Measures whose projections are absolutely continuous

Question

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely continuous with respect to the Lebesgue measure? I think this could be equivalent to the fact that $\mu e=0$ for any set $e$ of zero linear Hausdorff measure? References to related results will also be appreciated.