A review on fundamental bounds and estimators for photometry and astrometry of celestial point sources using array detectors, from first principles

TL;DR

This review discusses the evolution of observational models for celestial point sources, focusing on fundamental bounds like the CRLB and the performance of estimators such as ML, LS, and WLS, highlighting their theoretical and practical aspects.

Contribution

It provides a comprehensive summary of the transition from empirical to probabilistic models, analyzing key bounds and estimators in astronomical photometry and astrometry.

Findings

CRLB sets theoretical performance limits.

Practical estimators match CRLB only at certain SNR levels.

Joint flux and background estimation improves photometric accuracy.

Abstract

Precise astrometric and photometric measurements of celestial point sources are fundamental to modern astronomy. These measurements, used to determine object positions, motions, and fluxes, are based on observational models that have evolved from empirical centroiding rules to rigorous probabilistic formulations at the pixel level. This review summarizes key contributions that formalized this transition and analyzes seminal works addressing both the theoretical limits and the empirical performance of estimators. Central to these developments is the derivation of fundamental bounds, such as the Cram\'er-Rao Lower Bound (CRLB), and the assessment of widely used estimators, including Maximum Likelihood (ML), Least Squares (LS), and Weighted Least Squares (WLS). These studies show that, while the CRLB sets a theoretical benchmark, practical estimators achieve it only under specific signal-to-noise ratio (SNR) regimes, with notable discrepancies in high-SNR conditions. Moreover, recent results demonstrate that jointly estimating source flux and background significantly improves photometric precision compared to sequential approaches. Looking ahead, the increasing complexity of astronomical surveys, driven by massive data volumes, dynamic observational conditions, and the integration of machine learning, poses new challenges to reliable inference. In this context, tools from statistical theory, including performance bounds and theoretically grounded estimators, remain critical to guide algorithm design and ensure robust astrometric and photometric pipelines.