Instrument-limited pixel-level SNR bounds from optical throughput

TL;DR

This paper introduces a pixel-level optical throughput factor to explicitly relate scene radiance, photon budget, and SNR in imaging systems, providing fundamental bounds on image quality based on optical geometry.

Contribution

It defines a per-pixel optogeometric throughput factor that links scene radiance to photon counts and SNR, enabling precise, system-level SNR bounds at the pixel scale.

Findings

Photon count scales linearly with pixel throughput

SNR is limited by shot noise proportional to the square root of photon count

Additional noise sources reduce SNR below shot-noise limits

Abstract

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.