  In this paper, we propose a new two-dimensional directional discrete wavelet
transform that can decompose an image into 12 multiscale directional edge
components. The proposed transform is designed in a fully discrete setting and
thus is easy to implement in actual computations. The proposed transform is
viewed as a category of redundant discrete wavelet transforms implemented by
fast in-place computational algorithms by a lifting scheme that has been
modified to incorporate redundancy. The redundancy is limited to $(N \times
J+1)/4$, where $N=12$ is the directional selectivity and $J$ is a decomposition
level of the multiscale transform. Numerical experiments in edge detection
using various images demonstrate the advantages of the proposed method over
some conventional standard methods. The proposed method outperforms several
conventional edge detection methods in identifying both the location and
orientation of edges, and well captures the directional and geometrical
features of images.
