Computational study for PDE in image processing

Abstract

In general, it is necessary for noise to rise when one capture images with camera. Unfortunately, it is almost impossible to catch an uncontaminated images. Thus, we should hold on noises over contaminated images. It is both fortunate and well known that the distribution of noises is statistically under normal distribution. But all of natural noises are not in normal distribution, but almost. We call it Gaussian white noise. In that sense, Gaussian white noise can be regarded as frequent noise. Through noise level constrained regularization or Tikhonov Regularization, we can conclude that denoised images are very close to the solution of Euler Lagrange Equation. But this equation is ill posed problem when total variation norm is defined by norm. Thus, to change that problem into the well posed problem, we should replace norm of the characteristic part by semi norm. Although it is difficult to apply the approximate algorithm into the case, there are two algorithms: Primial dual method, Damped Newton method. I compared iterations by both algorithms, respectively. Consequently, Damped Newton method generate the monotone convergent iteration because of damping term. But its convergent velocity was very low. But just when we iterate during very long period, enough quality can be guaranteed.