On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations?

It is a well‐known result, due to Ryser, that if a binary picture on the square grid has the same x and y projections as another such picture, then the first picture can be transformed into the second by a series of switching operations, each of which changes the picture at just four grid points and preserves both projections. In this article, we show that if a grid [such as a two‐dimensional (2D) hexagonal grid or the 3D cubic grid] has three or more directions of projection, then Ryser's result has no analog for that grid. Specifically, we show that on any grid with three or more directions of projection there cannot exist any constant L such that every binary picture can be transformed to any other binary picture with the same projections by a series of projection‐preserving changes, each of which involves at most L grid points. This is proved for a very general concept of “grid” that encompasses virtually all practical grids in Euclidean n‐space, and even some grids in higher‐dimensional analogs of cylindrical and toroidal surfaces. (In fact, the set of grid points can be any finitely generated Abelian group of rank ≥ 2.) © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 118–125, 1998