  We address the astrophysical imaging of a family of deformed Kerr black holes
(BHs). These are stationary, asymptotically flat black hole (BH) spacetimes,
that are solutions of General Relativity minimally coupled to a massive,
complex scalar field: Kerr BHs with scalar hair (KBHsSH). Such BHs bifurcate
from the vacuum Kerr solution and can be regarded as a horizon within a
rotating boson star. In a recent letter, it was shown that KBHsSH can exhibit
very distinct shadows from the ones of their vacuum counterparts. The setup
therein, however, considered the light source to be a celestial sphere
sufficiently far away from the BH. Here, we analyse KBHsSH surrounded by an
emitting torus of matter, simulating a more realistic astrophysical
environment, and study the corresponding lensing of light as seen by a very far
away observer, to appropriately model ground-based observations of Sgr A*. We
find that the differences in imaging between KBHsSH and comparable vacuum Kerr
BHs remain, albeit less dramatic than those observed for the corresponding
shadows in the previous setup. In particular, we highlight two observables that
might allow differentiating KBHsSH and Kerr BHs. The first is the angular size
of the photon ring (in a Kerr spacetime) or lensing ring (in a KBHSH
spacetime), the latter being significantly smaller for sufficiently
non-Kerr-like spacetimes. The second is the existence of an edge in the
intensity distribution (the photon ring in Kerr spacetime). This edge can
disappear for very non-Kerr-like KBHsSH. It is plausible, therefore, that
sufficiently precise Very Long Baseline Interferometric observations of BH
candidates can constrain this model.
