Stability of the interior problem with polynomial attenuation in the region of interest

Abstract

In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function fa on an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution if fa is assumed to be piecewise polynomial on the ROI. In this paper, we tackle the related question of stability. It is well known that lambda tomography allows one to stably recover the locations and values of the jumps of fa inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, fa on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on fa, we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open set U on which fa is the restriction of a real-analytic function, then fa is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result.