Biorthogonal Wavelets with 6-fold Axial Symmetry for Hexagonal Data and Triangle Surface Multiresolution Processing

This paper studies the construction of highly symmetric compactly supported wavelets for hexagonal data/image and triangle surface multiresolution processing. Recently hexagonal image processing has attracted attention. Compared with the conventional square lattice, the hexagonal lattice has several advantages, including that it has higher symmetry. It is desirable that the filter banks for hexagonal data also have high symmetry which is pertinent to the symmetric structure of the hexagonal lattice. The high symmetry of filter banks and wavelets not only leads to simpler algorithms and efficient computations, it also has the potential application for the texture segmentation of hexagonal data. While in the field of CAGD, when the filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms for regular vertices have high symmetry which makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. In this paper we study the construction of 6-fold symmetric biorthogonal filter banks and the associated wavelets, with both the dyadic and √ 3 refinements. The constructed filter banks have the desirable symmetry for hexagonal data processing, and they result in multiresolution algorithms with the required symmetry for triangle surface processing. The constructed filter banks also result in very simple multiresolution decomposition and reconstruction algorithms which include the algorithms for regular vertices proposed in [3, 40].