Connectivity preserving transformations for higher dimensional binary images

Abstract

An N -dimensional digital binary image ( I ) is a function I : Z N → { 0 , 1 } . I is B 3 N − 1 , W 3 N − 1 connected if and only if its black pixels and white pixels are each ( 3 N − 1 )-connected. I is only B 3 N − 1 connected if and only if its black pixels are ( 3 N − 1 )-connected. For a 3-D binary image, the respective connectivity models are B 26 , W 26 and B 26 . 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 B 26 -connected 3-dimensional images I and J each having n black voxels are transformable using a sequence of O ( ( c 1 + c 2 ) n 2 ) 26-local interchanges. Here, c 1 and c 2 are the total number of 8-connected components in all 2-dimensional layers of I and J respectively. We also show bounds on B 26 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 B 26 , W 26 connectivity model and each having n black voxels are transformable using a sequence of O ( n 2 ) 26-local interchanges. We generalize this result to show that any two B 3 N − 1 , W 3 N − 1 -connected N -dimensional simply connected images each having n black pixels are transformable using a sequence of O ( N n 2 ) ( 3 N − 1 ) -local interchanges, where N > 1 .