Hilbert Transform Pairs of Wavelets

ABSTRACT The orthogonal projection of a real orthonormal wavelet basis onto the Hardy space—consisting of functions whose Fourier transform is supported only on the positive real axis—yields a Parseval frame of wavelets. Motivated by practical considerations, Kingsbury proposed the use of two real scaling functions from multiresolution analyses (MRAs) whose corresponding wavelets form a Hilbert transform pair, in order to construct such a Parseval frame. Despite the widespread use of the resulting dual‐tree complex wavelet transform (DT‐CWT) in signal and image processing, a rigorous mathematical foundation for the construction of Hilbert transform pairs is largely missing. This paper aims to fill this gap. We present a systematic operator–theoretic framework for the construction of Hilbert transform pairs of wavelets and discuss several examples in this context. Our main contribution is a generalization of a construction due to Kingsbury and Selesnick: we prove that any unitary convolution operator satisfying a refinement equation maps MRAs to MRAs. Moreover, we establish the convergence of the infinite series of sawtooth functions that underlies Selesnick's construction using Hermite's identity. Finally, we demonstrate that vanishing moments simultaneously improve the decay of the wavelet, the scaling function, and the associated filter coefficients, thus providing a significant enhancement of the construction.