  This article provides a mathematical analysis of singular (nonsmooth)
artifacts added to reconstructions by filtered backprojection (FBP) type
algorithms for X-ray CT with arbitrary incomplete data. We prove that these
singular artifacts arise from points at the boundary of the data set. Our
results show that, depending on the geometry of this boundary, two types of
artifacts can arise: object-dependent and object-independent artifacts.
Object-dependent artifacts are generated by singularities of the object being
scanned and these artifacts can extend along lines. They generalize the streak
artifacts observed in limited-angle tomography. Object-independent artifacts,
on the other hand, are essentially independent of the object and take one of
two forms: streaks on lines if the boundary of the data set is not smooth at a
point and curved artifacts if the boundary is smooth locally. We prove that
these streak and curve artifacts are the only singular artifacts that can occur
for FBP in the continuous case. In addition to the geometric description of
artifacts, the article provides characterizations of their strength in Sobolev
scale in certain cases. The results of this article apply to the well-known
incomplete data problems, including limited-angle and region-of-interest
tomography, as well as to unconventional X-ray CT imaging setups that arise in
new practical applications. Reconstructions from simulated and real data are
analyzed to illustrate our theorems, including the reconstruction that
motivated this work---a synchrotron data set in which artifacts appear on lines
that have no relation to the object.
