On The Numerical Methods Of Image Restoration

Images, as an important bridge of the human and the world, play an significant role in human’s obtaining information, expressing information and transmitting information. In the last few decades, we have witnessed an era of imaging sciences. From satellite imaging, X-ray imaging to modern medical computer tomography (CT), magnetic reso-nance imaging (MRI) and positron emission tomography (PET), digital images became more and more common and important. In general, image processing includes image enhancement, image restoration, image inpainting, image segmentation, etc.In this paper, we focus on the numerical methods in image restoration problems. First of all, we simply introduce some basic concepts of digital image and two basic mathematical models, then study the inverse problem and regularization method.Then, we apply the preconditioned conjugate gradient method to solve the half-quadratic regularization image restoration problem. Image restoration is the fundamental problem in image processing area. Except for many different filters applied to obtain an restored image, a degraded image can often be recovered efficiently by solving a mini-mization function which consists of a data-fidelity term and a regularization term. In spe-cific, half-quadratic regularization can effectively preserve image edges in the recovered images and a fixed-point iteration method is usually employed to solve the minimization problem. In this paper, Newton method is applied to solve the half-quadratic regular-ization image restoration problem. And at each step of the Newton method, a structured linear system of a symmetric positive definite coefficient matrix arises and preconditioned conjugate gradient method is applied to solve it with a product preconditioner.The exper-imental results demonstrate that the product preconditioned conjugate gradient method is efficient for the half-quadratic regularization image restoration in terms of the numerical performance and image recovering quality.At last, we propose two step method to handle the multiplicative noise restoration problem. Restoration of images corrupted by blur and multiplicative noise is a challeng-ing problem in applied mathematics and it has attracted much attention in recent years. Inspired by previous works on image recovery under impulse noise with blur, we pro-pose a two-step approach to handle the multiplicative noise restoration problem. In the proposed method, the multiplicative noise is first reduced by nonlocal filters, and then a convex variational model is further adopted on the result of the first step. The variational model is composed of an L1-L2data-fidelity term and a total variation (TV) regulariza-tion term. The alternating direction method (ADM) is utilized to solve this variational problem, and we also demonstrate that the ADM algorithm converges at least linearly. Experimental results are given to show that the proposed two-step approach performs bet-ter than the existing methods for multiplicative noise image restoration both in the quality of the restored images and the convergence speed of the algorithms.