Sufficient Conditions for Topology-Preserving Parallel Reductions on the BCC Grid

Abstract

Parallel reductions transform binary pictures only by changing a set of black points to white ones simultaneously. Topology preservation is a major concern of some topological algorithms composed of parallel reductions. For 3D binary pictures sampled on the body-centered cubic (BCC) grid, we propose a new sufficient condition for topology-preserving parallel reductions. This condition takes some configurations of deleted points into consideration, and it provides a method of verifying that formerly constructed parallel reductions preserve the topology. We present two further sufficient conditions that investigate individual points, directly provide deletion rules of topology-preserving parallel reductions, and allow us to construct parallel thinning algorithms.