Connectivity preserving transformations for higher dimensional binary images

An N-dimensional digital binary image (I) is a function I:Z^N→{0,1}. I is connected if and only if its black pixels and white pixels are each (3^N−1)-connected. I is only connected if and only if its black pixels are (3^N−1)-connected. For a 3-D binary image, the respective connectivity models are and . A pair of (3^N−1)-neighboring opposite-valued pixels is called interchangeable in a N-D binary image I, if reversing their values preserves the original connectedness. We call such an interchange to be a (3^N−1)-local interchange. Under the above connectivity models, we show that given two binary images of n pixels/voxels each, we can transform one to the other using a sequence of (3^N−1)-local interchanges. The specific results are as follows. Any two -connected 3-dimensional images I and J each having n black voxels are transformable using a sequence of O((c₁+c₂)n²) 26-local interchanges. Here, c₁ and c₂ are the total number of 8-connected components in all 2-dimensional layers of I and J respectively. We also show bounds on connectivity under a different interchange model as proposed in A. Dumitrescu, J. Pach, Pushing squares around, Graphs and Combinatorics 22 (1) (2006) 37-50]. Next, we show that any two simply connected images under the , connectivity model and each having n black voxels are transformable using a sequence of O(n²) 26-local interchanges. We generalize this result to show that any two , -connected N-dimensional simply connected images each having n black pixels are transformable using a sequence of O(Nn²)(3^N−1)-local interchanges, where N>1.