Wavelet image denoising: dual-tree versus double-density
The formalism of the continuous wavelet transform is relatively flexible. To make a practical tool out of it, you ought to discretize it, and here come the pain. It is quite easy to discretize it in a very redundant way (a discrete wavelet frame), but the price to pay when analyzing high-dimensional signals (2D images, 3D volumes) can quickly become too high: on the order or $J^d$, with $J$ the number of wavelet levels and $d$ the dimension.

So people have tried to obtain a critically sampled, discrete wavelet, without redundancy. This is the discrete wavelet transform (DWT). Nothing comes free. The price to pay is a narrow choice in the wavelets, and a processing sensitive to shifts. Ivan Selesnick, among others, worked on several intermediate designs: by allowing a bearable redundancy (typically $2\times$ or a little more in each dimension), one can obtain results close to, or even above the quality of fully redundant designs.

He worked especially:


on dual-tree wavelets and 
on the double-density wavelet, 


both  with a redundancy of $2$ in 1D, whatever the level. The first one compensates the poorer DWT choice by combining it with its Hilbert pair, allowing lower shift variance. The second allowed more degrees of freedom in the design of wavelet filters, hence a richer choice. Both have their merits, that may depend on the application. A little redundancy in wavelets often helps. 

And both can be further combined, as in  The Double-Density Dual-Tree Discrete Wavelet Transform, by the same author in 2004. The gain comes from the double advantage  of having more design freedom in the wavelets (allowing for instance better cutoff frequency and symmetry) and the Hilbert pair for shift-invariance.

This is confirmed at Double-Density Complex Wavelet Transform:


  Both the double-density DWT and the dual-tree DWT have their own
  distinct characteristics and advantages, and as such, it was only
  natural to combine the two into one transform called the
  double-density complex (or double-density dual-tree) DWT.