Highly Symmetric √ 3-refinement Bi-frames for Surface Multiresolution Processing

Multiresolution techniques for (mesh-based) surface processing have been developed and successfully used in surface progressive transmission, compression and other applications. A triangular mesh allows √ 3, dyadic and √ 7 refinements. The √ 3-refinement is the most appealing one for multiresolution data processing since it has the slowest progression through scale and provides more resolution levels within a limited capacity. The √ 3 refinement has been used for surface subdivision and for discrete global grid systems Recently lifting scheme-based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets (with either dyadic or √ 3 refinement) have certain smoothness, they will have big supports. In other words, the corresponding multiscale algorithms have large templates; and this is undesirable for surface processing. On the other hand, frames provide a flexibility for the construction of system generators (called framelets) with high symmetry and smaller supports. In this paper we study highly symmetric √ 3-refinement wavelet bi-frames for surface processing. We design the frame algorithms based on the vanishing moments and smoothness of the framelets. The frame algorithms obtained in this paper are given by templates so that one can easily implement them. We also present interpolatory √ 3 subdivision-based frame algorithms. In addition, we provide frame ternary multiresolution algorithms for boundary vertices on an open surface.