Topics in variational PDE image segmentation, inpainting and denoising

Variational methods have been extensively used and studied in image processing in the past decade because of their flexibility in modeling and various advantages in the numerical implementation. Examples of this include image segmentation, object tracking, texture synthesis, vector field visualization, etc. This dissertation contains the study of variational methods in image segmentation, inpainting and denoising. We propose a fast algorithm for level set based optimization. The main advantage of this method is that the gradient of the functional is not needed. This enables it to be applied to broader range of optimization problems. When applying this algorithm to Chan-Vese image segmentation model, it converges extremely fast. For image inpainting, we formalize a low-level global deterministic solution for texture synthesis and image inpainting. A global energy is defined as a function of correspondence map, which is defined as linking each blank or missing pixel to the pixel where its value is taken from, in the seed image. We demonstrate why they should not be seen as procedures to sample a probability distribution on the correspondence maps. Finally, we discuss the adaptive total variation denoising. This method is a modification of the original total variation denoising method. We prove that, after this modification, the numerical scheme for the Euler-Lagrange equation of the adaptive total variation model is stable. We show numerical experiment that this model can keep the edge of object as well as that of the TV model.