Localization accuracy in single molecule microscopy using electron-multiplying charge-coupled device cameras

The electron-multiplying charge-coupled device (EMCCD) is a popular technology for imaging under extremely low light conditions. It has become widely used, for example, in single molecule microscopy experiments where few photons can be detected from the individual molecules of interest. Despite its important role in low light microscopy, however, little has been done in the way of determining how accurately parameters of interest (e.g., location of a single molecule) can be estimated from an image that it produces. Here, we develop the theory for calculating the Fisher information matrix, and hence the Cramer-Rao lower bound-based limit of the accuracy, for estimating parameters from an EMCCD image. An EMCCD operates by amplifying a weak signal that would otherwise be drowned out by the detector's readout noise as in the case of a conventional charge-coupled device (CCD). The signal amplification is a stochastic electron multiplication process, and is modeled here as a geometrically multiplied branching process. In developing our theory, we also introduce a "noise coefficient" which enables the comparison of the Fisher information of different data models via a scalar quantity. This coefficient importantly allows the selection of the best detector (e.g., EMCCD or CCD), based on factors such as the signal level, and regardless of the specific estimation problem at hand. We apply our theory to the problem of localizing a single molecule, and compare the calculated limits of the localization accuracy with the standard deviations of maximum likelihood location estimates obtained from simulated images of a single molecule.