  We present a method to reconstruct pure spatial qudits of arbitrary dimension
$d$, which is based on a point diffraction interferometer. In the proposed
scheme, the quantum states are codified in the discretized transverse position
of a photon field, once they are sent through an aperture with $d$ slits, and a
known background is added to provide a phase reference. To characterize these
photonic quantum states, the complete phase wavefront is reconstructed through
a phase-shifting technique. Combined with a multipixel detector, the
acquisition can be parallelized, and only four interferograms are required to
reconstruct any pure qudit, independently of the dimension $d$. We tested the
method experimentally, for reconstructing states of dimension $d=6$ randomly
chosen. A mean fidelity values of $0.95$ is obtained. Additionally, we develop
an experimental scheme that allows to estimate phase aberrations affecting the
wavefront upon propagation, and thus improve the quantum state estimation. In
that regard, we present a proof-of-principle demonstration that shows the
possibility to correct the influence of turbulence in a free-space
communication, recovering mean fidelity values comparable to the propagation
free of turbulence.
