Complex linear-phase biorthogonal filterbanks with approximately analytic wavelets

In this paper, approximately analytic wavelets associated with a single filterbank tree are designed. It is shown that if the scaling filters are made analytic, the wavelets also become analytic. In other words, the real and imaginary parts of the scaling filters form a Hilbert transform pair. They have identical magnitude and phase shifts of π/2 and −π/2 for the frequency range (0,π) and (−π, 0), respectively. Conjugate symmetric filters are used in the biorthogonal setting so that the phase relationship is structurally guaranteed. The error between the magnitudes of real and imaginary parts of the scaling filters is then minimized subject to biorthogonality. As a result complex, linear-phase wavelet bases is obtained which have vanishing moments. The error level in the negative/positive frequency suppression for a length 10 filterbank with 5 vanishing moments is of the order 10−3. The designed wavelets may be potentially useful in applications such as face recognition, image segmentation and texture analysis.