  We investigate algorithms for reconstructing a convex body $K$ in $\mathbb
{R}^n$ from noisy measurements of its support function or its brightness
function in $k$ directions $u_1,...,u_k$. The key idea of these algorithms is
to construct a convex polytope $P_k$ whose support function (or brightness
function) best approximates the given measurements in the directions
$u_1,...,u_k$ (in the least squares sense). The measurement errors are assumed
to be stochastically independent and Gaussian. It is shown that this procedure
is (strongly) consistent, meaning that, almost surely, $P_k$ tends to $K$ in
the Hausdorff metric as $k\to\infty$. Here some mild assumptions on the
sequence $(u_i)$ of directions are needed. Using results from the theory of
empirical processes, estimates of rates of convergence are derived, which are
first obtained in the $L_2$ metric and then transferred to the Hausdorff
metric. Along the way, a new estimate is obtained for the metric entropy of the
class of origin-symmetric zonoids contained in the unit ball. Similar results
are obtained for the convergence of an algorithm that reconstructs an
approximating measure to the directional measure of a stationary fiber process
from noisy measurements of its rose of intersections in $k$ directions
$u_1,...,u_k$. Here the Dudley and Prohorov metrics are used. The methods are
linked to those employed for the support and brightness function algorithms via
the fact that the rose of intersections is the support function of a projection
body.
