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：我们考虑一类在过去被深入研究的离散断层成像问题：从水平及/或垂直X射线（即连续水平线与垂直线上点的数量）重建凸格点集。HV-凸多联骨的经典重建分两步进行：首先是通过填充操作完成的填充步骤，其次是交换组件的凸聚合步骤。关于凸聚合步骤，我们证明了三点结论：（1）用于重建HV-凸多联骨的凸聚合步骤并非总能给出有效解。这一反例被称为"the bad guy"，并否定了该领域的一个猜想。（2）仅从一个X射线重建数字凸格点集可在多项式时间内完成。我们通过将凸聚合问题编码为有向无环图来证明该结论。（3）运用相同策略，我们证明从水平与垂直X射线重建肥胖数字凸集可在多项式时间内求解。肥胖性是关于数字凸集中左、右、顶、底点相对位置的一种性质。非肥胖格点集的重建复杂度仍是一个未解问题。