  We present a proof-of-principle demonstration of a method to characterize
\textit{any} pure spatial qudit of arbitrary dimension $d$, which is based on
the classic phase shift interferometry technique. In the proposed scheme a
total of only $4 d$ measurement outcomes are needed, implying a significant
reduction with respect to the standard schemes for quantum state tomography
which require of the order of $d^2$. By using this technique, we have
experimentally reconstructed a large number of states ranging from $d=2$ up to
$14$ with mean fidelity values higher than $0.97$. For that purpose the qudits
were codified in the discretized transverse momentum-position of single
photons, once they are sent through an aperture with $d$ slits. We provide an
experimental implementation of the method based in a Mach-Zehnder
interferometer, which allows to reduce the number of measurement settings to 4
since the $d$ slits can be measured simultaneously. Furthermore, it can be
adapted to consider the reconstruction of the unknown state from the outcome
frequencies of $4d-3$ fixed projectors independently of the encoding or the
nature of the quantum system, allowing to implement the reconstruction method
in a general experiment.
