Several Image Restored Models Based On The Total Variation

Image processing technology is the effective analysis and processing for de-graded images. It can help mankind to have a better understanding of the world and has been widely applied to many important fields, such as aerospace engineer-ing, biomedical, object identification and geographical mapping. However, since the acquirement strategies of image have some limitations, there usually exist com-plex structure features in the degraded images. This brings enormous challenges for constructing effective mathematical models and numerical algorithms in order to get superior image from the degraded image.Among the numerous image processing technology, although the models based on the total variation have some limitations, they have given rise to the exten-sive attention of researchers in many fields of image processing. That is because this class of models has good mathematical properties and efficient numerical al-gorithms. In this dissertation, we start with several types of restoration models based on the total variation and propose the new restoration models and numerical algorithms for image denoising, image deblurring, image inpainting, multiplicative noise removal and so on. Our main innovations are as follows:(1) For the convex optimization problem with two nonsmooth terms, we transform it into a unconstrained problem by using variable substitution and propose the alternating direction method of multipliers. What’s more, we theoretically analyze the convergence and validity of this algorithm. However, when we nu-merically implement this algorithm, several equations need to be solved. This greatly reduces the efficiency of the proposed algorithm. Based on the relation of the projection operator and the shrinkage threshold operator, we provide a proximal point method to solve this convex optimization problem, and give the required conditions for the convergence of this method. In fact, the proximal point method means that the concise projection operator is introduced into the dual problem of the initial convex optimization problem. Hence, the dual variable can be projected into a convex unit ball. This makes the proximal point method effective and stable.(2) Based on the definition of the total variation function space, we give the weighted total variation function space and discuss the mathematical prop-erties of this space. The ROF model is known to keep edges well. But it has the undesirable staircase effect in the smooth regions. The LLT model can avoid this undesirable effect, but cause the edges blurring. In view of these facts, based on the weighted total variation function space, we suggest a hybrid model by using the convex combination of these two models. It has been used to image denoising, image deblurring, image inpainting and so on. Moreover, we solve this model by the alternating direction method of multipliers and the proximal point method respectively and compare the effectiveness of these two methods. Especially, with the help of variable substitution, we extend the solvable application range by converting the problem of image deblurring, image inpainting and multiplicative noise removal into the framework of the additive noise removal. In fact, we draw the edge indicator function into the hybrid model which can be seen as the convex combination parameter. Thus, the ROF model plays an important role at the edges and the LLT model does at the smooth region. That is to say, the hybrid model displays localization. Hence, it has adaptability and robustness.(3) The drawback of the conventional LOT model is that it needs to solve two partial differential equations (PDEs). Since the split Bregman method has the advantages of efficiency, stablility and small memory footprint, we extend this method to solve the second step of the LOT model. Therefore, the numerical results have been improved. Actually, it is difficult to look for other effective algorithms in order to solve the first step of the LOT model expect solving its corresponding PDE. That is because it is to seek the smoothed flow field of a nonconvex optimization problem. As a result of the equivalence relation between the unit normal vector in the optimality condition of the ROF model and the dual variable in its dual problem, we substitute the dual variable for the smoothed flow field in the first step of the LOT model. Consequently, we obtain a two step model based on the dual variable. In addition, we provide some examples to illustrate the efficiency of the proposed model.