A Fourth-order Partial Differential Equation-Based Split Bregman Method For Image Restoration

In 1992, Rudin, Osher and Fatemi in their pioneering paper put forward a well-known total variation-based image restoration model, which is called the ROF model. Since then, the variational partial differential equation-based image restoration methods began their rapid development process. The Euler-Lagrange equation derived from the ROF model is a second-order PDE, which can be utilized to obtain the restored image with a good quality:it can remove noise while preserving the edges of an image. How-ever, the ROF model has also some shortcomings such as "staircase effects" and more complicated computation, so a lot of modified or extended models and fast calculation methods are proposed to overcome these deficiencies.In this dissertation, we mainly study how to combine a class of fourth-order varia-tional partial differential equation models with good nature-LLT model and a class of fast optimization algorithms-split Bregman iterative method to recover an image so that we obtain better image restoration effect but take less computational cost. This thesis is composed of the following five chapters.In the first chapter, the basic knowledge of image restoration and the correlative achievements on the variational PDE-based image restoration are briefly presented.The second chapter is devoted to introducing some basic mathematical knowledge which are theoretical foundation of the variational PDE-based image restoration.In the third chapter, we introduce Bregman iterative methods and split Bregman iterative methods and give their convergence, which greatly inspire our works.In the fourth chapter, by combining split Bregman iterative method and the LLT model, we propose a fourth-order variational PDE-based split Bregman iterative method for image restoration. At the same time, we prove the convergence of the proposed method.In the fifth chapter, the numerical computational scheme of the proposed method is deduced in detail and the numerical experiments are implemented. Experimental resluts show that the proposed method is fast with good vision effect.