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Compressed sensing is an image reconstruction technique to achievehigh-quality results from limited amount of data. In order to achieve this, itutilizes prior knowledge about the samples that shall be reconstructed.Focusing on image reconstruction in nanotomography, this work proposesenhancements by including additional problem-specific knowledge. In moredetail, we propose further classes of algebraic inequalities that are added tothe compressed sensing model. The first consists in a valid upper bound on thepixel brightness. It only exploits general information about the projectionsand is thus applicable to a broad range of reconstruction problems. The secondclass is applicable whenever the sample material is of roughly homogeneouscomposition. The model favors a constant density and penalizes deviations fromit. The resulting mathematical optimization models are algorithmicallytractable and can be solved to global optimality by state-of-the-art availableimplementations of interior point methods. In order to evaluate the novelmodels, obtained results are compared to existing image reconstruction methods,tested on simulated and experimental data sets. The experimental data compriseone 360{\deg} electron tomography tilt series of a macroporous zeolite particleand one absorption contrast nano X-ray computed tomography (nano-CT) data setof a copper microlattice structure. The enriched models are optimized quicklyand show improved reconstruction quality, outperforming the existing models.Promisingly, our approach yields superior reconstruction results, particularlywhen information about the samples is available for a small number of tiltangles only

Improving reconstructions in nanotomography for homogeneous materials via mathematical optimization