Theoretical Minimum Uncertainty of Single-Molecule Localizations Using a Single-Photon Avalanche Diode Array

Abstract

Optical microscopes are fundamentally limited to a resolution of several hundreds of nanometers by the diffraction of light. Single-molecule localization microscopy (SMLM) circumvents this limit by sparsely exciting fluorescent molecules at different time instances. Single molecules can subsequently be localized with improved precision over the diffraction limit. Due to their high frame rate, single-photon avalanche diode (SPAD) arrays are imagers that can be used for SMLM. Most SPADs have to recharge after each photon arrival. During this recharging period, the SPAD is insensitive to more photon arrivals. As a result, SPAD arrays will measure zero or one photons for each pixel in each frame, whereas scientific complementary metal-oxide semiconductor (sCMOS) imagers and electron multiplying charge-coupled devices (EMCCD) have a discrete frame output. Here, we describe the photon arrivals in the image formation model of the SPAD array as a binomial process rather than as a Poissonian process. In addition, we quantify the minimum theoretical uncertainty of single-molecule localizations using a binomial Cramér-Rao lower bound and benchmark it with 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. Without saturation, which is the case for most SMLM applications, the binomial CRLB model gives 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.