  We consider the generalized Radon transform (defined in terms of smooth
weight functions) on hyperplanes in $\mathbb{R}^n$. We analyze general filtered
backprojection type reconstruction methods for limited data with filters given
by general pseudodifferential operators. We provide microlocal
characterizations of visible and added singularities in $\mathbb{R}^n$ and
define modified versions of reconstruction operators that do not generate added
artifacts. We calculate the symbol of our general reconstruction operators as
pseudodifferential operators, and provide conditions for the filters under
which the reconstruction operators are elliptic for the visible singularities.
If the filters are chosen according to those conditions, we show that almost
all visible singularities can be recovered reliably. Our work generalizes the
results for the classical line transforms in $\mathbb{R}^2$ and the classical
reconstruction operators (that use specific filters). In our proofs, we employ
a general paradigm that is based on the calculus of Fourier integral operators.
Since this technique does not rely on explicit expressions of the
reconstruction operators, it enables us to analyze more general imaging
situations.
