Resolvent Calculation And Its Application To Composite Maximal Monotone Operators

Image restoration is one of the most important issues in image processing research.It has been widely used in many practical problems,such as medical image reconstruction,computer vision,and mechanical industry.The image restoration model based on total variation has been widely concerned in recent years.The main idea is to an appropriate energy function for a specific image restoration problem,and recover the original image by minimizing the energy function.According to the convex optimization and monotone operator theory,solving these optimization problems is equivalent to solving a monotone inclusion problem.The operator splitting algorithm is an important method to solve the monotone inclusion problem.It is worth mentioning that the core problem of the operator splitting algorithm is how to effectively calculate the resolvent operator of the corresponding maximal monotone operator.In order to solve the constrained total variation image denoising model,this paper proposes a model for solving the more general composite maximal monotone operators and then discusses the monotone inclusion problem of the sum of three maximal monotone operators.Futher,We solve the optimization problem of the sum of three convex functions.The main work and related results of this paper are summarized as follows:(1)We introduce the research background of the thesis and the research works at home and abroad,and present the research content of this paper.(2)Exploring the resolvent calculation of the composite maximal monotone operator and its application in image denoising.Based on the analysis of the constrained total variation image denoising model,we propose the problem of computing resolvent of the composite maximal monotone operator.By transforming the solution of the resolvent operator into a fixed point equation,a fixed point iterative algorithm is established.In the infinite dimensional Hilbert space,we prove the strong convergence of the proposed fixed point iterative algorithm.The results obtained improve and generalize the existing results.Furthermore,we establish aniterative algorithm to solve the problem of proximity operators of the sum of two convex functions.In order to verify the effectiveness of the algorithm,we apply to solve the constrained total variation image denoising model and obtain satisfactory numerical results.In particular,in the numerical experiments section,the influence of iterative parameters on the proposed iterative algorithm is discussed and explained in detail,and the optimal parameter selection method is given.(3)An inner-outer iterative algorithm is proposed to solve the monotone inclusion problem of the sum of three maximal monotone operators.The fixed point method of the composite maximal monotone operator and the inexact forward-backward operator splitting algorithm are used to prove the weak convergence of the inner-outer iterative algorithm.At the same time,an inner-outer algorithm for solving the optimization problem of the sum of three convex functions is established.The objective function includes a differentiable convex function and a linear transformation composite convex function.The obtained results improve and generalize some know existing results.Finally,by applying the L2+TV image restoration model with constraints and comparing with other existing algorithms,the numerical results show that the proposed iterative algorithm recovers the image quality higher than the comparison algorithm,and the number of iterations is less than others.