A Method For Solving Linear Ill-Posed Problems With Its Application

With the promotion of the practical application, the area of application for the theory of inverse problem has extended to almost all fields of science. The inverse problem has become one of the most popular mathematical research areas. At the same time, the regularization theory is also developed for solving these kinds of problems.Among different methods for solving the ill-posed problems, the total variation(TV) regularization method is widely used because of its ability to keep the edge information of the original problem. The method is proved to be very effective in dealing with nonsmooth object boundary. In the field of image denoising, TV regularization has become one of the most important methods. In this paper, a parameter ? is introduced to overcome the non-differentiablity at zero point of TV norm. A homotopy curve is constructed using the homotopy technique. Thus a new method for solving linear ill-posed problems is derived. The convergence of the new iterative scheme is proved. If the data is not polluted by noise, the convergent of the iterative scheme can be guaranteed by the Hilbert space theory, the inequality theory,and the Cauchy principle. For the sake of practical application, the observed data always has certain level of perturbation. Based on the data with perturbation, the convergent of the iterative scheme is proved by the inequality theory and the Morozov discrepancy principle.In the field of medical imaging, bioluminescent tomography is a new kind of molecular imaging technique, which has attracted much attention, because it is non-invasive, convenient and economic. The BLT imaging could diagnose or predict the pathology of organization by fluorescing labeled target gene. In essence, it determines the position of the light-emitting cells inside the body through the measured optical knowledge of body surface. This process is a typical inverse problem of mathematical physics, and locating the unknown light source in the body is ill-posed. The commonly used mathematical model of the optical transmission in the body problem is the radiative transfer equation(RTE). However, most biomedical imaging research is developed for the diffusion approximation of RTE equation. In this paper, we propose a new iterative scheme to reconstruct the source term of the RTE equation. The experimental results show that the new iterative method is effective in restoring the shape and the location of the light source inside the body. Moreover, the method is effective in preserving the boundary information of light source. Thus the method is a satisfactory candidate for solving the linear ill-posed problems. The iterative scheme can be applied to other linear inverseproblems, and it has great application prospects.