Dual-Tree Tight Frame Wavelets with Symmetric Envelope

Wavelets based on Hilbert pairs have appealing properties when applied to image denoising and feature detection due to their directional sensitivity. In this paper we propose dual-tree 4-channel tight frame wavelets and scaling functions {phi
_h
, psi
_h1
, psi
_h2
, psi
_h3
} and {phi
_g
, psi
_g1
, psi
_g2
, psi
_g3
} based on FIR filterbanks of four filters, and downsampling by 2. Such wavelets closely approximate shift invariance, and the resulting complex wavelets lead to exactly symmetric envelope. The filters enjoy vanishing moments property.