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

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 highquality 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 450C 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.