Some Effective Numerical Methods For High Order Variational Models For Image Restoration

Image restoration is one of the most important basic studies in image processing, and it is a phase before doing further image processing tasks e.g. edge detection, segmentation, pattern recognition, and object tracking. It has a wide application in some fields such as astronomy, remote sensing image and medical imaging. One of the most important methods for image restoration is using variational models, which plays a vital role in preserving image edge, texture and so on. The common technique for variational models is solving their corresponding Euler-Lagrange equations. The nonlinearity of such equations is too high to solve easily. How to solve these equations fast and efficient is one of the most exciting topics in image restoration.In this thesis, several variational image restoration models and their algorithms are re-searched. For the sake of the high nonlinearity of the Euler-Lagrange equation, a homotopy equation based on gradually reducing the smooth parameter β is constructed, and several efficient curve tracking schemes are proposed. We start with solving total variation (TV: total variation) model by Rudin, Osher and Fatemi in1992, and then consider fourth or-der partial differential equation (PDE) from Lysaker, Lundervold and Tai's LLT (Lysaker, Lundervold and Tai) model and the combination of TV model and LLT model. Finally, we solve a more complex model-ZC (Zhu and Chan) model proposed by Zhu and Chan, which minimizes a nonconvex second order functional, and its Euler-Lagrange equation is fourth order too.The main achievements of this dissertation are put forward:(i) Since the Newton method does not work for TV model, we present a homotopy method. Firstly, we construct a homotopy equation based on gradually reducing the smooth parameter β to improve the nonlinearity of the Euler-Lagrange equation. Secondly, accord-ing to the local convergence of the Newton method we present a scheme to adjust the steplength, and for the sake of the geometrical property of the solution curve, a Lagrange interpolation formula is used to obtain the predictor point, and with these efforts we speed up the convergence of homotopy method efficiently.(ii) Due to the high nonlinearity of the LLT model with smallβ, the convergence of the fixed point is difficult. A homotopy method based on gradually reducing the smooth parameter is proposed. Numerical tests shows the homotopy method has efficiently improved the convergence of the fixed point method.(ⅲ) A modified convex combination of TV model and LLT model proposed Lysaker and Tai is presented to improve the quality of the restored image. Moreover, for the sake of the former homotopy work on TV model and LLT model, we proposed two new convex combination algorithms about these models based on the framework of homotopy method. Numerical tests show that our new convex combination algorithms not only preserve the advantages of the old algorithms, but also enhance the quality of the restored image.(ⅳ) The solution for nonconvex variational models is one of difficult tasks in image processing. Since ZC model is one of the members in the high order nonconvex variational models, the solution is quite difficult to find. In this thesis, we present a fixed point curvature method to solve the Euler-Lagrange equation directly. The method is composed of outer and inner iterations. The outer iteration is fixed point method, and the inner iteration is solving a fixed point equation which is similar to TV equation. Moreover, a relaxed fixed point method based on the fixed point curvature method is proposed; Finally, we construct a homotopy equation to enhance the convergence of the fixed point curvature method and the relaxed fixed point method by gradually reducing β.