In discrete tomography, ghosts represent indeterminate locations of a reconstruction when there is insufficient projection information to admit a unique solution. Our previous work presented maximal ghosts, which are tilings of \(2^N\) connected points of \(\pm 1\) values with zero line sums over N directions. These directions are given by the recursion \(v_{n+1}=v_n + 2\epsilon _{n+1}v_{n-1}\) with \(\epsilon _{n+1} \in \{-1,1\}\). By including one additional direction, interior points are cancelled leaving only a thin boundary of ghost errors. Here, we show that a simple modification to this recursion is not possible to generate boundary ghosts in three dimensions. Rather, we present a combination of three different recurrences to achieve this goal. We derive results pertaining to the connectivity, size and structure of these shapes.

在离散断层扫描中，当没有足够的投影信息来接受唯一的解决方案时，重影表示重建的不确定位置。我们之前的工作提出了最大重影，它们是\(\pm 1\)值的\(2^N\)连接点的平铺，在N 个方向上具有零线总和。这些方向由递归\(v_{n+1}=v_n + 2\epsilon _{n+1}v_{n-1}\)与\(\epsilon _{n+1} \in \{ -1,1\}\). 通过包括一个额外的方向，内部点被取消，只留下一个薄薄的重影错误边界。在这里，我们表明对这种递归的简单修改是不可能在三个维度上生成边界重影的。相反，我们提出了三种不同重复的组合来实现这一目标。我们推导出与这些形状的连通性、大小和结构有关的结果。