Theory of dual-tree complex wavelets

We study analyticity of the complex wavelets in Kingsbury's dual-tree wavelet transform. A notion of scaling transformation function that defines the relationship between the primal and dual scaling functions is introduced and studied in detail. The analyticity property is examined and dealt with via the transformation function. We separate analyticity from other properties of the wavelet such as orthogonality or biorthogonality. This separation allows a unified treatment of analyticity for general setting of the wavelet system, which can be dyadic or M-band; orthogonal or biorthogonal; scalar or multiple; bases or frames. We show that analyticity of the complex wavelets can be characterized by scaling filter relationship and wavelet filter relationship via the scaling transformation function. For general orthonormal wavelets and dyadic biorthogonal scalar wavelets, the transformation function is shown to be paraunitary and has a linear phase delay of omega/2 in [0, 2 pi).