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

Abstract

Abstract-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 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 , 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.

On the Hilbert Transform of Wavelets

“…Now, we show that (9) is sharp, by considering the special case of B-spline wavelets. It is known that if is a B-spline wavelet of degree , then is again a (fractional) B-spline wavelet of the same degree, and hence has the same decay of [7], [14]. This exactly what is predicted by (9), since is known to have vanishing moments.…”

“…The advantages of using Hilbert wavelet pairs for signal analysis had also been recognized by other authors [5], [6]. More recently, it was shown in [7] how a Gabor-like wavelet transform could be realized using such Hilbert pairs.…”

“…It was shown in [7] that this pathology can, however, be overcome by a careful design of the dual multiresolution in which the Hilbert transform is applied only on the wavelet, and never explicitly on the scaling function.…”

“…It is known that a particular family of spline wavelets, namely, the B-spline wavelets [11], converge to a function of the form with the increase in the order of the spline. In particular, it was shown in [7] that the Hilbert transform has comparable localization, smoothness, and vanishing moments for sufficiently large orders (cf. Fig.…”

“…The fundamental reasons why the Hilbert transform can be seamlessly integrated into the multiresolution framework of wavelets are its scale and translation invariances, and its energy-preserving (unitary) nature [7]. These properties are at once obvious from the Fourier-domain definition of the transform.…”