A note on tiling under tomographic constraints

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the projections. We are interested in the problem of reconstructing a tiling which has given projections. Some simple variants of this problem, involving tiles that are 1×1 or 1×2 rectangles, have been studied in the past, and were proved to be either solvable in polynomial time or NP-complete. In this note, we make progress toward a comprehensive classification of various tiling reconstruction problems, by proving NP-completeness results for several sets of tiles.