Modulus-based Iterative Methods For Constrained Fractional Tikhonov Regularization

This thesis mainly discusses the application of fractional Tikhonov regularization in image deblurring.As we all know,image acquisition inevitably undergoes quality degradation,so it is significantly important to restore degraded images.Image deblurring is an important part of image restoration,and it is also a typical ill-posed problem.In order to obtain high-quality images,in this thesis,image deblurring is transformed into solving non-negative linear least-squares problems.Tikhonov regularization in standard form,Newton method and gradient descent method are commonly used to solve linear least-squares problems.Therefore,in order to solve the non-negative linear least-squares problems,modulus iterative methods,based on Tikhonov regularization in standard form,are proposed to solve the non-negativeness of the solution.We call these methods modulus-based iterative methods for constrained Tikhonov regularization in standard form.However,a disadvantage of Tikhonov regularization in standard form is that the solution obtained will appear excessive smoothing problems,that is,many details of the real solution will be lost.Consequently,experts and scholars have proposed the fractional Tikhonov regularization to solve these problems,so the work of this thesis is to change modulus-based iterative methods for constrained Tikhonov regularization in standard form into modulus-based iterative methods for constrained fractional Tikhonov regularization to obtain a better solution.At the beginning of this thesis,the research background of image deblurring and some research work done by domestic and foreign scholars are introduced.Text firstly introduces some preliminary knowledge,and then introduces the classic method,that is,Tikhonov regularization in standard form to solve the least-squares problems,and non-negative least-squares problems due to the need of practical application are also introduced.Secondly,in order to solve these problems,the modulus-based iterative methods for constrained Tikhonov regularization in standard form are introduced in detail.To further improve efficiency of operation and facilitate the solution of matrix inverse,Golub-Kahan bidiagonalization transforms large matrix problems into Krylov subspace,which greatly reduces the amount of computation.Next,fractional Tikhonov regularization is introduced in detail to improve the over-smoothing problems of the solution brought by Tikhonov regularization in standard form.In view of the characteristics of fractional Tikhonov regularization,this thesis replaces the modulus-based iterative methods for constrained Tikhonov regularization in standard form with constrained fractional Tikhonov regularization.Moreover,Golub-Kahan bidiagonalization plays an important role in the calculation of fractional matrix,which transforms large matrix problems into Krylov subspace and improves the operation efficiency.Detailed proof and derivation are given at the same time,the selection of several parameters used in the whole process are also explained in detail.Besides,some boundary conditions are simply stated in this thesis.Finally,numerical experiments are carried out on one-dimensional non-negative signals and two-dimensional images,and the solutions obtained by these two methods are output and compared respectively,the experimental results also show that the modulus-based iterative methods for constrained fractional Tikhonov regularization are superior to modulus-based iterative methods for constrained Tikhonov regularization in standard form.