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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 \(BV_{\alpha }\) 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.

### MSC:

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 |

### Keywords:

fractional-order derivative; Sobolev space; image denoising; staircase effect; texture-preserving
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\textit{J. Zhang} and \textit{Z. Wei}, Appl. Math. Modelling 35, No. 5, 2516--2528 (2011; Zbl 1217.94024)

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### References:

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