Fast Numerical Methods For Image Processing Based On PDE Models

Image processing is an applied science closely related to the national econo-my and people’s livelihood. It has penetrated into all areas of people’s lives andwork, such as aerospace, biomedical engineering, industrial inspection, robot vi-sion, military guidance, geophysical and atmospheric environment. Furthermore,it has brought huge economic and social benefts to mankind. At the same time,image processing technology is still far from being unable to meet the needs of soci-ety. Therefore, image processing research has important signifcance and practicalvalue.In this dissertation, by using partial diferential equation method, two basicproblems in image processing are studied: image denoising, image segmentation.Image denoising belongs to the context of image restoration. It requires to denois-ing the observed image and restoring the original appearance of the ideal image.Image segmentation is a technology that separates the object of interest in theimage from the remaining portion in the image in order to serve higher-level imageprocessing. After briefy introducing some basic concepts and research status ofimage processing, this dissertation is devoted to studying in-depth image denoisingand image segmentation. The main work is described as follows:A nonlinear multigrid method for solving the LLT model(isotropic) is pro-posed. By using the LFA of the Chambolle’s dual iterations (called CDA) forsolving the LLT model and analyzing its smoothing rate, we know that the non-linear multigrid method with the Chambolle’s dual iterations as its smoother willconverge slowly. Furthermore, by using the LFA of a modifed smoother, it is seenthat choosing a suitable parameter contributes to much improved rates. Basedon these two observations, we propose to use an improved dual iteration as thesmoother of the multigrid method. Since we use the multigrid method to solvethe dual problem of the LLT model and then obtain the solution of the originalproblem, the numerical difculties caused by the non-diferentiability of the LLTmodel are also overcome. Using the proposed multigrid method to process grayscale image in experiments, the efect is signifcantly better than the CDA. Whenthe image size becomes larger, that is discretization becomes more sophisticated,multigrid method will increase less calculation amount than other methods.A nonlinear multigrid method for solving a general image denoising modelwith two L1-regularization terms is studied. In particular, we apply this method to solve two special models: the anisotropic ROF model and the anisotropic LLTmodel. By using the LFAs of the Chambolle’s dual iterations and a modifedsmoother for solving these two models respectively and analyzing their smoothingrates, we prove that the proposed multigrid algorithm with the improved dualiteration as its smoother is very reasonable. To overcome the numerical difcultiescaused by the non-diferentiability of the models, we use the multigrid algorithm tosolve the dual equation derived from the dual problem. Diferent from the previousstudies, we give a simpler derivation of the dual formulation of the general modelby augmented Lagrangian method. Numerical experiments verify the efciency ofthe proposed multigrid method for solving these two anisotropic image denoisingmodels and indicate that such a multigrid method is more suitable to deal withlarge-sized images.An improved dual algorithm for solving two-phase piecewise constant Mumford-Shah model for image segmentation is studied. The original problem is dividedinto three subproblems to solve. We propose to use an improved dual iterative tosolve one of these subproblems. In order to prove the rationality of the proposedmethod, we use the LFA to analyze the convergence rate of the Chambolle’s dualiterations and a modifed iterative scheme for solving the subproblem, respectively.The numerical experiments show that the proposed algorithm has maintained fastsegmentation ability of the level-set image segmentation method based on the dualiterations, and improves the segmentation quality.This dissertation is supported by the National Natural Science Foundation ofChina(Nos.60872129,60835004).This dissertation is typeset by software LATEX2ε.