  A number of practically important imaging problems involve inverting the
generalized Radon transform (GRT) $\mathcal R$ of a function $f$ in $\mathbb
R^3$. On the other hand, not much is known about the spatial resolution of the
reconstruction from discretized data. In this paper we study how accurately and
with what resolution the singularities of $f$ are reconstructed. The GRT
integrates over a fairly general family of surfaces $\mathcal S_y$ in $\mathbb
R^3$. Here $y$ is the parameter in the data space, which runs over an open set
$\mathcal V\subset\mathbb R^3$. Assume that the data $g(y)=(\mathcal R f)(y)$
are known on a regular grid $y_j$ with step-sizes $O(\epsilon)$ along each
axis, and suppose $\mathcal S=\text{singsupp}(f)$ is a piecewise smooth
surface. Let $f_\epsilon$ denote the result of reconstruction from the descrete
data. We obtain explicitly the leading singular behavior of $f_\epsilon$ in an
$O(\epsilon)$-neighborhood of a generic point $x_0\in\mathcal S$, where $f$ has
a jump discontinuity. We also prove that under some generic conditions on
$\mathcal S$ (which include, e.g. a restriction on the order of tangency of
$\mathcal S_y$ and $\mathcal S$), the singularities of $f$ do not lead to
non-local artifacts. For both computations, a connection with the uniform
distribution theory turns out to be important. Finally, we present a numerical
experiment, which demonstrates a good match between the theoretically predicted
behavior and actual reconstruction.
