We consider the problem of generating a map between two triangulated meshes, M and M', with arbitrary and possibly differing genus. This problem has rarely been tackled in its generality. Early schemes considered only topological spheres. Recent algorithms allow inputs with an arbitrary number of tunnels but require M and M' to have equal genus, mapping tunnel to tunnel. Other schemes which allow more general inputs are not guaranteed to work and the authors do not provide a characterization of the input meshes that can be processed successfully. Moreover, the techniques have difficulty dealing with coarse meshes with many tunnels.
In this paper we present the first robust approach to build a map between two meshes of arbitrary unequal genus. We also provide a simplified method for setting the initial alignment between M and M', reducing reliance on landmarks and allowing the user to select "landmark tunnels" in addition to the standard landmark vertices. After computing the map, we automatically derive a continuous deformation from M to M' using a variational implicit approach to describe the evolution of non-landmark tunnels. Overall, we achieve a cross parameterization scheme that is provably robust in the sense that it can map M to M' without constraints on their relative genus or on the density of the triangulation with respect to the number of tunnels. To demonstrate the practical effectiveness of our scheme we provide a number of examples of inter-surface parameterizations between meshes of different genus and shape.