A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising

作者：Jun Zhang^*; Wei Zhihui

来源：Applied Mathematical Modelling, 2011, 35(5): 2516-2528.

DOI：10.1016/j.apm.2010.11.049

• Fractional-order derivative
• Sobolev space
• Image denoising
• Staircase effect
• Texture-preserving

The total variation model proposed by Rudin, Usher 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 Grunwald-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.

• 出版日期2011-5
• 单位南京理工大学