Orthogonal and Biorthogonal $\sqrt 3$-Refinement Wavelets for Hexagonal Data Processing

The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.