The high beam current and sub-angstrom resolution of aberration-corrected scanning transmission electron microscopes have enabled electron energy loss spectroscopic (EELS) mapping with atomic resolution. These spectral maps are often dose-limited and spatially oversampled, leading to low counts/channel and are thus highly sensitive to errors in background estimation. Even more advanced techniques, such as principal component analysis (PCA) and multivariate curve resolution are affected by variance in the background. However, by taking advantage of redundancy in the dataset map one can often improve background estimation and ultimately the signal-to-noise ratio of the final extracted signal. In particular, this can be accomplished by reducing the variance in power law amplitude and exponent. Here we consider two approaches to maximize signal-to-noise: linear combination of power laws (LCPL) and local background averaging (LBA)[1].

The ubiquitous power law form of the EELS background [2] usually varies only minimally over a sample for atomic resolution maps, where sampling often exceeds five times the Nyquist limit, enabling a better fit to the background by utilizing the oversampled data. In LCPL, preselected exponents provide stability and improve signal to noise (Fig. 1b, d) whereas typical power laws fail for low counts where readout noise errors can introduce negative values (Fig. 1a). Additionally, by locally averaging over a small region before obtaining the power law exponents, as in LBA, the number of counts available for background estimation is effectively increased. Both techniques can be used in tandem with each other and/or with other component analysis approaches. In practice, LBA and LCPL perform comparably to PCA, which can be easily abused by neglecting significant components (Fig. 2). Background extrapolations of any form rely on the implementation of proper estimator techniques. While maximum likelihood estimators perform significantly better for data characterized by Poisson shot noise [3], for low count data where Gaussian readout noise dominates, the optimal estimator converges to the standard linear least squares.