  We study the tomography of spatial qudits encoded on structured light
photons. While direct position measurements with cameras do not provide an
informationally complete Positive Operator Valued Measure (POVM) in the space
of fixed order modes, we complement this POVM with an astigmatic
transformation. The enlarged POVM is informationally complete, allowing full
characterization of the spatial quantum state from simple intensity
measurements in both the intense and in the low photocount regimes. For intense
light, the standard technique of linear inversion is used. For the low
photocount regime, we employ Bayesian mean inference, and study how the quality
of the tomographic reconstruction behaves as we increase the photocounts. In
both cases, we also perform the tomography using a convolutional neural
network, which displays an increased flexibility in exchange for a slightly
lower quality reconstruction in some of the cases. These methods will be useful
for classical and quantum communication with structured light.
