A remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

最近引起离散几何界注意的一个显着的离散集族是凸多米诺族，它们是欧几里得凸集的离散对等体，并结合了凸性和连通性的约束。在本文中，我们依靠Barcucci等人定义的方法研究从正交投影重建它们的问题。（Theor Comput Sci 155（2）：321-347，1996）。特别是，在重建过程中，可能有必要通过在轮廓的特定位置添加点来扩展多米诺骨内部的凸子集（例如多米诺核），而不会失去其凸度。为了达到这个目标，我们根据编码多米诺骨牌的边界词的某些组合特性来考虑凸度。所以，我们首先显示一些条件，这些条件使我们能够扩展保持凸性的核。然后，我们提供了一个示例，其中添加一个或两个点会导致凸度损失，可以通过添加其他点（无法先验确定其数量和位置）来恢复凸度。