Reconstruction of convex polyominoes from orthogonal projections of their contours

The problem of reconstructing a convex polyominoes from its horizontal and vertical projections when the projections are defined as the number of cells of the polyomino in the different lines and columns was studied by Del Lungo and M. Nivat. In this paper, we study the reconstruction of any convex polyomino when the orthogonal projections are defined as the contour length of the object intercepted by the ray. We prove the NP-hardness of this problem for several classes of polyominoes: general, h-convex, v-convex. For hv-convex polyominoes we give a polynomial time algorithm for the reconstruction problem.