Abstract text (incl. figure legends and references)
Introduction

Modern STEM instruments now deliver routinely EELS and EDX spectrum-images of huge size. This opens the possibility to apply the methods of Multivariate Statistical Analysis (MSA) for improving the quality and interpretability of results. In particular, Principal Component Analysis (PCA) was routinely applied for denoising data [1]. However, PCA delivers results in the abstract, uninterpretable form. This issue can be addressed by another family of the MSA methods – unmixing algorithms. Unlike PCA, an unmixing approach decomposes the data on a linear combination of endmembers – the spectra with clear physical meaning, corresponding for instance to the real compounds constituting the object.

Objectives

In this work, we develop an unmixing algorithm based on the Vortex Component Analysis (VCA) [2]. The original VCA strategy is not well suited for STEM spectrum-imaging and must be significantly modified. First, the treatment should consider noise always present in STEM data. Second, the VCA results often appear non-unique while the reproducibility is a key requirement for a robust algorithm. We elaborate this issue by the combination of the VCA approach with Bayesian inference and clustering.

Materials & methods

The present unmixing algorithm was tested in a number of synthetic and experimental datasets. The experiments involved atomically resolved layered ceramic nanolayers and nanostructured semiconductor devices investigated by Electron Energy-Loss Spectroscopy (EELS) and Energy Dispersive X-ray Spectroscopy (EDX).

Results

The processing starts with the PCA reduction of data dimensionality down to typically 5-10 dimensions. This pre-treatment facilitates strongly the retrieval of reasonable endmembers while also implying a drastic denoising and compression of data. Analyzing data distribution in such a reduced factor space allows to retrieve the endmembers as shown in exemplary EELS dataset in Fig. 1. Namely, the VCA procedure is repeatably applied providing a set of potential endmembers as suggested by Spiegelberg [3]. Finally, the output is subjected to the statistical treatment, which includes Bayesian inference and clustering. The different modifications of the method, e.g. Bayesian priors and various clustering strategies are discussed.

Fig. 1: Data distribution in the factor space formed by (a) 1st and 2nd or, alternatively (b) 2nd and 3rd PCA components. The positions of the retrieved endmembers are denoted by the colored spots. (c) EEL spectra and abundancies of the extracted endmembers, which allow to identify them with the actual compounds constituting the device: TiN, HfO2, SiO2, Si and TaN.

Conclusion

The suggested method represents STEM spectrum-images as a compact linear mixture of few physically interpretable components, which greatly facilitates collection, storage, interpretation and quantification of huge volumes of spectrum-imaging data.

[1] see for example: M.Watanabe et al, Micr. Anal. (2009) p.6.

[2] J.M.P. Nascimento, J.M.B. Dias, IEEE trans. Geoscience Remote Sensing 43 (2005) p.898.

[3] J. Spiegelberg et al., Ultramicroscopy 182 (2017) 205-211.

The authors appreciate support from ERC (grant 715620 under the Horizon 2020 program) and DFG project 431448015.