An Image Smoothness Method Based On Two-Dimention Reproducing Kernel

This paper proposes an image smoothness method based on partial differential equation with reproducing kernel. Compared with other general methods, there is advantage such as image clearness, numerical steady. In solving the problem of image smoothness based on partial differential equation, the general methods are to construct difference equation, then solve it with initialization and edge conditions. These methods have low precision and diffuse error quickly.Firstly, this paper recalls the basic research instance of image smoothness, analyses the advantage , disadvantage of all kinds of methods, and then discusses the basic framework of image processing based on partial differential equation. Secondly, this paper recalls the basic nature of serval impotant reproducing kernel spaces, especially the approximate theory of two-dimension reproducing kernel space W₂ , and the convergence,error of the approximate method and so on. In solving image smoothness models based on partial differential equation, this paper approximates the image with W₂ space reproducing kernel. So we transform the problem of solving partial differential equation to solving linear equation and get the answer by iterative. Because the derivative of reproducing kernel function also is a wavelet function, the wavelet function possess a better local character, and approximate the function in W₂ space and its derivative very well, the reproducing kernel method to solve diffusion equation has higher precision, slower speed of error diffusion, and not sensitivity to time interval compared with the finite difference method. This paper proposes a method to solve both diffusion equation model and Alvarez model with W₂ space reproducing kernel, and this method get a higher precision. We extend this method to solve other image processing models based on diffusion equation.At last,the experiment shows the advantage of this method, and we discuss and give out the direction of further research. Compared with the finite difference method, the method proposed in this paper has higher precision, but also higher computational complexity. The method proposed in this paper can apply to the problem that is not real-time.