  Optimal estimation of signal amplitude, background level, and photocentre
location is crucial to the combined extraction of astrometric and photometric
information from focal plane images, and in particular from the one-dimensional
measurements performed by Gaia on intermediate to faint magnitude stars. Our
goal is to define a convenient maximum likelihood framework, suited to
efficient iterative implementation and to assessment of noise level, bias, and
correlation among variables. The analytical model is investigated numerically
and verified by simulation over a range of magnitude and background values. The
estimates are unbiased, with a well-understood correlation between amplitude
and background, and with a much lower correlation of either of them with
location, further alleviated in case of signal symmetry. Two versions of the
algorithm are implemented and tested against each other, respectively, for
independent and combined parameter estimation. Both are effective and provide
consistent results, but the latter is more efficient because it takes into
account the flux-background estimate correlation.
