On the shiftability of dual-tree complex wavelet transforms

The dual-tree complex wavelet transform (DT- WT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT- WT which, among other things, offers a direct explanation for the improvement in the shift-invariance. The representation is based on the shifting action of the group of fractional Hilbert transform (fHT) operators, which extends the notion of arbitrary phase-shifts from sinusoids to finite-energy signals (wavelets in particular). In particular, we characterize the shiftability of the DT- WT in terms of the shifting property of the fHTs. At the heart of the representation are certain fundamental invariances of the fHT group, namely that of translation, dilation, and norm, which play a decisive role in establishing the key properties of the transform. It turns out that these fundamental invariances are exclusive to this group. Next, by introducing a generalization of the Bedrosian theorem fo...

• Type: Journal
• Year: 2010
• Where: TSP
• Authors: Kunal Narayan Chaudhury, Michael Unser
• Keywords: Artificial Intelligence, Hilbert Transform, TSP 2010, Wavelet Transform, Wavelets