On Semismooth Newton Method For Total Variation-based Image Noise Removal

In the digital image denoising process,the frequently-used idea of mathematical knowledge for image denoising method is based on variation method.At present,the most classic model of this idea is the total variational model.The biggest advantage of it is to effectively keep the edge around object while we eliminate noise of the image.But the regularization term of TV model is nondifferentiable such that the whole model is nondifferentiable,thus the traditional optimization method based on differential defi-nition is no longer applicable.The promotion of the classical differential definition,the establishment of generalized differential definition and the corresponding optimization theory and algorithm have become a significance study on nonsmooth optimization.Researchers proposed a Semismooth Newton method in the study process,and then quickly became one of the popular research methods in nonsmooth optimization.This paper is continue to study the Semismooth Newton method to eliminate noise,re-covery images and improve quality and visual effects of image.Taking the TV model as the mathematical model,one dimensional signal images,two dimensional inverse source problems and two dimensional image problems are as study objects,the numer-ical optimization algorithm is developed.The work of this thesis are:(1)some basic knowledge of denoising in digital image processing theory are introduced,and the research status of image denoising at home and abroad in recent years and the significance of image denoising in practical application are reviewed.(2)The basic theory and total variational model are introduced.According to the basic optimization theory,we transform it into the optimization problem.And we describe the primal dual algorithm and the alternating direction method of multiplier which are widely used to solve TV problem.(3)We introduce the study of constrained least squares problem algorithm,nar-rate the optimization problem with constraints is transformed to the unconstrained op-timization problem by projecting and put forward a gradient descent method based on projection at last.(4)The nonlinear and nonsmooth equations and Semismooth Newton method are introduced.Furthmore,we point out a new algorithm:the Semismooth Newton method combined with fixed-point iteration,and describe locally superlinear convergence of it.(5)Compiling these algorithms' code in Matlab which involved in the article to do numerical simulation experiment.The experimental results are obtained by compar-ing the experimental phenomena and the results,with the proposed new algorithm is superior to the other three algorithms in the case of the same noise and parameters.