New Models And Fast Algorithms In Image Restoration And Segmentation

In this thesis, we mainly use partial differential equation(PDE) and varia-tional methods to study some mathematical models and fast algorithms in image processing including image restoration and segmentation. Generally speaking, there are two trains of thought to establish mathematical models. The first one is to use PDE methods to construct the needed PDEs directly, and the sec-ond one is to use optimization methods to establish variational models, i.e., the minimization problems of energy functionals. Then, we need some effective al-gorithms to solve the considered models. For PDE-based models, we usually need to choose appropriate finite difference schemes to discretize the PDEs and get the numerical solutions. For variational models, on one hand, we can first derive the Euler-Lagrange equations associated with the energy functionals, and get their gradient descent flows, then discretize them to obtian the numerical solutions; on the other hand, we can transform the minimization problems and employ good iterative algorithms such as Chambolle's projection approach, split Bregman iteration and so on to speed up the computation.The main research results are as follows:1. An improved LOT model and fast algorithmROF model is one of the most famous image denoising models, but it leads to undesirable "staircase effect". LOT model can alleviate this effect. However, in the case of unknown noise information, its results may be undesirable for preserving fine structures such as image edges and textures. We propose an improved LOT model to solve this problem. The model is based on variational methods and consists of two steps. We first adopt total variation (TV) model to smooth the orientation (a variable, the computational speed is two-fold faster) of the unit normal vectors of an image. Then, we reconstruct the unit normals according to the smoothed orientation and make it sufficiently close to the unit normals of the desirable image. Here, we introduce an edge detector function so that the proposed model can preserve the repetitive structures such as image edges and textures. Moreover, we adopt L1 data fidelity term instead of the original L2 one in LOT model, which can keep the contrast better. Besides, there are three nonlinear PDEs to be solved in LOT model, but the finite difference methods take much time to compute. For the improved model, we employ the fast dual-based method (Chambolle's projection alogrithm), and hence dramatically improve the computational speed (about ten times).2. Second-order PDE models with new gradient fidelity term for avoiding staircase effectSimilar to ROF model, some classical second order nonlinear PDEs often produce staircase effect during noise removal. To remove this effect, we introduce a new gradient fidelity term and construct a minimization problem, and then combine associated Euler-Lagrange equation with those second order PDEs. The construction of the gradient fidelity term is composed of two steps. First, we directly regularize the gradient of noisy image by the vectorial TV model. Second, we minimize the L2 norm between the gradient of desirable image and the one got from the first step. The use of TV model can preserve image gradient better in image edges, and thereby keep the gradient fidelity better. In the numerical computation, we apply the vectorial dual-based method to solve the vectorial TV model, which will save more time than the usual gradient descent flow and hence speed up the pre-processing step for the gradient of noisy image.3. Nonlocal TV-based models for multiplicative noise removalMultiplicative noise reduction is also an important problem in image restora-tion. In this thesis, we make use of nonlocal TV norm for multiplicative noise removal and propose two nonlocal TV-based models. A main advantage of non-local TV norm over classical TV norm is the superiority in dealing with better textures and repetitive structures. The Bregman and split Bregman iterations are used to implement the two new models. Here, Bregman iteration helps to preserve more details in restored images such that they look clearer, and the split Bregman iteration can avoid the regularization of nonlocal TV norm and speed up the computation.4. A fast algorithm for vectorial image restoration Vectorial TV-based models are a kind of effective mathematical model for the vectorial image restoration such as color image denoising, image colorization, color image inpainting and so on. Therefore, it is very necessary to study the corresponding effective and fast algorithms. We first extend a fast algorithm based on the split Bregman iteration proposed by Jia and Zhao to the vecto-rial TV model, which will quicken up the computation. The proposed vectorial algorithm does not need to solve any difference equations, and has very simple iterative scheme which is favorable to making code. The most important of all, it converges to the solution of vectorial TV model, and the number of iterations to reach the solution is low. Then we extend the application of the proposed vec-torial algorithm to some color image restoration including color denoisings in the red-green-blue (RGB) and chromaticity-brightness (CB) color representations, image colorization based on CB color model and image inpainting.5. Image segmentation based on fuzzy region competition and the corre-sponding fast algorithmLevel set method is often used to solve some mathematical models in im-age segmentation, but this method is not only slow but also dependent on the initialization. We summarize an image segmentation model based on fuzzy re-gion competition, which combine the edge information and the intensities in each region. It is a uniform segmentation framework which can unify some famous segmentation models for example C-V model, nonparametric statistical model, etc. This kind of model has a global solution, and its segmentation results are independent of the initial guess of the curve. Moreover, there are many effective algorithms to solve it. This thesis extends Jia and Zhao's fast algorithm which solves the anisotropic ROF model to isotropic weighted TV model, and then applies it to image segmentation model based on fuzzy region competition and proposes a simple and fast algorithm for image segmentation.