Numerical Methods For Several Classes Of PDE Models In Image Processing

With the digitalization and informationization of the society, digital image processing is becoming a more and more important subject and has a very wide range of applications in biology, industrial production, health care, communications and so on. In general, there are many problems in image processing including image restoration, image enhancement, image segmentation, image coding, image com-pression, image zooming, et al. In this dissertation, we focus on image restoration and zooming.Usually, the image processing methods can be summarized into three cate-gories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equation (PDE). In this dissertation, we study the methods based on PDE including local models and nonlocal models. To local models, we research the ROF model, LLT model and TV-Stokes model respectively. As for nonlocal models, we research the nonlocal variational model. One advan-tage using PDE in image processing is there are many theories and methods for solving the PDE in numerical computation. available. Experiments demonstrated that PDE plays an important role in the development of image science.In Chapter3, we propose a modified fixed point iterative algorithm for solv-ing the fourth-order PDE model in image restoration. Compared with the stan-dard fixed point algorithm, the proposed algorithm needn’t to compute inverse matrices so that it can speed up the convergence and reduce the roundoff er-ror. Theoretically, we give the convergence analysis for the isotropic diffusion and anisotropic diffusion models respectively by means of different strategies. Using the Cauchy-Schwarz inequality and some properties of inequality, we give the con-vergence analysis for the anisotropic diffusion model. For the isotropic diffusion model, a sophisticated strategy on spectral properties of matrices is analyzed in convergence theorems so that a simpler and better bound is deduced. Furthermore, we give some experimental results to illustrate the effectiveness and advantages of the proposed algorithms by comparing with the standard fixed point algorithm, the time marching algorithm and the split Bregman algorithm.In Chapter4, we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle’s dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. In the second method, we propose an image zooming method based on the TV-Stokes model using the dual formulation. By imposing the divergence free condition on the tangential vector field, we got a nonlinear TV-Stokes image zooming model. Once the regularized tangent vector is obtained in the first step, the corresponding regularized normal vector can be computed. So, in the second step, we solve the same problem as in the LOT model. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.In Chapter5, we study the nonlocal total variation (NL-TV) regularization technique for image zooming, which exploits the spatial interactions in images. Due to using the nonlocal regularization in image zooming, our model preserves the smooth region, fine structure and texture well. To solve the nonlinear Euler-Lagrange equation associated with the NL-TV regularization framework, we pro-pose a split Bregman NL-TV image zooming method. Furthermore, we update the weight function during the image zooming which contains a better similarity information between two pixels. Then, we give the convergence analysis of the pro-posed algorithm. Experimental results illustrate the effectiveness and reliability of our method by comparing it with some previous methods.This dissertation is supported by the National Natural Science Foundation of China (60872129).This dissertation is typeset by software LATEX2ε.