Discrete tomography for inscribable lattice sets

In this paper we deal with uniqueness and reconstruction problems in Discrete Tomography. For a finite set $D$ of directions in $\mathbb{Z}^2$, we introduce the class of $D$-inscribable lattice sets, and give a detailed description of their geometric structure. This shows that such sets can be considered as the natural discrete counterpart of the same notion known in the continuous case, as well as a kind of generalization of the class of the so called $L$-convex polyominoes (or moon polyominoes). In view of reconstruction from projections along the directions in $D$, two related questions of tomographic interest are investigated, namely uniqueness and additivity. We show that both properties are fulfilled by $D$-inscribable lattice sets. Moreover, concerning the case $D=\{e_1,e_2\}$, we provide an explicit reconstruction algorithm from the knowledge of directed horizontal and vertical $X$-rays, jointly with a few preliminary results towards a possible sharp stability inequality.