Topology preservation of template-based 6-subiteration reduction operations

Generally, a reduction operation (e.g., thinning and shrinking) on 3D binary images can be represented as a set of reduction templates where every object voxel of the image satisfying any template is turned to a background voxel. Generally, it is rather difficult, error-prone and time-consuming for verifying the topological soundness of a 3D parallel reduction operation. This paper proposes sufficient conditions of time complexity O(n) for verifying the topological soundness of 3D parallel 6-subiteration reduction operations of n templates where such a kind of 3D reduction operations is performed alternatively from the six orthogonal directions to turn object voxels to background voxels. By such sufficient conditions, the topology soundness of a 3D 6-subiteration parallel reduction operation can be verified by checking each and every of its templates.