On Higher Order Variational Models And Algorithms For Image Restoration

Image restoration has been one of the important contents in image processing tech-nology.Compared with other methods,partial differential equation based method in image restoration shows a strong vitality.Image restoration is divided into denoising,deblurring,repainting,and more and more popular in recent years,dehazing and de-raining.Using the methods of partial differential equation,the variational problem has been set up.Then,the variational model can be solved in a unified way by realizing a primal-dual method for associated saddle point problems.We first put forward the haze removal model based on generalized generalized variation(TGV)and the rain streaks removal model based on m-fold infimal convolution of oscillation TGV.We demonstrate the superiority of the proposed models through a large number of experiments.And then we introduce the finite element method to discrete the denoising problem.The main work and innovation of this paper can be listed as follows:1.We propose a new variational model for removing haze from a single input image.The proposed model combines two Total Generalized Variation(TGV)regularizations,which are related to the image intensity and the transmission map respectively,to build an optimization problem.Actually TGV functionals are more appropriate for describing a natural color image and its transmission map with slanted plane.By minimizing the en-ergy functional with double-TGV regularizations,we obtain the final haze-free image and the refined transmission map simultaneously instead of the general two-step framework.The existence and uniqueness of solutions to the proposed variational model are further obtained.Moreover,the variational model can be solved in a unified way by realizing a primaldual method for associated saddle point problems.A number of experimental results on natural hazy images are presented to demonstrate our superior performance,in comparison to some state-of-the-art methods in terms of the subjective and objec-tive visual quality assessments.Compared with the Total Variation(TV)based models,the proposed model can generate a haze-free image with less staircasing artifacts in the slanted plane and more details in the remote scene of an input image2.Since the regularization based on m-fold infimal convolution of oscillation TGV can be applied to texture decomposition.The observation of rainy images is that the rain streaks in real scenes are multidirectional and multiscale.As a result,we proposed a derain model with the regularization of m-fold infimal convolution of oscillation TGV.The proof of the existence is also given in detail.In addition,we introduce a primal-dual algorithm to solve the proposed model.Experiment results show that the proposed model is superior in competition.The experiments include synthetic images and natural images,and the subjective and objective quality evaluations are adopted.3.Finite Difference Method(FDM)is a well-established variational computational technique to solve problems in image analysis.Compared to the extensively discussed finite difference schemes,very few work has been devoted to Finite Element Method(FEM),which motivates the proposed work.On one hand,FEM has strong physical backgrounds,which allows clear and physically meaningful derivation of difference equa-tions that are easy to implement.On the other hand,combined with the variational methods,the semi-discrete FEM scheme in time scale can give favorable stability and efficiency properties of computations.Thus,in this paper,we give the finite element numerical approach of two important image denoising models,namely,the You-Kaveh(Y-K)model and the Perona-Malik(P-M)model,and the numerical analysis for FEM is also given systematically.Compared results demonstrate the superiority of the pro-posed FEM over FDM,in terms of suppressing blocky effects while maintaining the visual quality.