High-order TGV Variational Models And Algorithms In Image Restoration

Variational models are alaways a popular topic in inverse problems and computer vision,which have been widely used in a variety of image processing fields,for exam-ple,image restoration,fusion,registration,segmentation and reconstruction in medical images and so on.Regularization technique has strong theoretical analysis,and we can analyze the performances of the models in infinite dimensional space,e.g.,the well-posedness and the propert,ies of solutions.There are always staircasing effects in early total variation(TV)models,since the bounded variation(BV)can admit piece-wise constant functions.In addition,the classical regularization term can not describe the structural textures of an image well and lose some detailed information.Then,this thesis proposes some improved high-order variational models,including the total generalized variation(TGV)model which can reduce the staircasing,and two infimal convolution models based on Shearlet transform and oscillation TGV,which can rep-resent textures well.The main work and innovation of this paper can be listed as follows:1.Due to the drawback of staircase effects of low-order TV based models,this thesis proposes two high-order models based on second-order TGV to process the deblurring problems with Inpulsive and Poisson noises.Since the models with non-quadratic fi-delity terms are non-differentiable and non-linear,we introduce alternating direction method of multipliers(ADMM)algorithm.Usually,the directly extended ADMM for the minimization problem with multi-block variables is not necessarily convergent,so we introduce Prediction-Correction scheme to guarantee the convergence,and each subproblems can be solved simply.The numerical experiments show that our proposed models can reduce the staircasing efficiently,and can also obtain better visual results and higher evaluation standards.Moreover,the modified ADMM algorithm can achieve the convergence fast.We also give the dual form of multi-channel TGV and apply this regularizer to color images.2.Since the traditional variational models,e.g.,TV and TGV,can not represent the texture part of an image well,this thesis proposes a novel infimal convolution regularizer based on TGV and Shearlet transform,which can be applied to image denoising,inpainting and under-sampled MRI reconstruction.Compared to original Wavelet,Shearlet is sensitive to orientation and has the good property of representing the positions and orientations of singularities(edges),so we use TGV to represent the piecewise smooth cartoon component and use the L1 norm of Shearlet transform to regularize the oscillatory texture component.Numerical experiments show that our model can recover not only the piecewise smooth part of an image,but also the detailed texture part.In the theoretical analysis part,we also give the solution existence of the general model in L2 space.In addition,we extend the new regularizer to JPEG decompression problem and show its efficiency in texture preservation compared to a state-of-the-art learning approach.3.Since the sinusoidal function is periodic oscillatory,the regularization with it as the kernel space can detect oscillatory textures.Inspired by the second-order TGV,this thesis proposes a new type of regularization functional called oscillation TGV which can represent structured textures in a specified direction and scale.According to dual the-ory,we construct the basic properties of oscillation TGV,e.g.,the lower semi-continuity and coercivity in LP space,and the equivalence with BV space.For images with different scales and directions of textures,we design a m-fold infimal convolution regularization term.As the same with oscillation TGV,we also prove the lower semi-continuity and coercivity of the infimal convolution term,and give the solution existence of the general model in functional space.Then,we introduce a first-order primal-dual algorithm to solve our general variational imaging problems associated with this infimal convolution term,and estimate the operator norm which controls the convergence condition.Nu-merical experiments show that our proposed models can recover textures well and are competitive with some existing state-of-the-art methods.Furthermore,considering the complexity of m-fold infimal convolution term in applications,we propose an adaptive oscillation TGV,which can adapt to different directions and scales of textures according to local information,and simplify the model.