Theoretical minimum uncertainty of single-molecule localizations using a single-photon avalanche diode array

Abstract

Single-photon avalanche diode (SPAD) arrays can be used for single-molecule localization microscopy (SMLM) because of their high frame rate and lack of readout noise. SPAD arrays have a binary frame output, which means photon arrivals should be described as a binomial process rather than a Poissonian process. Consequentially, the theoretical minimum uncertainty of the localizations is not accurately predicted by the Poissonian Cramér-Rao lower bound (CRLB). Here, we derive a binomial CRLB and benchmark it using simulated and experimental data. We show that if the expected photon count is larger than one for all pixels within one standard deviation of a Gaussian point spread function, the binomial CRLB gives a 46% higher theoretical uncertainty than the Poissonian CRLB. For typical SMLM photon fluxes, where no saturation occurs, the binomial CRLB predicts the same uncertainty as the Poissonian CRLB. Therefore, the binomial CRLB can be used to predict and benchmark localization uncertainty for SMLM with SPAD arrays for all practical emitter intensities.

Fundamental limits to depth imaging with single-photon detector array sensors

“…Further, semi-analytical single pixel models that characterise the SPADs using Poisson or Erlang distributions together with confidence intervals 48 , 49 and incorporate effects such as SPAD dead time 50 , 51 have been presented. However, the Poisson (and by extension Erlang) distributions have been found to diverge from observed SPAD behaviour in both the small and large photon number limits 52 , 53 . Additionally, accurate simulations of SPAD images have been presented in the context of Fluorescence Lifetime Imaging Microscopy (FLIM) and lifetime estimation 53 , 54 .…”

“…However, the Poisson (and by extension Erlang) distributions have been found to diverge from observed SPAD behaviour in both the small and large photon number limits 52 , 53 . Additionally, accurate simulations of SPAD images have been presented in the context of Fluorescence Lifetime Imaging Microscopy (FLIM) and lifetime estimation 53 , 54 .…”

Simulating single-photon detector array sensors for depth imaging

“…Further, semi-analytical single pixel models that characterise the SPADs using Poisson or Erlang distributions together with confidence intervals 48,49 and incorporate effects such as SPAD dead time 50,51 have been presented. However, the Poisson (and by extension Erlang) distributions have been found to diverge from observed SPAD behaviour in both the small and large photon number limits 52,53 . Additionally, accurate simulations of SPAD images have been presented in the context of Fluorescence Lifetime Imaging Microscopy (FLIM) and lifetime estimation 53,54 .…”

“…However, the Poisson (and by extension Erlang) distributions have been found to diverge from observed SPAD behaviour in both the small and large photon number limits 52,53 . Additionally, accurate simulations of SPAD images have been presented in the context of Fluorescence Lifetime Imaging Microscopy (FLIM) and lifetime estimation 53,54 .…”

Quantifying the minimum localization uncertainty of image scanning localization microscopy

“…To calculate the theoretical minimum uncertainty that can be attained with SpinFlux localization, we need a model to describe the amount of photons collected by a camera pixel. Existing models for (me)SMLM ( 5 , 8 , 14 , 26 , 27 ) do not suffice for this, as they do not include a pinhole in the illumination and emission paths. In this subsection, we therefore develop a statistical image formation model for SpinFlux.…”