The Application And Improvement Of Tikhonov Regularization Method For Electrical Impedance Tomography

This thesis introduces a medical imaging technology-Electrical Impedance Tomography(EIT),which is a new-emerging functional imaging technology.In recent decades,EIT has been widely studied by domestic and foreign scholars,because it has the advantages of portable,high speed,non-invasive and so on.Because of the conductivity properties of different objects,we can inject current and measure the voltage value at surface electrodes to get the image of the conductivity distribution or variation inside the object.It has excellent application prospect in medical,biological and so on.However,this technology also has difficulties.The quality of imaging,instability,and data errors are the main reasons that restrict its development.From the mathematical point of view,EIT can be regarded as a kind of parameter identification problem of second-order elliptic partial differential equation.So it often has the characteristics of inverse problem with ill-posedness.Therefore,this paper discusses the forward and inverse problems of EIT:The first chapter is the Introduction.It mainly introduces the basic principles of EIT and the research status at home and abroad.Then we introduce the value and technical difficulties of the research.Lastly,we introduce the related concepts of inverse problem and ill-posedness and list the structure of this paper.In the second chapter,the forward problems of EIT are studied.Firstly,the working mode of EIT imaging(current injection and voltage measurement)is introduced.Then the mathematical model of forward problem is derived by Maxwell's equations and related boundary conditions.We select the complete-electrode model and solved it by finite element method(FEM).In the finite element splitting,two models of sparse and dense are obtained.The third chapter discusses the inverse problem of EIT,which is the focus of this thesis.In this chapter,we solve the problem by the ordinary least squares method and find that the solution is unstable or loses its practical significance.Therefore,the regularization method is introduced,including definition,principle and the Tikhonov regularization method which is mainly studied in this paper.The convergence and stability of Tikhonov regularization method are discussed theoretically.In order to improve the quality of imaging by Tikhonov regularization method,we import variation function to the regularization penalty term and propose the Total variation regularization method.Then we derive the iterative format of Newton s Method.And we compare the imaging quality of the two regularization methods based on EIDORS 2D software.Then,we propose the combination regularization method with new penalty term to further improve Tikhonov regularization method.After that,we introduce the methods to select a regularization parameter and we get the iterative format.Lastly,we give two numerical experiments.In the last chapter,we draws the conclusions of the study and analyzes the shortcomings of this paper and the direction of future research.