Total Variation Image Denoising Bases On Homotopy Continuation

We mainly study the total variation image denoising using homotopy continuation in this paper. Total variation image denoising is regarded as one of the main methods for image denoising. Its solution belongs to variation function, so it allows to have discontinuities in the solution. Therefore, we can keep edges well if we use this model for image restoring, and it is also benefit for the postprocessing. But it is a hard task to solve this equation, one reason is that the TV-norm is nondifferentiable when |â–½_u| = 0, so we can't apply a linearizationtechnique such as the Newton method. The other reason is that the Euler-Lagrange equation has a highly nonlinear term. The Newton method for such equation is known to have a very small domain of convergence, and it is very difficult to make sure that the initial value such as the observed image belongs to this domain. So the Newton method is not an ideal method for solving this equation. We analyse and compare the traditional timemarching method, the fixed point iteration method, Chan, Chan and Zhou proposed such an approach using the Newton method combined with continuation and the primal-dual method. Due to these methods' drawback, we propose using the homotopy continuation method for solving total variation image denoising problem in this paper. It is a global convergence method, so it can avoid the drawback of the Newton method. The main idea is to perturb the TV-norm functional term of the Euler-Lagrange equation so large that it can be solved by the Newton method when the observed image is an initial value, and we regard this equation as an auxiliary equation. Then we construct a homotopy equation to combine the auxiliary equation with the Euler-Lagrange equation. We make the solution of the auxiliary equation as the solution curve's initial value and trace the curve until it gets to the target point. During the path-following we use secant predictor. Because of the path's regularization, the solution curve does not turn back, so we do correct on the hyperplane that the homotopy parameter t is unchanged. We have implement this method in the numerical test, experiment results show that the method can denoise better.