ON TOPOLOGY PRESERVATION IN 2-D AND 3-D THINNING

Abstract

An (m, n)-simple 1 in a binary image I has the property that its deletion “preserves topology” when m-adjacency is used on the 1’s and n-adjacency on the 0’s of I. This paper presents new, easily visualized, necessary and sufficient conditions for a 1 in I to be (m, n)-simple, for (m, n)=(26, 6), (18, 6), (6, 26) or (6, 18) when I is a 3-d image and (m, n)=(8, 4) or (4, 8) when I is a 2-d image. Systematic and fairly general methods of verifying that a given parallel thinning algorithm always preserves topology are described, for the cases where 8-/26-adjacency is used on the 1’s and 4-/6-adjacency on the 0’s, or vice versa. The verification methods for 2-d algorithms are mainly due to Ronse and Hall; the methods for 3-d algorithms were found by Ma and Kong. New proofs are given of the correctness of these verification methods, using the characterizations of simple 1’s presented in this paper.