On the Construction of a Generalized Voronoi Inverse of a Rectangular Tessellation

We introduce a new concept of constructing a generalized Voronoi inverse (GVI) of a given tessellation ${\cal T}$ of the plane. Our objective is to place a set $S_i$ of one or more sites in each convex region (cell) $t_i \in {\cal T}$, such that all the edges of ${\cal T}$ coincide with the edges of Voronoi diagram $V(S)$, where $S = \bigcup_i S_i$, and $\forall i, j, i \ne j, S_i\bigcap S_j = \emptyset$. In this paper, we study the properties of GVI for the special case when $\cal T$ is a rectangular tessellation and propose an algorithm that finds a minimal set of sites $S$. We also show that for a general tessellation, a solution of GVI always exists.