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Solving the interior problem of computed tomography using a priori knowledge

A case of incomplete tomographic data for a compactly supported attenuation function is studied. When the attenuation function is a priori known in a subregion, we show that a reduced set of measurements are enough to uniquely determine the attenuation function over all the space. Furthermore, we found stability estimates showing that reconstruction can be stable near the region where the attenuation is known. These estimates also suggest that reconstruction stability collapses quickly when approaching the set of points that is viewed under less than 180°. This paper may be seen as a continuation of the work ‘Truncated Hilbert transform and image reconstruction from limited tomographic data’ (Defrise et al 2006 Inverse Problems 22 1037). This continuation tackles new cases of incomplete data that could be of interest in applications of computed tomography.