Dual mesh resampling

The dual of a 2-manifold polygonal mesh without boundary is commonly defined as another mesh with the same topology (genus) but different connectivity (vertex-face incidence), in which faces and vertices occupy complementary locations and the position of each dual vertex is computed as the center of mass (barycenter or centroid) of the vertices that support the corresponding face. This barycenter dual mesh operator is connectivity idempotent but not geometrically idempotent for any choice of vertex positions, other than constants. In this paper we construct a new resampling dual mesh operator that is geometrically idempotent for the largest possible linear subspace of vertex positions. We look at the primal and dual mesh connectivities as irregular sampling spaces, and at the rules to determine dual vertex positions as the result of a resampling process that minimizes signal loss. Our formulation, motivated by the duality of Platonic solids, requires the solution of a simple least-squares problem. We introduce a simple and efficient iterative algorithm closely related to Laplacian smoothing, and with the same computational cost. We also characterize the configurations of vertex positions where signal loss does and does not occur during dual mesh resampling, and the asymptotic behavior of iterative dual mesh resampling in the general case. Finally, we describe the close relation existing with discrete fairing and variational subdivision, and define a new primal-dual interpolatory recursive subdivision scheme.