This paper is focused on image reconstructions from incomplete projecting data. Based on the assumption of continuity among image pixels

the linear system which the optimal solution satisfies is derived for incomplete projecting data. An artifitial parameter is then introduced and a new linear system is formed from the original one. First

the new system is solved by perturbation method and the expansion cofficients are computed by an iteration; second

the solution to the original system is achieved by applying rational approximation of vector-valued funcation to the perturbation solution. The difficulty of the high computation amount of the direct solution of the original linear system is overcome by the proposed method. Numerical examples indicate that when only three items in the perturbation expansion are used

satisfactory approximations to the original images can be achieved by the rational approximation method.