A New Method For Solving Linear Ill-posed Problems With Application In Medical Imaging

In recent decades, inverse problems for “tracing the input from the output”have appeared in many scientific fields, such as physics, medicine, geology and biology. We call such inverse problems mathematical physics inverse problems. As most of the inverse problems cannot be solved analytically, numerical methods play a fundamental role in the study of the inverse problem. The difficulty of solving the inverse problem comes from the theoretical research and the realization of the algorithm. Despite these difficulties, there are many important applications in fields like engineering and technology, which make the mathematical physics inverse problem a hot research direction.This paper focuses on the bioluminescence tomography imaging problem in medical image, and consider a linear approximation method of this problem. Instead of using the Tikhonov regularization method, we construct a new iterative method for solving the linear ill-posed problems based on the total variation regularization and homotopy perturbation. Compared with the traditional regularization methods,the new method can reconstruct the parameter with less iterations. In particular, the total variational regularization could effciently reconstruct the parameters with piecewise smooth boundary.Theoretically, We introduce the Bregman distance to substitute the total variational penalty, the discrepancy and error, which prove to be decreasing monotonously, of the problem, and indicate the algorithm to be convergent. In the numerical simulations, we compare the new method with2L-regularization to deal with the bioluminescence tomography imaging problem. The simulation results show that the reconstructed image of the new method is more clear than the2L-regularization method result, and the reconstruction of the abnormal body boundary is better.