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Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

K.N. Chaudhury, M. Unser

IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3411-3425, September 2009.

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions—the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L2(ℝ) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L2(ℝ), we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.

AUTHOR="Chaudhury, K.N. and Unser, M.",
TITLE="Construction of {H}ilbert Transform Pairs of Wavelet Bases and
        {G}abor-Like Transforms",
JOURNAL="{IEEE} Transactions on Signal Processing",

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