# Nonnegative matrix factorization incorporating domain specific constraints for four dimensional scanning transmission electron microscopy

## TL;DR

This paper introduces a new nonnegative matrix factorization method for electron microscopy data that incorporates domain-specific constraints to improve material analysis.

## Contribution

A novel constrained NMF technique for 4D STEM that integrates domain-specific knowledge to enhance decomposition and classification.

## Key findings

- The constrained NMF successfully decomposed simulated and experimental 4D STEM data into interpretable diffractions and maps.
- Nanometer-sized crystalline precipitates in ZrCuAl were detected and classified using the proposed method.
- The method outperforms PCA and primitive NMF by incorporating domain-specific constraints and reducing artifacts.

## Abstract

Modern electron microscopy enables the acquisition of extremely large datasets, necessitating optimized machine learning techniques, such as dimensionality reduction and clustering, to extract material insights. We propose a novel nonnegative matrix factorization (NMF) technique that integrates domain-specific constraints inherent to electron microscopy, including spatial resolution and continuous intensity features without downward-convex peaks. This constrained NMF was applied to four-dimensional (4D) scanning transmission electron microscopy (STEM). Using the constrained NMF, both simulated and actual experimental data were successfully decomposed into interpretable diffractions and maps that cannot be achieved using principal component analysis (PCA) and primitive NMF methods. Additionally, hierarchical clustering was optimized based on diffraction similarity, which is a combination of a polar coordinate transformation and uniaxial cross-correlation. Then, nanometer-sized crystalline precipitates embedded in an amorphous metallic glass, ZrCuAl, were successfully detected and classified according to their diffraction patterns. The present scheme is broadly applicable across various characterization techniques, including hyperspectral imaging, and effectively mitigates the known artifacts found in conventional machine learning techniques that rely solely on mathematical constraints without domain-specific knowledge.

The online version contains supplementary material available at 10.1038/s41598-025-23541-7.