An Algorithm In Image Processing Based PDE

The paper studies a type of energy functionals which often occurred in recent articles on PDE image processing. First, we review the source, feature and research outcome systematically, then present an Euler-Lagrange equation by accurate computation. According to the discreteness of image processing problems, we simplify the equation slightly and proposed a highly nonlinear and degenerate second-order PDE of parabolic type.here g(x) =1/(1+k|x|~2) G{x) is a Gauss function, k > 0, d > 0, β > 0 are constants, andI(x) is the origin image.It is difficult to discuss the classical solution of the above equation, so we introduce a sort of PDE' generalized solution — the viscosity solution. We redefine strictly the notion of so-called viscosity solution. At the same time, some unsolved conclusions of the past articles are done by using the classical theory of second-order parabolic PDE and the knowledge of viscosity solution. And we demonstrate the existence, uniqueness and stability for the viscosity solution of the proposed model in detail. At last, numerical experiments are completed with the upwind finite difference scheme for image denoising and image segmentation. From the result, the model in this paper can preserve local important structures such as edges and corners effectively, and it has good quality of noise removal with high speed.