从一个或两个X射线重建凸集
Reconstruction of Convex Sets from One or Two X-rays
AI总结：
本文研究了通过单个或两个X射线重建凸晶格集的问题，证明了单X射线重建可在多项式时间内完成，并探讨了脂肪凸集的多项式时间重建方法。
摘要：
我们考虑了一类在离散断层成像中深入研究的问题：通过水平和/或垂直X射线重建凸晶格集，即通过连续水平和垂直线序列中点数的数目。HV-凸多边形的重建通常分为两个步骤，首先是填充步骤，包括填充操作，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-凸多边形的凸聚合步骤并不总能提供解决方案。该例子被称为“坏人”例子，推翻了该领域的猜想。(2) 仅通过一个X射线重建数字凸晶格集可在多项式时间内完成。我们通过将凸聚合问题编码为有向无环图来证明这一点。(3) 采用相同策略，证明了通过水平和垂直X射线重建脂肪数字凸集可在多项式时间内完成。脂肪性是数字凸集的属性，涉及集合左右上下点的相对位置。非脂肪晶格集的重建复杂性仍是一个开放问题。
英文摘要：
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.
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