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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 B-splines; (2) characterization of the dual-tree complex wavelet transform using the formalism of fractional Hilbert transforms.

Introduction

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., piecewise-smooth signals) are concerned, wavelet representations employing dilated-translated copies of a fast-decaying oscillating waveform for signal analysis have proven to be more efficient. Complex wavelets, derived from the combination of non-redundant 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 dual-tree complex-wavelet transform is a particular instance where the components bear a specific HT correspondence.
Kingsbury introduced the dual-tree complex-wavelet 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 dual-tree 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 finite-impulse 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 AM-FM analysis.

Main Contributions

We proposed a novel method for constructing exact HT pairs of wavelet bases using B-splines, and formulated necessary and sufficient conditions for generating such wavelet pairs. In particular, we constructed HT pairs of biorthogonal wavelet bases using well-localized scaling functions having identical Riesz bounds. Moreover, we identified a family of analytic spline wavelets that resemble the "optimally-localized" Gabor function.
The analytic wavelets were then used to construct a family of two-dimensional complex wavelets based on a tensor-product 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 2-D complex wavelets constructed using B-spline wavelet bases converge to 2-D Gabor-like functions (similar to the ones proposed by Daugman). We also showed how the the corresponding transforms can be implemented efficiently using perfect-reconstruction filterbank algorithms.

Collaborations: Prof. Michael Unser

Period: 2006-ongoing

Funding: Swiss National Science Foundation

Major Publications

[1] 

K.N. Chaudhury, M. Unser, "On the Hilbert Transform of Wavelets," IEEE Transactions on Signal Processing, in press.

[2] 

K.N. Chaudhury, M. Unser, "Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms," IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3411-3425, September 2009.

[3] 

K.N. Chaudhury, M. Unser, "On the Shiftability of Dual-Tree Complex Wavelet Transforms," IEEE Transactions on Signal Processing, vol. 58, no. 1, pp. 221-232, January 2010.

[4] 

K.N. Chaudhury, M. Unser, "Gabor Wavelet Analysis and the Fractional Hilbert Transform," Proceedings of the SPIE Optics and Photonics 2009 Conference on Mathematical Methods: Wavelet XIII, San Diego CA, USA, August 2-6, 2009, vol. 7446, pp. 74460T-1/74460T-7.

[5] 

K.N. Chaudhury, M. Unser, "The Fractional Hilbert Transform and Dual-Tree Gabor-Like Wavelet Analysis," Proceedings of the Thirty-Fourth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'09), Taipei, Taiwan (Republic of China), April 19-24, 2009, pp. 3205-3208.

[6] 

K.N. Chaudhury, M. Unser, "Construction of Hilbert Transform Pairs of Wavelet Bases and Optimal Time-Frequency Localization," Proceedings of the Thirty-Third IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'08), Las Vegas NV, USA, March 30-April 4, 2008, pp. 3277-3280.

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