Slice and BlockwiseWell-Composed Sets

An infinite or closed continuous surface partitions space R² or R³ into two disjoint sub-spaces, an "inside" and an "outside". Notions of voxel set separability describe an analogous partitioning of discrete space Z² or Z³ by a surface voxelisation. Similar concepts, 2D and 3D well-composed sets, define the manifold nature of the boundary between a voxel set and its complement embedded in R² or R³. Cohen-Or and Kaufman define separating sets and present theorems for slicewise construction of 3D separating voxel sets from a group of 2D separating slices. This paper presents similar theorems for 3D well-composed sets. This allows slicewise construction to be applied in a wider range of situations, for example, where the manifold nature of a voxel set boundary is of vital importance or where we are considering solid voxelisa- tions. Theorems for blockwise construction of 2D and 3D well-composed sets from a pair of smaller well- composed sets are also presented, providing further tools for piecewise analysis of voxel sets.