The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections

Abstract

Abstract. The number of line-convex directed polyominoes with given horizontal and vertical projections is studied. It is proven that diagonally convex directed polyominoes are uniquely determined by their orthogonal projections. The proof of this result is algorithmical. As a counterpart, we show that ambiguity can be exponential if antidiagonal convexity is assumed about the polyomino. Then, the results are generalised to polyominoes having convexity property along arbitrary lines.

Reconstructing Some hv-Convex Binary Images from Three or Four Projections

“…Furthermore, for certain classes it was shown that only a polynomial number of discrete sets with the same projections can belong to the given class [1,4]. Finally, for some classes it is known that ambiguity in those classes can be exponentially large [2,7,9]. In this paper we are going to investigate the problem of ambiguity in the class of hvconvex discrete sets having decomposable configurations.…”

“…One of the most frequently used techniques to reduce ambiguity is to suppose that the set to be reconstructed belongs to a certain class of discrete sets having some geometrical properties. There are classes of discrete sets where ambiguity is completely eliminated (see [2,7]). Furthermore, for certain classes it was shown that only a polynomial number of discrete sets with the same projections can belong to the given class [1,4].…”