The Dual‐Tree Discrete Wavelet Transform

Summary

This chapter introduces the dual‐tree discrete wavelet transform (DTDWT), another version of the discrete wavelet transform (DWT) that is nearly shift invariant but with much less redundancy. The DTDWT is designed similar to the Fourier transform, but with the local nature of the DWT retained. For orthogonal transforms, Parseval's theorem states that the sum of the squared‐magnitude of a time‐domain sequence equals the sum of the squared‐magnitude of the corresponding transform coefficients. In the case of DTDWT, the sum of the squared‐magnitude of two sets of coefficients has to be taken. The wavelet functions of the real and imaginary trees approximately form a Hilbert transform pair. Each set of lowpass and highpass filters is orthogonal or biorthogonal and satisfies the perfect reconstruction (PR) property. Synthesis filters are the time‐reversed and shifted versions of analysis filters.