The Fast Solutions To Fractional Partial Differential Equation And Its Application In Image Restoration

This paper studies the fast solutions to fractional partial differential equation and its application in image restoration.And in image restoration,this paper proposes the three-dimensional fractional total variation regularized tensor optimized model for threedimensional image deblurring.Briefly,this paper respectively applies a second-order accurate finite difference scheme for temporal and spatial discretization to fractional partial differential equation,which results in series of linear systems.And those linear systems is solved simultaneously by a designed strategy.In addition,the discretized forms of the fractional partial differential operator named fractional total variation is utilized to image deblurring in the tensor optimized framework.Firstly,we give the research background and the present development.Next,this paper studies the numerical solutions to fractional diffusion equations.The second-order accurate finite difference scheme utilizes the Crank-Nicholson formula and the weighted and shifted Grünlward-Letnikov approximation for the temporal and spatial discretization,respectively.This numerical scheme is proved to be unconditionally stable.However,it results in series of linear systems and this paper designs a block linear equation to process these linear systems at the same time.Further,the proposed block linear system is then solved by iterative methods with two proposed block diagonal and block stair preconditioners,which are highly suitable for parallel computations.Finally,numerical results show the effectiveness of two block preconditioners.Secondly,the discretized forms of the fractional partial differential operator named fractional total variation is utilized to image deblurring problem.Researchers have utilized the fractional partial differential operator to preserve texture and alleviate staircase effects by exploiting the nonlocal smoothness of images.However,for three-dimension(3D)images,the existing fractional-order derivative-based models only exploit the nonlocal smoothness of spatial dimensions and fail to consider the other dimensional information such as the nonlocal smoothness along the spectral dimension for multispectral image(MSI).To tackle this issue,we propose a three-dimensional fractional total variation(3DFTV)based-model,which exploits the nonlocal smoothness along all three dimensions of 3D images.By viewing 3D images as third-order tensors,we mathematically formulate the proposed model under the tensor algebra.Furthermore,we develop an efficient algorithm based on alternating direction method of multipliers(ADMM)to solve the proposed model.Experimental results demonstrate the superiority of our model against comparing models in terms of quality metrics and visual effects.Finally,we conclude this paper and give the prospect about the future work in the corresponding research.