### The Double-Density Dual-Tree DWT

#### Abstract

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, psi_{h,i}(t) and psi_{g,i}(t), i=1,2. The two wavelets psi_{h,1}(t) and psi_{h,2}(t) are off-set from one another by one half, psi_{h,1}(t) \approx psi_{h,2}(t-0.5) and psi_{g,i}(t) likewise. Therefore, the integer translates of one pair of wavelets fall midway between the integer translates of the other pair. Simultaneously, the two wavelets psi_{g,1}(t) and psi_{h,1}(t) form an approximate Hilbert transform pair, psi_{g,1}(t) \approx H{psi_{h,1}(t)} , and similarly for psi_{g,2}(t) and psi_{h,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.

The double-density dual-tree DWT. IEEE Trans. on Signal Processing, 52(5):1304-1314, May 2004.