A Nonlinear Multigrid Method For The LLT Model

It is inevitable that the images will be degraded, such as image blur,noise interference, etc, in the processes of image imaging, copying, scanning, transmit-ting, displaying, and so on. However,in many application areas, we need clear, high-quality images. So image restoration including noise removal and de-blurring is of great significance. The objective of image restoration is to reconstruct the original image and to recover the key features of the image as many as possible. Image restoration is a very important and challenging topic in image processing, and many related problems have not been resolved completely so far.In this paper,we apply an improved dual algorithm to a fourth-order PDE model-the LLT model.For solving the LLT model, we use a non-linear multi-grid method with FAS whose smooth iteration adopts the improved Chambolle's dual algorithm,that is,a fixed point iteration algorithm. This dissertation is divided into five chapters:In the first chapter,we introduce the history of the development of image processing, the research background and progress of image restoration, particularly the progress made in image de-noising. At the same time we briefly review the major work of this article and give the relevant preliminary knowledge.Chapter 2 is devoted to introducing the ROF model and two denoising models based on fourth-order PDEs,analyzing their advantages and disadvantages,and recalling the general algorithms for solving these models at the same time.In the third chapter, we review the ideas of the linear multi-grid method and the nonlinear multi-grid method, describe the general form of the algorithms.Chapter 4 is the main work of this dissertation. We apply the nonlinear multi-grid method to solve the LLT model whose smooth iteration adopts the improved Chambolle's dual algorithm. This algorithm has a faster convergence speed.In Chapter 5 we give the relevant numerical examples.Through numerical experiments we test our proposed algorithms.Finally, we conclude the dissertation and point out the topics of future re-searches.