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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 $B{V}_{\alpha }$ 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 $B{V}_{\alpha }$ 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.
##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) 68U10 Image processing (computing aspects) 49K10 Free problems in several independent variables (optimality conditions) 26A33 Fractional derivatives and integrals (real functions)
##### References:
 [1] Rudin, L.; Osher, S.; Fatemi, E.: Nonlinear total variation based noise removal algorithms, Physica D 60, No. 1-4, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F [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, No. 11, 2793-2800 (2010) [3] Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations: the 15th Dean jacqueline B, Lewis memorial lectures (2001) [4] Vese, L.; Osher, S.: Image denoising and decomposition with total variation minimization and oscillatory functions, J. math. Imaging vision 20, 7-18 (2004) [5] Osher, S.; Solé, A.; Vese, L.: Image decomposition and restoration using total variation minimization and H - 1 norm, Multi. model. Simul. 1, No. 3, 349-370 (2003) · Zbl 1051.49026 · doi:10.1137/S1540345902416247 [6] L. Linh, L. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev space, UCLA CAM Report 05-33, 2005. [7] Aubert, G.; Aujol, J. F.: Modeling very oscillating signals. Application to image processing, Appl. math. Opt. 51, No. 2, 163-182 (2005) · Zbl 1162.49306 · doi:10.1007/s00245-004-0812-z [8] Chan, T. F.; Marquina, A.; Mulet, P.: High-order total variation based image restoration, SIAM J. Sci. comput. 22, No. 2, 503-516 (2000) · Zbl 0968.68175 · doi:10.1137/S1064827598344169 [9] You, Y. L.; Kaveh, M.: Fourth-order partial differential equations for noise removal, IEEE trans. Image process. 9, No. 10, 1723-1730 (2000) · Zbl 0962.94011 · doi:10.1109/83.869184 [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, No. 12, 1579-1590 (2003) [11] Lysaker, M.; Tai, X. C.: Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput.vis. 66, No. 1, 5-18 (2006) [12] Bai, J.; Feng, X. C.: Fractional-order anisotropic diffusion for image denoising, IEEE trans. Image process. 16, No. 10, 2492-2502 (2007) [13] Pu YF. Fractional calculus approach to texture of digital image, in: Proceedings of 8th International Conference on Signal Processing, IEEE, Beijing, 2006, pp. 1002 – 1006. [14] Pu, Y. F.: Application of fractional differential approach to digital image processing, J. sichuan univ. (Eng. Sci. ed.) 39, No. 3, 124-132 (2007) [15] Pu, Y. F.: Fractional differential analysis for texture of digital image, J. algorithms comput. Technol. 1, No. 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, No. 9, 1319-1339 (2008) · Zbl 1147.68814 · doi:10.1007/s11432-008-0098-x [17] Podlubny, I.: Fractional differential equations [M], (1999) [18] Chambolle, A.: An algorithm for total variation minimization and applications, Jmiv 20, 89-97 (2004)