The Double-Density Dual-Tree Discrete Wavelet Transform

This paper introduces the double-density dual-tree DWT, a discrete wavelet transform that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets, ψh,i(t) and ψg,i(t), i = 1, 2. The two wavelets ψh,1(t) and ψh,2(t) are off-set from one another by one half, ψh,1(t) ≈ ψh,2(t− 0.5) and ψg,i(t) likewise. Therefore, the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, the two wavelets ψg,1(t) and ψh,1(t) form an approximate Hilbert transform pair, ψg,1(t) ≈ H{ψh,1(t)}, and likewise ψg,2(t) ≈ H{ψh,2(t)}. Therefore they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain FIR filters which satisfy the numerous constraints imposed. This design procedure employs a fractional-delay all-pass filter, spectral factorization, and filter bank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant. Permission to publish this abstract separately is granted. Submitted to IEEE Transactions on Signal Processing.