Fast numerical methods for image restoration

Abstract

In the computer vision field, most problems can be described as energy functionals. The optimums of these energy functionals are the solutions of the computer vision problems. The fast numerical methods seeking the solutions are fundamentally important and highly demanded.

We mainly solve three different essential computer vision problems: image denoising problem, image segmentation problem and surface reconstruction problem. We will review the critical models such as the Rudin, Osher and Fatemi (ROF) model, TV-L1 model and Euler's elastica model for denoising and related problems. The Mumford-Shah model and the Chan-Vese model are also included for solving segmentation problem. In surface reconstruction problem, the weighted minimal surface model is introduced as background.

In this thesis, we use two types of fast numerical methods for

solving these energy minimization problems. The first one is

multiplier based method to the augmented Lagrangian function of TV-L1 model, for image denoising and image fusion problems. The other one is graph cuts technique for fast solving higher order curvature based models. It has been applied to solve the image denoising, segmentation and surface reconstruction problems.