On the Shiftability of Dual-Tree Complex Wavelet Transforms

TL;DR

This paper introduces an amplitude-phase representation of the dual-tree complex wavelet transform (DT-CWT) that explains its enhanced shift-invariance through fractional Hilbert transform properties, extending to multi-dimensional directional analysis.

Contribution

It provides a novel amplitude-phase framework for DT-CWT based on fractional Hilbert transforms, linking shiftability to fundamental invariances and extending the analysis to directional multi-dimensional wavelets.

Findings

The representation explains the shift-invariance of DT-CWT.

The fractional Hilbert transform characterizes wavelet shiftability.

Directional extension improves understanding of shift-invariance along specific directions.

Abstract

The dual-tree complex wavelet transform (DT-CWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-CWT 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-CWT 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 for the fHT operator, we derive an explicitly understanding of the shifting action of the fHT for the particular family of wavelets obtained through the modulation of lowpass functions (e.g., the Shannon and Gabor wavelet). This, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. Finally, we extend these ideas to the multi-dimensional setting by introducing a directional extension of the fHT, the fractional directional Hilbert transform. In particular, we derive a signal representation involving the superposition of direction-selective wavelets with appropriate phase-shifts, which helps explain the improved shift-invariance of the transform along certain preferential directions.