Fast numerical method for nonlinear image processing

Abstract

Total Variation(TV)regularization method is very effective for reconstructing "blocky", discontinuous, images from contaminated image with noise. But TV is represented by highly nonlinear integro-differential equation that is hard to solve. There have been many efforts to obtain stable and fast methods. C. Vogel[25] introduced "the Fixed Point Lagged Diffusivity Iteration", which solves the nonlinear equation by linearizing. In this thesis, we apply the Multigrid(MG) method for Cell Centered Finite Difference (CCFD) to solve system arise at each step of this fixed point iteration. We test many images varying noises and regularization parameter α and smoothness parameter β which appear in TV method. In this experiment, we know some facts. First, β is not sensitive. Second, all optimal values of αs at each picture are different. Third, the optimal value of α is directly proportional to the amount of noise. Fourth, If we lay great emphasis on removal noise, to use gradient norm($W^1_1$ semi-norm) is more suitable than to use $L^2$ norm for our purpose. Fifth, since a sharp and delicate edges tend to blunt, our denoising experiment is more effective board part than delicate part of picture.