On the Construction of Discrete Directional Wavelets: HDWT, Space-Frequency Localization, and Redundancy Factor

We address the issue of constructing directional wavelet bases. After considering orthonormal directional wavelets whose Fourier transforms are indicator functions, we give a construction of directional wavelets with fast decay that is based on an hexagonal filter bank tree. An implementation for squarely sampled images and numerical results are presented. Then we discuss the frequency localization of directional wavelet bases. We analyze the incompatibility between a proper localization and the non-redundancy constraint, and show that the non permissibility condition can be extended to general wavelet bases that are not necessary generated by a filter bank tree. At last, we show that there exist directional wavelet tight frames that are well localized and have a redundancy factor arbitrary close to 1.