Fully fractional anisotropic diffusion for image denoising

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...

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Bibliographic Details
Published in:Mathematical and computer modelling 2011-07, Vol.54 (1), p.729-741
Main Authors: Janev, Marko, Pilipović, Stevan, Atanacković, Teodor, Obradović, Radovan, Ralević, Nebojša
Format: Article
Language:English
Subjects:
Anisotropic diffusion
Exact sciences and technology
Fractional derivatives
Fractional linear multistep methods
Fractional order differences
Fractional ordinary differential equations
Image denoising
Mathematical analysis
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Real functions
Sciences and techniques of general use
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Summary: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.
ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2011.03.017