Directional Hypercomplex Wavelets for Multidimensional Signal Analysis and Processing
Chan, Wai Lam; Choi, Hyeokho; Baraniuk, Richard G.
We extend the wavelet transform to handle multidimensional signals that are smooth save for singularities along lower-dimensional manifolds. We first generalize the complex wavelet transform to higher dimensions using a multidimensional Hilbert transform. Then, using the resulting <i>hypercomplex wavelet transform</i> (HWT) as a building block, we construct new classes of nearly shift-invariant wavelet frames that are oriented along lower-dimensional subspaces. The HWT can be computed efficiently using a 1-D dual-tree complex wavelet transform along each signal axis. We demonstrate how the HWT can be used for fast line detection in 3-D.