A discrete projection analogue to Pick’s theorem

We consider a symmetric convex polygon with 2N sides such that all vertices are integer lattice points and the sides are the vectors (pi, qi) and (−pi,−qi) The area A can be expressed in terms of the total number of discrete projections B, for all directions (pi, qi), as A=B/2−N/2 [Display omitted] 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.