Abstract
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 nontrivial 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.
Original language  English 

Article number  101066 
Number of pages  3 
Journal  Graphical Models 
Volume  109 
DOIs  
Publication status  Published  May 2020 
Keywords
 Discrete tomography
 Ghosts
 Lattice polygon
 Mojette transform
 Pick's theorem
Equipment

Centre for Electron Microscopy (MCEM)
Peter Miller (Manager)
Office of the ViceProvost (Research and Research Infrastructure)Facility/equipment: Facility