Optimal estimation of signal amplitude, background level, and photocenter location is crucial to the combined extraction of astrometric and photometric information from focal plane images, 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 that is suited to the efficient iterative implementation and 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.