Directional Hypercomplex Wavelets for Multidimensional Signal Analysis and Processing
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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 hypercomplex wavelet transform (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.
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W. L. Chan, H. Choi and R. G. Baraniuk, "Directional Hypercomplex Wavelets for Multidimensional Signal Analysis and Processing," vol. 3, 2004.