Improving reconstructions in nanotomography for homogeneous materials via mathematical optimization

Compressed sensing is an image reconstruction technique to achieve high-quality results from limited amount of data. In order to achieve this, it utilizes prior knowledge about the samples that shall be reconstructed. Focusing on image reconstruction in nanotomography, this work proposes enhancements by including additional problem-specific knowledge. In more detail, we propose further classes of algebraic inequalities that are added to the compressed sensing model. The first consists in a valid upper bound on the pixel brightness. It only exploits general information about the projections and is thus applicable to a broad range of reconstruction problems. The second class is applicable whenever the sample material is of roughly homogeneous composition. The model favors a constant density and penalizes deviations from it. The resulting mathematical optimization models are algorithmically tractable and can be solved to global optimality by state-of-the-art available implementations of interior point methods. In order to evaluate the novel models, obtained results are compared to existing image reconstruction methods, tested on simulated and experimental data sets. The experimental data comprise one 360{\deg} electron tomography tilt series of a macroporous zeolite particle and one absorption contrast nano X-ray computed tomography (nano-CT) data set of a copper microlattice structure. The enriched models are optimized quickly and show improved reconstruction quality, outperforming the existing models. Promisingly, our approach yields superior reconstruction results, particularly when information about the samples is available for a small number of tilt angles only