Some Applications Of Fractional Partial Differential Equations In Image Denoising

Image denoising,as one of the most basic research topics in image processing,has been concerned and studied by many scholars.However,the traditional denoising algorithms often destroy the intrinsic fine structure of the image,such as texture,edge and cartoon.To overcome this shortcoming,we adopt an image denoising algorithm based on fractional partial differential equation,which can not only selectively smooth noisy images,but also make a trade-off between denoising and over-smoothing.At the same time,other problems of image denoising are discussed and analyzed.Finally,based on the variational method and the non-local properties of fractional calculus operators,we establish some new mathematical models and propose corresponding solving algorithms.The main research contents are as follows:Firstly,We construct a fractional-order reaction-diffusion equation system model for image decomposition and restoration,which decomposes noisy images into cartoon components in fractional Sobolev space and texture components in negative Hilbert space for denoising.The model consists of two diffusion equations,one is fractional 1-Laplace flow and the other is Laplace flow.In addition,we list the definition of weak solutions of systems of equations and the theorem guaranteeing the existence and uniqueness of weak solutions,and use regularization method to deal with singularities in equations.The numerical results show that the new model is more effective than ROF model and OSV model.Secondly,For multiplicative noise,we construct a non-homogeneous fractional-order 1-Laplace evolution equation model based on the maximum posteriori estimation and the long-range interaction of non-local diffusion operators,and give the definition of the weak solution of the model and its existence and uniqueness theorem.The experimental results show that the removal effect of the new model is better than that of AA model.Finally,A fast denoising algorithm for fractional anisotropic explicit diffusion equation is constructed based on the long-range interaction property of fractional calculus nonlocal operators and the time step of periodic variation.In order to balance the efficiency and accuracy of the model,we present a truncated matrix method to deal with the discrete problem of fractional calculus and estimate the error.By estimating the stability condition of iteration through spectrum analysis,we implement an explicit iterative numerical algorithm with periodic variation of time step size.The numerical results show that the new algorithm has considerable efficiency growth and can achieve satisfactory denoising results and efficiency faster.