Connectivity preserving transformations for higher dimensional binary images

An N -dimensional digital binary image (I) is a function I : Z → {0, 1}. I is B3N−1,W3N−1 connected if and only if its black pixels and white pixels are each (3 − 1)-connected. I is only B3N−1 connected if and only if its black pixels are (3 − 1)-connected. For a 3-D binary image, the respective connectivity models are B26,W26 and B26. 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 − 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 − 1)local interchanges. The specific results are as follows. Any two B26-connected 3dimensional images I and J each having n black voxels are transformable using a sequence of O((c1 + c2)n 2) 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 B26 connectivity under a different interchange model as proposed in [3]. Next, we show that any two simply connected images under the B26,W26 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 B3N−1,W3N−1-connected N -dimensional simply connected images each having n black pixels are transformable using a sequence of O(Nn2) (3 − 1)-local interchanges, where N > 1.