Minimal non-deletable sets and minimal non-codeletable sets in binary images

The concepts of strongly 8-deletable and strongly 4-deletable sets of 1s in binary images on the 2D Cartesian grid were introduced by Ronse in the mid-1980s to formalize the connectivity preservation conditions that parallel thinning algorithms are required to satisfy. In this paper we call these sets deletable and codeletable, respectively. To establish that a proposed parallel thinning algorithm for binary images on the 2D Cartesian grid preserves 8-(4-)connected foreground components and 4-(8-)connected background components, it is enough to prove that the set of 1s which are changed to 0s at each pass of the algorithm is always a deletable (codeletable) set. Ronse established results that are very useful in this context for proving that a finite set D of 1s is deletable or codeletable. In particular, he showed that D and its proper subsets are all codeletable in a binary image if each singleton and each pair of 8-adjacent pixels in D is codeletable. He further showed that D and its proper subsets are all deletable in a binary image if (1) each singleton and each pair of 4-adjacent pixels in D is deletable, and (2) no set of 2, 3, or 4 pairwise 8-adjacent pixels that is an 8-connected foreground component of the image is entirely contained in D. In the 1990s and early 2000s analogous results were obtained by Hall, Ma, Gau, and the author for binary images on the 2D hexagonal grid, the 3D Cartesian and face-centered cubic grids, and the 4D Cartesian grid. This paper extends the above-mentioned work to binary images on almost any polytopal complex whose union is n-dimensional Euclidean space, for n ≤ 4. Our main results generalize and unify the corresponding results of the earlier work. c © 2008 Elsevier B.V. All rights reserved.