A New Split Bregman Method With Variate Step For Image Deblurring And Denoising With Impulse Noise And Application

In the era of big data,In various application problems such as engineering,information,biology and so on,the underlying model can be summarized as the equality constrained optimization problem.Since the objective function may not be smooth and the problem size is often particularly large,the traditional interior point algorithms may not be suitable.Alternatively,it may be more preferable to use a first-order algorithm such as split Bregman,which uses the objective function's separability of two variables to transform a larger subproblem into two smaller subproblems,and simplifies the calculation of subproblems.Due to the important value of application problems,scholars have proposed a series of improved algorithms in the past decade.Among them,the variable step size split Bregman algorithm(BOSVS)combined with linearization,variable step size,and nonmonotone techniques has significantly accelerated the convergence rate of classical split Bregman algorithm,and applies it to image deblurring and denoising problem with Gaussian noise,It has been proved to be an effective algorithm for solving image deblurring and denoising problem.However,the algorithm can only deal with Gaussian noise and cannot handle impulse noise.In this thesis,we improve the BOSVS and propose a new split Bregman method with variate step size for solving image deblurring and denoising problem with impulse noise.On one hand,it retains the advantages of BOSVS such as linearization technique,variate and nonmonotone step size;on the other hand,by adding a L1 regularization term to the objective function,the model can deal with Gaussian noise as well as impulse noise.And under certain assumptions,we prove the convergence of the new algorithm,Therefore,the new method has wider range of application than the BOSVS.The thesis is organized as follows:The first chapter mainly discusses the research background and significance,the domestic and international research status of the algorithm for solving TV model and so on.In the second chapter,We introduce the theoretical knowledge of Bregman distance,Bregman iteration and its convergence.The third chapter mainly introduces the theory of splitting Bregman algorithm and its convergence,and the relation with the augmented Lagrange method.In the fourth chapter,We introduce the split Bregman algorithm with variate step size.The theoretical results will be introduced in the fifth chapter,involving a new designed model with impulse noise and a new proposed split Bregman method with variate step size,then analyze its convergence.The numerical experimental results are given and validity of the algorithm applied to the Image deblurring and denoising is verified in the sixth chapter.The last chapter is a summary of this thesis and comments on the prospect study for the future research.