  In this paper we relate the matrix $S_B$ of the second moments of a
spherically truncated normal multivariate to its full covariance matrix
$\Sigma$ and present an algorithm to invert the relation and reconstruct
$\Sigma$ from $S_B$. While the eigenvectors of $\Sigma$ are left invariant by
the truncation, its eigenvalues are non-uniformly damped. We show that the
eigenvalues of $\Sigma$ can be reconstructed from their truncated counterparts
via a fixed point iteration, whose convergence we prove analytically. The
procedure requires the computation of multidimensional Gaussian integrals over
a Euclidean ball, for which we extend a numerical technique, originally
proposed by Ruben in 1962, based on a series expansion in chi-square
distributions. In order to study the feasibility of our approach, we examine
the convergence rate of some iterative schemes on suitably chosen ensembles of
Wishart matrices. We finally discuss the practical difficulties arising in
sample space and outline a regularization of the problem based on perturbation
theory.
