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 √ 3, dyadic and √ 7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The √ 3-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 √ 3-refinement surface subdivision, and it is also the main reason for the recommendation to use the √ 3-refinement for discrete global grid systems. However, there is little work on compactly supported √ 3-refinement wavelets. In this paper we study the construction of compactly supported orthogonal and biorthogonal √ 3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with 2-fold symmetry and construct the associated orthogonal √ 3-refinement wavelets. We study the 6-fold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of 6-fold symmetric √ 3-refinement filter banks and construct the associated biorthogonal wavelets.