Compactly Supported Orthogonal and Biorthogonal $\sqrt 5$ -Refinement Wavelets With 4-Fold Symmetry

Recently √ 5-refinement hierarchical sampling has been studied and √ 5-refinement has been used for surface subdivision. Compared with other refinements such as the dyadic or quincunx refinement, √ 5-refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates. This special property has been shown to be very useful to represent adaptively and render complex and procedural geometry. When √ 5-refinement is used for multiresolution data processing, √ 5- refinement filter banks and wavelets are required. While the construction of 2-dimensional nonseparable (bi)orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers, the construction of (bi)orthogonal wavelets with √ 5-refinement has not been investigated. The main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with √ 5-refinement. In this paper we obtain block structures of orthogonal and biorthogonal √ 5- refinement FIR filter banks with 4-fold rotational symmetry. We construct compactly supported orthogonal and biorthogonal wavelets based on these block structures.