Three-Dimensional Maximal and Boundary Ghosts

Abstract

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.