An unbiased ADMM-TGV algorithm for the deconvolution of STEM-EELS maps

Electron-energy-loss-spectroscopy (EELS) spectra in the scanning transmission electron microscope (STEM) are affected by various types of noise. Additionally, they are convolved with the detector point spread function and the energy distribution of the electron source. Often, iterative deconvolution is employed to sharpen peaks and improve the data. However, since the Richardson–Lucy algorithm (RLA) has become the standard deconvolution algorithm in EELS, little progress has been made in terms of technique. In this paper, the authors aim to provide an update to STEM-EELS deconvolution and demonstrate how to significantly improve results compared to those achievable with the RLA. The major limitation of the RLA is that it does not guarantee convergence. Furthermore, the RLA is restricted to pure Poisson noise and lacks adaptability due to limitations in its general structure, particularly when compared to more modern algorithms. A new and versatile approach is the Alternating Direction Method of Multipliers (ADMM), which is based on Lagrangian methods and enables to overcome these restrictions. The generality of ADMM allows us to develop a deconvolution algorithm tailored to EELS maps and incorporate a recent noise model. We extend the standard Bayesian maximum likelihood of the RLA to a maximum a-posteriori approach in ADMM, which enables us to leverage the principles of total general variation (TGV) to enforce convergence. Furthermore, we define the algorithm such that it operates unbiased of the user. To demonstrate the superiority of the ADMM, it is tested against the RLA using simulated data. Eventually, our algorithm is successfully applied to experimental data as well.