Research Of Variational Regularization And Euler-Lagrange Equation For Image Processing

In recent years, variational method、partial differential equation (PDE) and sparsereprensentations play active roles in many image processing fields. They have becomethree basic tools for image processing and computer vision. This dissertation mainlystudies the models and applications of variation、PDE and sparse reprensentations inimage processing. The main work can be summarized as follows:1. Two image inpainting models are proposed. One is the image inpainting modelbased on nonlocal diffusion. This model can inpaint the damaged domains by using theglobal information of the image in the diffusion process, it can overcome theshortcomings that the PDE-based models tend to produce some blur and cannotpreserve the texture. The other is the variational inpainting model based on alternateiteration. There are two variables in the new model, so it is firstly turned into twosimple submodels by using the alternative minimization method in the new algorithm,and then the two submodels are solved using split Bregman method respectively. Dueto the fast convergence property of the split Bregman method, the new algorithm canimprove the inpainting speed, thus reducing the runtime. The experimental resultsshow that the two image inpainting models can both obtain the better inpainting effect.2. As the ROF model for additive noise removal tends to produce the staircaseeffect, a novel variational denoising model is proposed based on the study of the LOTmodel. The model can be turned into two simple submodels by using the alternativeminimization method, one is used to construct an angle, and the other is used toreconstruct the image. In the numerical computation, we apply the split Bregmanmethod to solve the two submodels alternately. The experimental results show that thenew algorithm not only has faster convergence rate, but also can alleviate the staircaseeffect and preserve the edge information better.3. Aiming at the problem of multiplicative noise removal, combining thevariational method、PDE、sparse reprensentations and dictionary learning, a sparsityregularization method for multiplicative noise removal is proposed in log-domain. Theproposed method mainly contains three steps. Firstly, a better log-image is obtained byusing the sparse representation based on the dictionary learning. Secondly, the totalvariation (TV) model is used to amend the log-image. Finally, via an exponentialfunction and bias correction, the result is transformed back from the log-image domain to the real one. The experimental results show that the new method is more effective tofilter out the multiplicative noise while well preserving the texture.4. TV can preserve the edge information better, but it tends to produce the staircaseeffect. In order to overcome the drawback, a novel variational regularization model formultiplicative noise removal is proposed in log-domain based on the total generalizedvariation (TGV) introduced recently, and the existence and uniqueness of a minimizerfor the proposed model are proven. TGV has many advantages, which make the newmodel to be more effective for multiplicative noise removal while avoiding thestaircase effect and well preserving the feature and details. In the numericalcomputation, we use the first-order primal-dual algorithm and the Newton method tosolve the new model. The experimental results show that the proposed algorithm canget the better results from the visual effect and the peak signal to noise ratio (PSNR).5. Multiplicative noise removal is of momentous significance in coherent imagingsystems and various image processing applications. So far, most multiplicative noiseremoval models focus on the regularization method, these models require knowing theprior level of noise beforehand, however, the information isn’t obtained in some case,to overcome the drawback, two projection methods for multiplicative noise removalmodels are presented, and three fast numerical algorithms are given by using theduality technique and the variable-splitting. The experimental results show that theproposed algorithms not only have faster convergence rate, but also can effectivelyfilter out the multiplicative noise when the prior of noise is unknown.