Applications Of A Coupled PDE Model In Image Processing

This thesis makes an in-depth study and accurate calculation on a type of energy functional used for image processing which often appeared in recent papers on image processing based on partial differential equations. Based on a special method for discrete problem which is proposed in [1] and the idea of [2] of using a coupled PDE to simplify the numerical implementation, this thesis proposed a new PDE model which incorporates time-delay regularization and curvature-based diffusion. Our model takes the following form:where I(x) is the image to be processed, u is its smoothed version, /3 is a weightingparameter, r is a time-scale factor, exp is a Gaussion smoothingkernel with a pre-specified σ, g(s) with parameter K > 0, c is a constant.The model is highly nonlinear and degenerate, therefore we can't hope it has a solution which is differential in the classical sense. So we introduce the definition of so-called viscosity solutions. We make a suitable revision of the definition which has been used in the past papers on the similar problems and re-define the viscosity solutions strictly. Meanwhile, applying the theory of classical partial differential equation and viscosity solutions, a series of estimations of viscosity solutions are derived, some details and deficiencies not proved in relevant papers are supplemented and the existence, uniqueness and stability of the viscosity solution of the coupled PDE model are proved. Finally we give the actual results of our model in image processing. The experimental results show that our model can incorporate the past information of the gradient of u into the diffusion process and has a superiority in preserving sharp edgesand fine structures. So our model is a meaningful, reasonable and effective PDE image processing model not only in theory but also in practice.