[Paper Review] Do fast stellar centroiding methods saturate the Cramér-Rao lower bound?

Mohammadjavad Vakili, David W. Hogg|arXiv (Cornell University)|Oct 19, 2016

Adaptive optics and wavefront sensing3 citations

TL;DR

This paper evaluates whether fast stellar centroiding methods—specifically polynomial fitting after PSF smoothing or Gaussian smoothing—can achieve the Cramér-Rao lower bound (CRLB) for astrometric precision in low signal-to-noise, sky-limited conditions. It finds that when the PSF is known, PSF-matched polynomial centroiding saturates the CRLB; when unknown, fixed-Gaussian smoothing with polynomial fitting comes very close, while center-of-light methods fail badly.

ABSTRACT

One of the most demanding tasks in astronomical image processing---in terms of precision---is the centroiding of stars. Upcoming large surveys are going to take images of billions of point sources, including many faint stars, with short exposure times. Real-time estimation of the centroids of stars is crucial for real-time PSF estimation, and maximal precision is required for measurements of proper motion. The fundamental Cramér-Rao lower bound sets a limit on the root-mean-squared-error achievable by optimal estimators. In this work, we aim to compare the performance of various centroiding methods, in terms of saturating the bound, when they are applied to relatively low signal-to-noise ratio unsaturated stars assuming zero-mean constant Gaussian noise. In order to make this comparison, we present the ratio of the root-mean-squared-errors of these estimators to their corresponding Cramér-Rao bound as a function of the signal-to-noise ratio and the full-width at half-maximum of faint stars. We discuss two general circumstances in centroiding of faint stars: (i) when we have a good estimate of the PSF, (ii) when we do not know the PSF. In the case that we know the PSF, we show that a fast polynomial centroiding after smoothing the image by the PSF can be as efficient as the maximum-likelihood estimator at saturating the bound. In the case that we do not know the PSF, we demonstrate that although polynomial centroiding is not as optimal as PSF profile fitting, it comes very close to saturating the Cramér-Rao lower bound in a wide range of conditions. We also show that the moment-based method of center-of-light never comes close to saturating the bound, and thus it does not deliver reliable estimates of centroids.

Motivation & Objective

• To assess whether fast, computationally efficient centroiding methods can achieve the theoretical precision limit set by the Cramér-Rao lower bound (CRLB) in astrometry.
• To investigate the performance gap between optimal maximum-likelihood estimators and practical fast methods under realistic low signal-to-noise (SNR) and varying full-width at half-maximum (FWHM) conditions.
• To compare centroiding methods in two scenarios: when the PSF is known versus when it is unknown, focusing on sky-limited, Gaussian-noise images.
• To quantify the root-mean-squared error (RMSE) of various methods relative to the CRLB across a range of SNR and FWHM values.
• To determine whether commonly used methods like center-of-light centroiding are reliable for high-precision astrometry in large surveys.

Proposed method

• Simulates sky-limited, uncorrelated Gaussian-noise images with non-overlapping, faint stars using a specific analytic PSF model (e.g., Moffat-like).
• Computes the theoretical Cramér-Rao lower bound (CRLB) for centroiding error as a function of SNR and FWHM using the Fisher information matrix derived from the PSF model.
• Applies four centroiding methods: (1) maximum-likelihood fitting to the true PSF, (2) 2D second-order polynomial fitting to a 3×3 patch after PSF-matched smoothing, (3) polynomial fitting after fixed-Gaussian smoothing (no PSF knowledge), and (4) center-of-light (first-moment) estimation.
• Measures the root-mean-squared error (RMSE) of each method’s centroid estimates across many simulated stars and compares it to the CRLB via the ratio RMSE / CRLB.
• Performs two simulation sets: one varying SNR with fixed FWHM, and one varying FWHM with fixed SNR, to assess robustness across conditions.
• Uses scatter plots and median ratios to visualize and quantify how close each method comes to saturating the CRLB across parameter space.

Experimental results

Research questions

• RQ1 Can fast, non-iterative centroiding methods such as polynomial fitting after smoothing saturate the Cramér-Rao lower bound (CRLB) for stellar centroids in low-SNR, sky-limited conditions?
• RQ2 How does the performance of PSF-matched polynomial centroiding compare to maximum-likelihood estimation when the PSF is known?
• RQ3 How well does fixed-Gaussian smoothing followed by polynomial fitting perform when the PSF is unknown, relative to the CRLB?
• RQ4 To what extent does the center-of-light method fail to approach the CRLB, and is it suitable for high-precision astrometry?
• RQ5 How do the RMSE-to-CRLB ratios of these methods vary with signal-to-noise ratio and FWHM of the star's PSF?

Key findings

• When the PSF is known, PSF-matched polynomial centroiding after smoothing achieves a RMSE-to-CRLB ratio of approximately 1.0 across a wide range of SNR and FWHM values, indicating near-perfect saturation of the CRLB.
• With unknown PSF, fixed-Gaussian smoothing followed by polynomial fitting achieves a RMSE-to-CRLB ratio of about 1.05–1.15 across most tested conditions, showing it is nearly optimal and highly efficient.
• The center-of-light method consistently yields RMSE-to-CRLB ratios exceeding 2.0, even at high SNR, indicating it is fundamentally suboptimal and unsuitable for high-precision astrometry.
• The performance of all methods degrades with increasing FWHM and decreasing SNR, but the relative gap to the CRLB remains smallest for PSF-matched and fixed-Gaussian polynomial methods.
• Maximum-likelihood fitting to the true PSF achieves the lowest RMSE-to-CRLB ratio (≈1.0), confirming it as the optimal benchmark.
• The RMSE-to-CRLB ratio for PSF-matched and fixed-Gaussian polynomial methods remains within 10% of the bound across the tested range of SNR (5–40) and FWHM (2–5.6 pixels), demonstrating robustness and practical optimality.