Instrument-limited pixel-level SNR bounds from optical throughput

The radiometric integral is the fundamental radiance--to--flux relation in imaging, whereas \'etendue is typically used as a compact system-level descriptor. For quantitative imaging and calibration, however, the operative mapping must be explicit at the level of individual detector pixels, including pixel acceptance and field-dependent pupil visibility. This work packages the pixel-restricted radiometric integral into a reusable geometric throughput factor by defining a per-pixel optogeometric (optical-throughput) factor $F_{\mathrm{opg},i}$ (units \si{m^2.sr}) such that, under weak radiance variation, $\Phi_i \approx L_i\,F_{\mathrm{opg},i}$. Making throughput explicit at the pixel scale yields an optics-delivered photon budget in which the incident photon count at the detector, $N_{\mathrm{inc},i}$ (before quantum efficiency), scales linearly with geometry: $N_{\mathrm{inc},i}\propto F_{\mathrm{opg},i}$ for a given scene radiance distribution and fixed acquisition settings (bandwidth, integration time, and optical transmission). The corresponding optics-delivered (pre-detection) shot-noise ceiling is set by the incident photon count $N_{\mathrm{inc},i}$, with $\mathrm{SNR}_{\mathrm{inc},i}\le \sqrt{N_{\mathrm{inc},i}}\propto \sqrt{F_{\mathrm{opg},i}}$, while in photoelectron units one has $\mathrm{SNR}_i \le \sqrt{N_{\mathrm{ph},i}}=\sqrt{\eta(\bar\nu)\,N_{\mathrm{inc},i}}\propto \sqrt{F_{\mathrm{opg},i}}$, where $N_{\mathrm{ph},i}$ is the detected photoelectron count and $\eta(\bar\nu)$ is the (narrowband) quantum efficiency; additional detector/electronics noise sources (e.g.\ dark current and read noise) can only reduce the achieved SNR below these shot-noise limits.