The Numerical Solution Study Of Inverse Problems Based On The L1-norm And The L2-norm

In the inverse problem,the maximum a posteriori(MAP)estimation of model parameters by the data observation value is often based on quadratic programming,corresponding to the least square fitting of data and using the L2-norm for regularization term.The realization of this way is simple and often based on the Gauss-Newton method.Because the noise models of the observed value and the effect requirements of the reconstructed image are different,as an alternative,use of the L1-norm on the data term,the regularization term,or on both of them are desirable,we study in this paper.At the same time,the use of the L1-norm results in the non-smoothness of target functions,which is more complicated and challenging compared to quadratic algorithm on implementations.In methods for L1-norm minimization,the Primal Dual-Interior Point Method(PD-IPM)have been shown to be particularly efficient.In this paper,we derive a PD_IPM framework for selecting the L1-norm independently on the data term or regularization term of an inverse problem,solving the computational obstacles caused by the non-differentiable properties of L1-norm.We use Electrical Impedance Tomography(EIT)as an example of the inverse problem to vertify the implementation of the proposed algorithm is efficient,and the different effects of choosing the L2-norm or the L1-norm on the data term and regularization term of the inverse problem.It can identify that the L2-norm estimates are sensitive to outliers and the spatial distributions are smooth,and the fitting results are not good in the case of non-gaussian noise.As a contrast,use of the L1-norm on the data term renders the estimates robust to outliers and can also get satisfactory results without rejecting a greater error data,and the L1-norm for the regularization term allows reconstructing sharper space images.