Gradient-based and wavelet-based compressed sensing approaches for highly undersampled tomographic datasets

Highlights: • Comparison between TV-based and wavelet-based compressed sensing methods. • Higher-order TV and biorthogonal wavelets outperform the classical TV approach. • These approaches produce high quality spectroscopic reconstructions. • An open-source Python package is provided with both approaches. Electron tomography is widely employed for the 3D morphological characterization at the nanoscale. In recent years, there has been a growing interest in analytical electron tomography (AET) as it is capable of providing 3D information about the elemental composition, chemical bonding and optical/electronic properties of nanomaterials. AET requires advanced reconstruction algorithms as the datasets often consist of a very limited number of projections. Total variation (TV)-based compressed sensing approaches were shown to provide high-quality reconstructions from undersampled datasets, but staircasing artefacts can appear when the assumption about piecewise constancy does not hold. In this paper, we compare higher-order TV and wavelet-based approaches for AET applications and provide an open-source Python toolbox, Pyetomo, containing 2D and 3D implementations of both methods. A highly sampled STEM-HAADF dataset of an Er-doped porous Si sample and a heavily undersampled STEM-EELS dataset of a Ge-rich GeSbTe (GST) thin film annealed at 450°C are used to evaluate the performance of the different approaches. We show that polynomial annihilation with order 3 (HOTV₃) and the Bior4.4 wavelet outperform the classical TV minimization and the related Haar wavelet.