Complex Wavelets and the Hilbert Transform 
Investigators: Kunal Narayan Chaudhury 

Summary: The main goals of this project are: (1) investigation of novel methods of constructing exact Hilbert transform (HT) pairs of wavelet bases using the formalism of Bsplines; (2) characterization of the dualtree complex wavelet transform using the formalism of fractional Hilbert transforms. 

Fourier analysis provides an efficient way of encoding the relative location of information in signals through the phase function, which has a very straightforward interpretation. As far as signals with isolated singularities (e.g., piecewisesmooth signals) are concerned, wavelet representations employing dilatedtranslated copies of a fastdecaying oscillating waveform for signal analysis have proven to be more efficient. Complex wavelets, derived from the combination of nonredundant wavelet bases, provide an attractive means of restoring the phase information, where the the phase relation between the components (of the complex wavelet) is used to encode the relative signal "displacements" (besides offering robustness to interference). The dualtree complexwavelet transform is a particular instance where the components bear a specific HT correspondence.
Kingsbury introduced the dualtree complexwavelet transform and demonstrated its use in a number of applications. There is now good evidence that these complex wavelets exhibit better shift invariance than the traditional wavelet bases and that they tend to perform better in a variety of tasks such as denoising, texture analysis, and deconvolution. Selesnick made the crucial observation that the dualtree wavelets form an approximate Hilbert transform (HT) pair. He also derived a phase relation between the scaling filters that ensures exact HT pair correspondence while noticing that this correspondence cannot be achieved exactly using finiteimpulse filterbanks. The advantage of viewing these wavelets as a HT pair is that we can make a direct connection with the analytical signal formalism which is a powerful tool for AMFM analysis. 

We proposed a novel method for constructing exact HT pairs of wavelet bases using Bsplines, and formulated necessary and sufficient conditions for generating such wavelet pairs. In particular, we constructed HT pairs of biorthogonal wavelet bases using welllocalized scaling functions having identical Riesz bounds. Moreover, we identified a family of analytic spline wavelets that resemble the "optimallylocalized" Gabor function.
The analytic wavelets were then used to construct a family of twodimensional complex wavelets based on a tensorproduct approach; similar to Kingsbury's design, these wavelets can also be expressed as a linear combination of separable biorthogonal wavelet bases and exhibit better directionality than the ones obtained via the standard separable construction. In particular, the family of 2D complex wavelets constructed using Bspline wavelet bases converge to 2D Gaborlike functions (similar to the ones proposed by Daugman). We also showed how the the corresponding transforms can be implemented efficiently using perfectreconstruction filterbank algorithms. 

Collaborations: Prof. Michael Unser 


Funding: Swiss National Science Foundation 


[5]  K.N. Chaudhury, M. Unser, "The Fractional Hilbert Transform and DualTree GaborLike Wavelet Analysis," Proceedings of the ThirtyFourth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'09), Taipei, Taiwan (Republic of China), April 1924, 2009, pp. 32053208.

[6]  K.N. Chaudhury, M. Unser, "Construction of Hilbert Transform Pairs of Wavelet Bases and Optimal TimeFrequency Localization," Proceedings of the ThirtyThird IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'08), Las Vegas NV, USA, March 30April 4, 2008, pp. 32773280.


