Pick’s theorem expresses the area of a polygon on a grid in terms of the number of boundary and interior integer lattice points. Here, we present an analogous theorem for the area of a symmetric, convex polygon in terms of the number of polygon edges and total projection bins. These polygons arise naturally through discrete projection ghosts. Ghosts are geometric objects that define locations in discrete tomographic systems which are not uniquely determinable. In this work, we show that the area A of a ghost’s convex hull is related to the number of non-trivial discrete projection bins B over the ghost image for any set of N 2D discrete projections by $A=B/2-N/2$. The ratio B/A has a strong upper bound of exactly 2. This relation is analogous to Pick’s theorem for polygons with lattice point vertices.

皮克定理以边界和内部整数晶格点的数量表示网格上多边形的面积。在这里，我们根据多边形边的数量和总投影箱数，为对称，凸多边形的面积提供了一个类似的定理。这些多边形通过离散的投影重影自然产生。鬼影是几何对象，它们定义了无法唯一确定的离散层析系统中的位置。在这项工作中，我们表明，对于任何N 2D离散投影集，幻影凸壳的面积A与幻影图像上非平凡的离散投影仓B的数量有关，即$\mathrm{一种}=乙/2-ñ/2$。比值B / A具有正好为2的强上限。此关系类似于具有格点顶点的多边形的匹克定理。