Discrete Tomography pp 59-84 | Cite as

# Tomographic Equivalence and Switching Operations

## Abstract

A binary picture on an arbitrary grid is a mapping f from the set of all grid points to {0,1} such that f (x) = 1 for only finitely many grid points x. If two binary pictures f
_{1} and f
_{2} on the same grid have the property that for every grid line P the sets {p ∈ l| f
_{1}(p)= 1} and{ p ∈ l|f
_{2}(p)= 1} contain exactly the same number of grid points, then we say that f
_{1} and f
_{2} are tomographically equivalent. Given a binary picture f on the usual 2-dimensional square grid, there may exist an upright rectangle R (of any size) whose sides are grid lines, such that f = 1 at two diagonally opposite corner points of R and f= 0 at the other two corner points. If so, then we call the process of changing the value of the picture f from 1 to 0 and 0 to 1 at the four corner points of R (without changing the value of f at any other grid point) a rectangular 4-switch. Ryser showed in the 1950s that two binary pictures on the square grid are tomographically equivalent if and only if one picture can be transformed to the other by a finite sequence of rectangular 4-switches. We present a few different versions of this theorem, describe an application, and also give a proof of the result. We then show that the result has no analog on grids that have grid lines in three or more directions (such as the 3-dimensional cubic grid),because on such grids one can find for every integer L two tomographically equivalent binary pictures that differ at more than L grid points and are not tomographically equivalent to any other binary picture.

## Keywords

Grid Point Induction Hypothesis Corner Point Grid Line Switching Operation## Preview

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