The reconstruction problem is considered in those classes of discrete sets where the reconstruction can be performed from two projections in polynomial time. The reconstruction algorithms and complexity results are summarized in the case of hv-convex sets, hv-convex 8-connected sets, hv-convex polyominoes, and directed h-convex sets. As new results some properties of the feet and spines of the hv-convex 8-connected sets are proven and it is shown that the spine of such a set can be determined from the projections in linear time. Two algorithms are given to reconstruct hv-convex 8-connected sets. Finally, it is shown that the directed h-convex sets are uniquely reconstructible with respect to their row and column sum vectors. ` `