  Scanning gate microscopy images from measurements made in the vicinity of
quantum point contacts were originally interpreted in terms of current flow.
Some recent work has analytically connected the local density of states to
conductance changes in cases of perfect transmission, and at least
qualitatively for a broader range of circumstances. In the present paper, we
show analytically that in any time-reversal invariant system there are
important deviations that are highly sensitive to imperfect transmission.
Nevertheless, the unperturbed partial local density of states can be extracted
from a weakly invasive scanning gate microscopy experiment, provided the
quantum point contact is tuned anywhere on a conductance plateau. A
perturbative treatment in the reflection coefficient shows just how sensitive
this correspondence is to the departure from the quantized conductance value
and reveals the necessity of local averaging over the tip position. It is also
shown that the quality of the extracted partial local density of states
decreases with increasing tip radius.
