# The maximum a posteriori probability rule for atom column detection from HAADF STEM images

## TL;DR

This paper introduces a Bayesian MAP probability rule for detecting atom columns in HAADF STEM images, demonstrating superior performance over existing criteria and proposing a new image quality measure, ICNR, for better atom detectability assessment.

## Contribution

The paper details the derivation and implementation of the MAP probability rule for atom detection, showing its advantages over AIC and BIC, and introduces ICNR as a new image quality metric.

## Key findings

- MAP rule outperforms AIC and BIC in atom detection accuracy
- Simulation results confirm superior detection performance of the MAP rule
- ICNR correlates better with atom detectability than traditional measures

## Abstract

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.