Application to image processing, Appl. Math. Opt., 51, 2, 163-182 (2005) · Zbl 1162.49306
[8] Chan, T. F.; Marquina, A.; Mulet, P., High-order total variation based image restoration, SIAM J. Sci. Comput., 22, 2, 503-516 (2000) · Zbl 0968.68175
[9] You, Y. L.; Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9, 10, 1723-1730 (2000) · Zbl 0962.94011
[10] Lysaker, M.; Lundervold, A.; Tai, X. C., Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12, 12, 1579-1590 (2003) · Zbl 1286.94020
[11] Lysaker, M.; Tai, X. C., Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput.Vis., 66, 1, 5-18 (2006) · Zbl 1286.94021
[12] Bai, J.; Feng, X. C., Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16, 10, 2492-2502 (2007) · Zbl 1119.76377
[14] Pu, Y. F., Application of fractional differential approach to digital image processing, J. Sichuan Univ. (Eng. Sci. Ed.), 39, 3, 124-132 (2007)
[15] Pu, Y. F., Fractional differential analysis for texture of digital image, J. Algorithms Comput. Technol., 1, 3, 357-380 (2007)
[16] Pu, Y. F.; Wang, W.; Zhou, J.; Wang, Y.; Jia, H., Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation, Sci. China Ser. F: Inform. Sci., 51, 9, 1319-1339 (2008) · Zbl 1147.68814
[17] Podlubny, I., Fractional Differential Equations [M] (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[18] Chambolle, A., An algorithm for total variation minimization and applications, JMIV, 20, 89-97 (2004) · Zbl 1366.94048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.