Techniques Based On Partial Differential Equations For Image Restoration

In this dissertation, we extend and improve the existing theory of partial differen-tial equation (PDE) based models and algorithms for image restoration, by syntheticallycombing functional analysis theory, theory of convex analysis, optimization theory, andthe basic theory of partial differential equations. To quickly obtain the numerical solutionsof the proposed variational models, we investigate two numerical algorithms in detail:steepest descent scheme, and split Bregman iteration. The corresponding numerical ex-perimentations obviously demonstrate the superiority of the improved variational modelsand numerical methods. All these innovatory improvements enhance the quality of therecovered images evidently, and ground for the higher level image processing in imageprocessing and computer vision well, such as image analysis and understanding, patternrecognition, etc. The content of this thesis comprises four chapters listed as follows.Firstly, in Chapter 1, we introduce the research background and the practical signif-icance, and review the development of PDE based schemes for image restoration. Then,we present the primary coverage and the innovation. Lastly, the framework of this work isdisplayed in this chapter also.In Chapter 2, some necessary real analysis and functional analysis knowledge, onefrequently-used orthogonal transform in transform domain: Fourier transformation andFourier inverse transformation, and the classical numerical method of numerical simula-tion: finite difference method are retrospected here in brief.Then, Chapter 3 focuses on two improved variational PDE models: Weberized TV-L1 model, and adaptive fourth-order PDE filter for image restoration, and adopting thesteepest descent scheme for obtaining their numerical solutions. As for the Weberized TV-L1 model, which is a variational model based on the noted TV-L1 model, and combinedwith Weber law closely. Adequately considered the in?uence of human vision psychol-ogy, the proposed model can accurately recover the degraded images, and satisfy humanrequirements well. On the other hand, the classical LLT model effectively overcomes the"staircasing effects"in ?at regions while removing noise, however it results in edge blurryfrequently. In order to conquer this drawback, our proposed adaptive fourth-order PDE fil-ter can do it excellently. According to the distinctiveness of the provincial characteristicsof the image, the edge-stopping function adaptively regulates the diffusivity coefficient.Thus our novel model not only surmounts the"staircasing effects"well while removing noise, but availably avoids the blurring effects. Synchronously, the related numerical re-sults distinctly show its superiority and robustness over the traditional method.Finally, in Chapter 4, the application and generalization of the fast numerical strat-egy: split Bregman iteration, in PDE based models for image restoration are investigateddetailedly. First of all, we reiterate the fundamental theory of the related computationalmethods, i.e. Bregman iteration, linearized Bregman iteration, and split Bregman itera-tion. Secondly, replacing total variation by total bounded variation for regularization, wepropose three improved variational PDE models: total bounded variation regularization L2fidelity based scheme for image deblurring, nonlocal total bounded variation regularizationL2 fidelity based model for image denoising, and total bounded variation regularizationtechnique based Poissonian images recovery. Compared with the total variation based ver-sions, the total bounded variation regularization L2 fidelity based image deblurring model,and total bounded variation regularization technique based Poissonian images recoverymodel markedly accelerate the convergence speed of split Bregman iteration, and have theunexampled superiority. As for the nonlocal total bounded variation regularization L2 fi-delity based model for image denoising, we also demonstrate the efficiency by comparingwith its corresponding total variation based one. Numerical simulations illustrate that ourproposed scheme not only overcomes the"staircasing effects"well, but effectively speedsup the calculating speed of split Bregman iteration. Lastly, we employ the fast split Breg-man iteration for solving the H?1 fidelity based image decomposition and restoration.Contrasted with the primary steepest descent scheme, experimental results demonstratethat the proposed algorithm can reduce the iterations drastically, and obviously shorten theCPU time for computing.