  Recently, the maximum a posteriori (MAP) probability rule has been proposed
as an objective and quantitative method to detect atom columns and even single
atoms from high-resolution high-angle annular dark-field (HAADF) scanning
transmission electron microscopy (STEM) images. The method combines statistical
parameter estimation and model-order selection using a Bayesian framework and
has been shown to be especially useful for the analysis of the structure of
beam-sensitive nanomaterials. In order to avoid beam damage, images of such
materials are usually acquired using a limited incoming electron dose resulting
in a low contrast-to-noise ratio (CNR) which makes visual inspection
unreliable. This creates a need for an objective and quantitative approach. The
present paper describes the methodology of the MAP probability rule, gives its
step-by-step derivation and discusses its algorithmic implementation for atom
column detection. In addition, simulation results are presented showing that
the performance of the MAP probability rule to detect the correct number of
atomic columns from HAADF STEM images is superior to that of other model-order
selection criteria, including the Akaike Information Criterion (AIC) and the
Bayesian Information Criterion (BIC). Moreover, the MAP probability rule is
used as a tool to evaluate the relation between STEM image quality measures and
atom detectability resulting in the introduction of the so-called integrated
CNR (ICNR) as a new image quality measure that better correlates with atom
detectability than conventional measures such as signal-to-noise ratio (SNR)
and CNR.
