Texture Image Processing Model Based On Fractional Diffusion Equations

With the rapid development of artificial intelligence and big data,image processing has received more and more attention and research.At present,image processing technology has been widely used in clinical medicine,remote sensing,criminal investigation and other fields.Among the many features of image,texture is an important and difficult to describe feature.The texture image processing model based on fractional order partial differential equation comes into being because of the non-locality and self-similarity of texture.Unfortunately,most analytical solutions of fractional differential equations contain special functions or complex series.Therefore,it is very important to study the numerical solution of these equations.The nonlocality of fractional derivatives makes the numerical discretization of these equations more complicated,so it is urgent and meaningful to construct efficient algorithms for solving fractional partial differential equations.In this dissertation,we study the design of efficient numerical algorithm for fractional-order partial differential equations and fractional-order modeling in texture image processing.The main research contents are as follows:For a class of time fractional-order convection-diffusion equations,the initial boundary value problem is transformed into a time fractional-order diffusion wave equation by using the substitution of variables to eliminate the convection term in the original equation.In order to construct an efficient numerical scheme,high-order compact scheme is used for numerical discretization in spatial direction,and alternation direction implicit diference method is used for scheme construction in temporal direction.The numerical method decomposes the high-dimensional problem into several one-dimensional problems for solution,which effectively improves the computational efficiency.A new discrete norm is constructed,and the unconditional stability and convergence of the scheme are proved by discrete green's formula.Finally,the correctness of theoretical analysis and the high efficiency of numerical solution are verified by some examples.In view of the problem of image super-resolution reconstruction,the traditional variational model based on bounded variational function space cannot reconstruct the texture information effectively because the sampling and blurring process of image will destroy a lot of texture details.This dissertation proposes to use the fractional-order bounded variational function space to model the image texture features,and then build a variationalmodel based on fractional-order derivatives to reconstruct the image with super resolution.The existence of the extreme point of energy functional are proved by nonlinear analysis.In terms of numerical simulation,the scalar auxiliary variable method is used to solve the model effectively.This algorithm has the characteristics of unconditional stability,simple format construction and high computational efficiency.In order to further improve the computational efficiency,this dissertation proposes a time step adaptive iterative updating criterion,which effectively reduces the number of iterations.The results of numerical experiments show that the method presented in this dissertation has obvious advantages over other methods in terms of texture information and computational efficiency.For image speckle noise removal,it is very difficult to protect the texture during speckle suppression because speckle noise seriously damages the texture information of image.Therefore,based on the internal relationship between image texture and fractionalorder calculus operator,this dissertation designs a reasonable and effective speckle noise suppression model under the framework of fractional-order diffusion equation,and realizes the effective protection of image texture details.The gray detection operator is introduced into the diffusion coefficient to make the diffusion behavior affected by the gray value,so as to protect the image features of the region with low gray value.In theory,Stampacchia's truncation method is used to demonstrate the extremum principle of the equation.The frequency domain algorithm based on discrete Fourier transform and the spatial domain algorithm based on fractional difference are designed respectively.Using the statistical information of speckle noise,a new iterative stopping condition is proposed.Experimental results show that the model can effectively remove noise and better protect the texture information and low gray level information of the image.Aiming at the problem of image deblurring,a method based on fractional diffusion equation is proposed.In order to fully recover the texture details of the image,this paper designs the diffusion term and source term of the fractal diffusion equation.By virtue of Lebesgue's control convergence theorem,it is proved that when the equilibrium parameter is large enough,the restoration effect is close to the real image.Based on the definition of fractional difference,the finite difference scheme is designed to solve the equation numerically.The experimental results show that the model can effectively remove the blur and restore the image texture more fully.In addition,different from other methods,the method in this dissertation is not limited to blurring kernel with symmetry,so it ismore applicable.