Representation and reconstruction of polygons and polyhedra using Hierarchical Extended Gaussian Images

In this paper, two new representation schemes called the Branched Gaussian Image (BGI) and the Hierarchical Extended Gaussian Image (HEGI) are proposed. They can be used for uniquely representing both concave and convex, two-dimensional figures and three-dimensional objects. The representation by the Branched Gaussian Image involves several Gaussian spheres connected by branches where the traversing directions might be different. Examples showing how the use of the BGI enables us to achieve the mapping and inverse mapping of concave polygons and polyhedra uniquely are presented. The Gaussian spheres in the BGI can be organized into a tree-like hierarchical structure known as the Hierarchical Extended Gaussian Image. A general procedure for mapping a polygon or polyhedron to its HEGI representation is described. An approach for reconstructing concave polygons from Branched Gaussian Images is presented. The reconstruction of a concave polyhedron from its Hierarchical Extended Gaussian Images by using an iterative method or a closed-form method for each convex component is also described.