Study On PDE-based Image Denoising, Inpainting And Decomposition

With the development of computer technology,image processing has been more and more popular in the last two decades.There are three kinds of basic research methods, which are based on probability and statistics,wavelets and PDEs respectively.Image processing based on PDEs has achieved great development in the last two decades,be-cause of its strong adaptability,the anisotropy diffusion characteristic and the property of keeping the edge and texture details simultaneously.Its research areas include:image segmentation,image denoising,deblurring,image decomposition,image inpalnting and image reconstruction et al.This thesis focuses on the study of image denoising,image inpainting,as well as image decomposition based on PDEs and their applications.The main contributions include the following aspects.In chapter 3,three robust denoising models are proposed for different kinds of noise. For the additive noise,a local adaptive variational model based on L^p norm is proposed. In this model,the image space is considered as a Sobolev space W^l,p.An integral energy function about image denoising is defined and then a diffusion partial differential equation about time can be obtained by variational method.Then the existence and uniqueness of the solution of the model are proved.At last an intuitionistic explanation of this process from the aspect of geometric is given.For the multiplicative noise,the existed variational methods are usually non-convex globally,which pose the difficulty for theoretical proving and numerical computing.Due to this,a novel globally convex variational denoising model is proposed.The strictly convex property and as well as the existence and uniqueness of the solution of the variational problem are proved.At last,the experiment results show that our method can achieve reasonable results.Salt & pepper noise is a special kind of random noise,and the traditional variational methods,such as ROF model,can not remove this kind of noise.Motivated by Raymond Chants work,a new two stage denoising method is proposed.Finally,the stable property of the denoising model is analyzed and some numerical experiments are given.In chapter 4,due to fact that the traditional variational inpainting algorithms don't work well for texture images,a local texture-oriented variational inpainting method is proposed in this thesis.Firstly,the approximate direction of texture will be detected based upon the neighbors of the inpainting pixels,then an inpainting model will be defined in the set of coordinates generated by the local texture direction and its normal vector. The theoretical property of the solution of the inpainting model is analyzed.Finally the propose model is compared with the TV model and the OABE algorithm.In chapter 5,a nonlinear face recognition algorithm based on image decomposition is proposed.Firstly,the face images with illuminations will be normalized by the L^p+SQI algorithm,then for the normalized face images,a novel nonlinear topology-preserving face recognition algorithm is proposed.Finally,the experiment results show that this algorithm is stable to the variation of the illumination.