Fully fractional anisotropic diffusion for image denoising.

*(English)*Zbl 1225.94003Summary: This paper introduces a novel Fully Fractional Anisotropic Diffusion Equation for noise removal which contains spatial as well as time fractional derivatives. It is a generalization of a method proposed by Cuesta which interpolates between the heat and the wave equation by the use of time fractional derivatives, and the method proposed by Bai and Feng, which interpolates between the second and the fourth order anisotropic diffusion equation by the use of spatial fractional derivatives. This equation has the benefits of both of these methods. For the construction of a numerical scheme, the proposed partial differential equation (PDE) has been treated as a spatially discretized Fractional Ordinary Differential Equation (FODE) model, and then the Fractional Linear Multistep Method (FLMM) combined with the discrete Fourier transform (DFT) is used. We prove that the analytical solution to the proposed FODE has certain regularity properties which are sufficient to apply a convergent and stable fractional numerical procedure. Experimental results confirm that our model manages to preserve edges, especially highly oscillatory regions, more efficiently than the baseline parabolic diffusion models.

##### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

34A08 | Fractional ordinary differential equations and fractional differential inclusions |

35R11 | Fractional partial differential equations |

45K05 | Integro-partial differential equations |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |

68U10 | Computing methodologies for image processing |

##### Keywords:

anisotropic diffusion; fractional derivatives; fractional ordinary differential equations; fractional linear multistep methods; fractional order differences; image denoising
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\textit{M. Janev} et al., Math. Comput. Modelling 54, No. 1--2, 729--741 (2011; Zbl 1225.94003)

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