We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.

翻译：本文研究离散断层成像中一类已被深入探讨的问题：从水平方向和/或垂直方向X射线（即从连续水平线与垂直线上点的数量）重建凸晶格集。HV-凸多联骨牌的重建通常分两步进行：首先是通过填充操作进行填充步骤，其次是切换组件的凸聚合。我们证明了关于凸聚合步骤的三个结果：（1）用于重建HV-凸多联骨牌的凸聚合步骤并不总能给出解。产生该结果的实例被称为"坏家伙"，并否定了该领域的一个猜想。（2）仅从一个X射线重建数字凸晶格集可在多项式时间内完成。我们通过将有向无环图编码凸聚合问题来证明这一点。（3）采用相同策略，我们证明了从水平与垂直X射线重建胖数字凸集可在多项式时间内求解。胖度是数字凸集的一种性质，涉及集合左、右、上、下点的相对位置。非胖晶格集的重建复杂性仍是未解问题。