Analysis of stability of tomographic reconstruction of X-ray medical images

Slice reconstruction in X-ray computed tomography is reduced to the solution of integral equations, or a system of algebraic equations in discrete case. It is considered to be an ill-posed problem due to the inconsistencies in the number of equations and variables and due to errors in the experimental data. Therefore, determination of the best method of the slice reconstruction is of great interest. Furthermore, all available methods give approximate results. The aim of this article was two-fold: i) to compare two methods of image reconstruction, viz. inverse projection and variation, using the numerical experiment; ii) to obtain the relationship between image accuracy and experimental error. It appeared that the image obtained by inverse projection is unstable: there was no convergence of the approximate image to the accurate one, when the experimental error reached zero. In turn, the image obtained by variational method was accurate at zero experimental error. Finally, the latter showed better slice reconstruction, despite the low number of projections and the experimental errors.