On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations?
✍ Scribed by T. Yung Kong; Gabor T. Herman
- Book ID
- 101265156
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 220 KB
- Volume
- 9
- Category
- Article
- ISSN
- 0899-9457
No coin nor oath required. For personal study only.
✦ Synopsis
It is a well-known result, due to Ryser, that if a binary
In this article, we consider the question of whether Ryser's picture on the square grid has the same x and y projections as another result has analogs for grids other than the square grid. For a very such picture, then the first picture can be transformed into the second simple example in which the answer is yes, define (still in R 2 ) by a series of switching operations, each of which changes the picture the sets ވ Å {m 1 (01, 0) / n 1 (0.5, 0.75)Ém, n √ Z } and at just four grid points and preserves both projections. In this article, P H Å {(01, 0), (0.5, 0.75)}. Then ,ވ( P H ) is a ''hexagonal'' we show that if a grid [such as a two-dimensional (2D) hexagonal grid. Our definitions of lines, binary pictures and tomographic grid or the 3D cubic grid] has three or more directions of projection, equivalence for this grid are analogous to the definitions of these then Ryser's result has no analog for that grid. Specifically, we show concepts for the square grid. The two grids are in fact isomorphic, that on any grid with three or more directions of projection there and so Ryser's result does indeed carry over to the new grid. We cannot exist any constant L such that every binary picture can be transformed to any other binary picture with the same projections by illustrate this in Figure 1.
a series of projection-preserving changes, each of which involves at Looking at Figure 1, we see immediately that there is a third most L grid points. This is proved for a very general concept of ''grid'' vector which is symmetrical with the two vectors £ H 1 Å (01, 0)
that encompasses virtually all practical grids in Euclidean n-space, and £ H 2 Å (0.5, 0.75) in P H -namely, the vector £ H 3 Å (0.5, and even some grids in higher-dimensional analogs of cylindrical and 0 0.75). Consider the grid ,ވ( {£ H 1 , £ H 2 , £ H 3 }). This hexagonal toroidal surfaces. (In fact, the set of grid points can be any finitely grid has more lines than ( ,ވ P H ) Å ,ވ( {£ H 1 , £ H 2 }); its concept generated Abelian group of rank ¢ 2.)