Incomplete data problems in x-ray computerized tomography (Q1066651)

The reconstruction of an object from its x-ray scan is achieved by applying the inverse Radon transform on the measured data. For fast algorithms and stable inversion the directions of the x-rays have to be equally distributed over the semicircle. In the paper the intrinsic problem is studied that arises when the directions are restricted to a limited range of views. Stability considerations are based on the singular value decomposition of that transform. It shows that the smallest singular values decay exponentially indicating a severe ill- posedness of the problem. But in each group of singular values there is a subset of cardinality proportional to the size of the given range which contains singular values of the same size as in the full range case. This part of the spectrum can be reliably recovered. Pictures of singular functions are presented allowing to identify artifacts stemming from limiting the range of given data.