Filtered backprojection method and the interior problem in 2D tomography

We address here the interior problem in local tomography, by means of filtered backprojection (FBP). This algorithm, traditionally used in the context of complete data, is usually not considered as valuable for the interior problem. However, in this article, we prove that as well as more sophisticated methods, the FBP algorithm ensures that the difference between the original and the reconstructed functions is continuous. We verify numerically that the FBP method can supply satisfactory images of discontinuities (on Shepp and Logan phantom and real data). Nevertheless, we also show limits of FBP, by pointing up examples on which the dependence on exterior structures damages the reconstruction quality in the Region Of Interest.