A Class Of Reaction-diffusion Systems With Variable Exponent Growth In Image Decomposition

In the past two decades, nonlinear diffusion equations have been widelyused in the work of image processing. In this paper, we propose a variableexponents-growth reaction-diffusion system applied to image decomposition intocartoon and texture, based on Osher et.at.’s model [S. Osher, A. Sole, L. Vese,Image decomposition and restoration using total variation minimization and theH1norm, SIAM Journal on Multiscale Modeling and Simulation1(3)(2003)349–370]. The new model is coupled by a fast diffusion p(x)-Laplace equationand a slow diffusion p(x)-Laplace equation. The fast diffusion equation preservesthe structure information of images while the slow diffusion equation revises thesource term of the fast diffusion equation. These two equations interact with eachother and separate the structure information and texture information of images.In the aspect of theory, we study properties of solutions with different initialdata. In the second section, we discuss the properties of solutions of the newmodel withL2initial data. We first define the weak solution of the problem inthe space of variable exponents-growing Sobolev space. Then, by the Galerkin’smethod, we establish the existence of weak solutions of the model for Neumannboundary conditions withL2initial data. At last, we obtain the uniqueness ofthe weak solution by the monotonous of p(x)-Laplace equation.Based on the result of section2, we continue to study the problem withL1initial data. It is unable to define weak solutions for this problem when the initialdata belongs toL1space. A commonly used technique is to define entropysolution by the virtue of truncation function. According to the regularized method,we approximate theL1initial data and construct an approximate problem. Theresult of second section indicates that the approximate problem exists weaksolutions. Based on the strong convergence of the truncations of approximatesolutions, we obtain the existence of entropy solutions for our reaction-diffusionsystem withL1data. The uniqueness is obtained by choosing particular testfunction in the definition of entropy solution.In the aspect of numerical experiments, the finite different scheme is used toobtain the discrete form of the proposed model. The numerical results show the new model is not causing the staircasing effect compared with the TV model.Besides, the new model separates better the textured details from large region.Experimental results also illustrate the effectiveness of the new model in edgepreserving and noise removing.