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In this paper we study the joint determination of source and background fluxfor point sources as observed by digital array detectors. We explicitly computethe two-dimensional Cram\'er-Rao absolute lower bound (CRLB) as well as theperformance bounds for high-dimensional implicit estimators from a generalizedTaylor expansion. This later approach allows us to obtain computableprescriptions for the bias and variance of the joint estimators. We comparethese prescriptions with empirical results from numerical simulations in thecase of the weighted least squares estimator (introducing an improved version,denoted stochastic weighted least-squares) as well as with the maximumlikelihood estimator, finding excellent agreement. We demonstrate that theseestimators provide quasi-unbiased joint estimations of the flux and background,with a variance that approaches the CRLB very tightly and are, hence, optimal,unlike the case of sequential estimation used commonly in astronomicalphotometry which is sub-optimal. We compare our predictions with numericalsimulations of realistic observations, as well as with observations of abona-fide non-variable stellar source observed with TESS, and compare it to theresults from the sequential estimation of background and flux, confirming ourtheoretical expectations. Our practical estimators can be used as benchmarksfor general photometric pipelines, or for applications that require maximumprecision and accuracy in absolute photometry.

Optimal photometry of point sources: Joint source flux and background determination on array detectors -- from theory to practical implementation