Several Splitting Methods For Image Restoration

Image restoration is the most basic and preliminary image processing problem. The aim of image restoration is to remove the noise efficiently and recover the "true" image. In recent years, research in the field of image restoration mainly falls into four categories:variational method:partial differential equations; stochastic modeling and wavelet and space-scale analysis. Partial differential equations(PDEs) have been achieved successful applications in the image processing, since PDEs in classical mathematical physics are powerful tools to describe, model, and analyse many dynamic, and many variational problem or their regularized approximations can be effectively computed from the formal Euler-Lagrange equations. In this thesis, we extend and improve the existing higher order mean curvature model by combining optimization theory, convex analysis, variational principles and partial differential equations. The new model ensure the quality of the image restoration and reduce strong nonlinearity and regularization in the original model. In order to quickly solve the variational model, we design a nonlinear multigrid algorithm for solving the partial differential equations. Experimental results show that the new numerical algorithm is effective and superior. This thesis including four main contents.(1) In the first chapter and the second chapter we briefly introduces develop-ment and status of image processing, and jargons in image processing and multigrid algorithm and relevant concepts.(2) The third chapter mainly studies a splitting mean curvature-based model for gray-scale images corrupted by Gaussian noise. Inspired by augmented Lagrangian algorithm, here we consider one-level splitting method in this work. Instead of using the usual Neumann boundary conditions (BCs), we derived and implemented the precise BCs using staggered grids and locally coupled iterative solvers. Based on this model we propose specially linearizing global Gauss-Seidel method and local Gauss-Seidel method to solve the associated Euler-Lagrange equations. In order to accelerate the solution by a nonlinear multigrid method we found two efficient smoothers are necessary and constructed new transfer operators on staggered grid. Numerical results show that our model can deliver better quality of restoration than the previously fast method, and use less regularization of the nonlinearity than the nonlinear multigrid method proposed by Brito and Chen in2010. In particular, this model can avoid the staircase effect in the TV model.(3) The fourth chapter we apply the splitting mean curvature-based model to vectorial (color) image denoising. By combining channel coupling and splitting model, we present three mean curvature-based models:channel by channel model, local coupling and global coupling model. Through the comparison of experiments, the global case can obtain an optimal performance. And also designed the nonlinear multigrid algorithm for this model.(4) The fifth chapter studies a total variation minimization model for recovering color images corrupted by blurring and impulsive noise. In variational functional we use the l1fidelity term and local multichannel total variation regularization to denoise and deblur. An alternating minimization(AM) method is used to solve proposed the minimization problem. This procedure is divided into three-stage: the first step is deblurring, the second step is reducing the outliers and the third step is denoising. We show the covergence of the AM method and demonstrate that the algorithm is very efficient. experimental results show that the quality of restored color images are superior to some existed methods...