A discrete projection analogue to Pick's theorem

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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 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.

Original languageEnglish
Article number101066
Number of pages3
JournalGraphical Models
Volume109
DOIs
Publication statusPublished - May 2020

Keywords

  • Discrete tomography
  • Ghosts
  • Lattice polygon
  • Mojette transform
  • Pick's theorem

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