Optimal biorthogonal wavelet decomposition of wire-frame meshes using box splines, and its application to the hierarchical coding of 3-D surfaces

Optimal mechanisms are determined for the hierarchical decomposition of wire-frame surfaces generated by box splines. A family of box splines with compact support, suitable for the approximation of wire-frames is first defined, generated by arbitrary sampling matrices with integer eigenvalues. For each such box spline, the optimal positioning of the wire-frame nodes is determined for each level of the hierarchical wire-frame decomposition. Criterion of optimality is the minimization of the variance of the error difference between the original surface and its representation at each resolution level. This is needed so as to ensure that the wire mesh produces at each resolution as close a replica of the original surface as possible. Several such combinations of box spline generated meshes and the corresponding optimal node lattice sequences are examined in detail with a view to practical application. Their specific application to the hierarchical coding of three-dimensional (3-D) wire meshes is experimentally evaluated.