  Wavefront sensing is a set of techniques providing efficient means to
ascertain the shape of an optical wavefront or its deviation from an ideal
reference. Due to its wide dynamical range and high optical efficiency, the
Shack-Hartmann is nowadays the most widely used of these sensors. Here, we show
that it actually performs a simultaneous measurement of position and angular
spectrum of the incident radiation and, therefore, when combined with
tomographic techniques previously developed for quantum information processing,
the Shack-Hartmann can be instrumental in reconstructing the complete coherence
properties of the signal. We confirm these predictions with an experimental
characterization of partially coherent vortex beams, a case that cannot be
treated with the standard tools. This seems to indicate that classical methods
employed hitherto do not fully exploit the potential of the registered data.
