Generation of Three Dimensional Structuring Elements Over 3×3×3 Rectangular Grid

Three dimensional (3-D) structuring elements plays a vital role in the processing of three dimensional volumetric images such as 3-D medical images. These structuring elements are used in 3-D mathematical morphological operations such as erosion, dilation etc. There is no algebraic framework as such to systematically generate the 3-D structuring elements. The work discussed in this paper is systematically generating the 3-D structuring elements by using geometric filters (G-Filters).
This paper initially gives a novel concept of what we call as geometric filters (G-Filters) defined over a 3-D rectangular grid of pixels. By using these G-Filters, 256 convex Polyhedrons are generated. The 256 different structuring elements could be created by using the 256 convex polyhedrons. These G-Filters have potential applications to processing of volumetric images. In addition, the paper describes in brief the algebra of G-Filters by formulating a lattice of convex polyhedrons constructed in a 3×3×3 grid of pixels.
This paper proposes a new algorithm for automatic generation of 255 convex polyhedrons from its initial set named A containing all eight corners of the 3×3×3 rectangular grid. This paper also proposes an algorithm for hierarchal path enumeration used in visualizing the relationships between convex polyhedron sets and their corresponding subsets of other convex polyhedrons. Finally, an algorithm is proposed to generate 3D structuring element from a selected convex polyhedron.