The impact of the point spread function fitting radius on photometric uncertainty based on the Fisher information matrix

TL;DR

This work reframes PSF photometry aperture selection from maximizing signal-to-noise to maximizing information, deriving a CRLB-based criterion for the PSF fitting radius. By formulating the joint Fisher information for flux and background and providing high-resolution closed-form approximations, it shows how the information content evolves with aperture size and demonstrates monotonic growth toward a saturation limit. Numerical experiments reveal that S/N-based aperture choices can yield substantial precision losses (up to ~70–180%) compared with Fisher-information-guided selections, while the CRLB-based approach yields estimates that closely approach the theoretical limits. The results offer a data-driven, flexible framework for aperture design that improves photometric precision in crowded fields and under contamination, with practical implications for precision astronomy tasks such as exoplanet transit photometry and variable-star studies.

Abstract

In point spread function (PSF) photometry, the selection of the fitting aperture radius plays a critical role in determining the precision of flux and background estimations. Traditional methods often rely on maximizing the signal-to-noise ratio (S/N) as a criterion for aperture selection. However, S/N-based approaches do not necessarily provide the optimal precision for joint estimation problems as they do not account for the statistical limits imposed by the Fisher information in the context of the Cramér-Rao lower bound (CRLB). This study aims to establish an alternative criterion for selecting the optimal fitting radius based on Fisher information rather than S/N. Fisher information serves as a fundamental measure of estimation precision, providing theoretical guarantees on the achievable accuracy for parameter estimation. By leveraging Fisher information, we seek to define an aperture selection strategy that minimizes the loss of precision. We conducted a series of numerical experiments that analyze the behavior of Fisher information and estimator performance as a function of the PSF aperture radius. Specifically, we revisited fundamental photometric models and explored the relationship between aperture size and information content. We compared the empirical variance of classical estimators, such as maximum likelihood and stochastic weighted least squares, against the theoretical CRLB derived from the Fisher information matrix. Our results indicate that aperture selection based on the Fisher information provides a more robust framework for achieving optimal estimation precision.

Figures (4)

• Figure 1: Discrepancy analysis of the joint Fisher information as a function of the PSF aperture fitting radius in photometry (blue line, left ordinate on each plot, in %). The discrepancy in the source flux (left column) is defined as $\Delta_{max}\mathcal{I}_F=\frac{ \mathcal{I}_{F}(\mathcal{N})- \mathcal{I}_{F}(\mathcal{J}_{u})}{\mathcal{I}_{F}(\mathcal{N})}$. The discrepancy in the background (right column) is defined as $\Delta_{max}\mathcal{I}_B=\frac{ \mathcal{I}_{B}(\mathcal{N})- \mathcal{I}_{B}(\mathcal{J}_{u})}{\mathcal{I}_{B}(\mathcal{N})}$. The black line (right ordinate on each plot) indicates the corresponding S/N value. The vertical line shows the aperture at which the S/N is maximized. Results are reported for different representative values of $F, B$, and a FWHM of 1.0 arcsec.
• Figure 2: Discrepancy analysis of the joint Fisher information (in %) as a function of the halo coverage in photometry. The left panel is for the source flux; the right panel is for the background. Results are reported for different representative values of $F/ B$.
• Figure 3: Normalized discrepancy (vertical axis, Eq. (\ref{['normEQ']})) as a function of aperture radius and separation between the contaminating star (SBS) and the target (horizontal axis), for two contamination scenarios. The left panel corresponds to a faint contaminant ($F=2000$ [photo-e$^{-}$], $F_K=1000$ [photo-e$^{-}$], $K=1$), while the right panel shows the effect of a brighter one ($F=2000$ [photo-e$^{-}$], $F_K=2500$ [photo-e$^{-}$], $K=1$). Colors indicate the fractional loss of Fisher information relative to the full-pixel case.
• Figure 4: Discrepancy analysis of the estimators as a function of the PSF aperture fitting radius in photometry (orange line, left ordinate on each plot, in %). The discrepancy in the source flux (left column) is defined in Eq. (\ref{['eq:discr_f']}). The discrepancy in the background (right column) is defined in Eq. (\ref{['eq:discr_b']}). The broad black line (right ordinate on each plot) indicates the corresponding S/N value. The vertical black line shows the aperture at which the S/N is maximized. The vertical blue line corresponds to the value of the aperture radius following the Fisher information methodology proposed in Eq. (\ref{['optiV1']}), while the green vertical line corresponds to the value of the aperture radius following the approximated methodology in Eq. (\ref{['optiI']}) (in some cases, the blue and green vertical lines overlap). The achieved discrepancy is observed to be close—but not necessarily equal—to the threshold value of $20\%$ due to the discrete nature of the summation. Specifically, $\Delta_{\mathrm{max}} I_F$ denotes the last value of the normalized discrepancy that remains below the selected threshold $\delta$ as the aperture radius increases, which may not exactly match $\delta$. The choice of $\delta = 20\%$ is arbitrary and is adopted here purely for illustrative purposes. Results are reported for different representative values of $F, B$ and a FWHM of 1.0 arcsec.

Theorems & Definitions (7)

• theorem 1
• lemma 1
• proposition 1
• proposition 2
• theorem 2
• remark 1
• remark 2