Digitization of non-regular shapes in arbitrary dimensions

The preservation of topological properties during digitization is a hard problem in 3 and higher dimensions. Only for the very restricted class of r-regular shapes it is known that the connectivity and inclusion properties of shape components do not change.In a previous paper it was shown for the 2D case, how a much wider class of shapes, for which the morphological open–close and the close–open-operator with an r-disc lead to the same result, can be digitized correctly in this sense by using an additional repairing step.This paper extends this to the arbitrary dimensions and analyses the difficulties which occur in 3 or higher dimensional spaces.The repairing step is easy to compute, parallelizable and does not change as much hyper-voxels as a preprocessing regularization step. The results are applicable for arbitrary, even irregular, sampling grids in arbitrary dimensions.