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

TL;DR

This paper introduces a new method for creating Hilbert transform pairs of wavelet bases using B-spline factorization, leading to complex wavelets that resemble Gabor functions and enabling efficient 2D directional wavelet transforms.

Contribution

It develops a novel approximation-theoretic approach to construct Hilbert transform pairs of wavelets, including complex and directional wavelets, with practical FFT-based implementation.

Findings

Constructed Hilbert transform pairs of wavelet bases from B-spline scaling functions.

Designed complex wavelets resembling Gabor functions for improved localization.

Developed an efficient FFT-based filterbank algorithm for complex wavelet transforms.

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 L^2(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 L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional 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 FFT-based filterbank algorithm for implementing the associated complex wavelet transform.