# Wavelet image denoising: dual-tree versus double-density

What is the main difference between the dual-tree DWT and the double density DWT image denoising techniques?

How does the combination of them improve the quality of the denoised image?

• I'll go ahead and remove everything that is unrelated to your question. Please take care, next time, to focus on asking your question and supplying the information necessary to answer it. – Marcus Müller Nov 18 '16 at 12:07
• Your question claims there's a benefit of combining them, but you don't cite any sources, which would probably make it easier to explain. So, what is your metric for "quality"? – Marcus Müller Nov 18 '16 at 12:25
• the used metric is psnr, there are many papers which use double density dual tree DWT as the most approprate technique for image denoising ijert.org/view-pdf/3483/… – user24907 Nov 18 '16 at 13:23

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:

1. on dual-tree wavelets and
2. 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.

• A lot of time when Denoising using Wavelets there are ringing effects around edges. Will those mitigate it? What's the best approach to handle it? Thank You. – Royi Oct 4 '17 at 0:21
• Ringing artifacts can indeed be reduced by proper coefficient selection (scalar thresholding is often too crude) and adequate transform choice. Redundant, directional transforms indeed help to that task – Laurent Duval Oct 4 '17 at 10:30
• What would be the best combination (Transform and Denoising)? Given we do soft threshold, which Wavelet Transform would be the least "Ringing Generating"? Thank You. – Royi Oct 4 '17 at 11:07
• That seems to me a full integral question per se, that I cannot answer in a comment. Feel free to ask a separate question – Laurent Duval Oct 4 '17 at 11:32
• I wish we could gather few people for a discussion on that in a chat or something. I have nice image to experiment with. – Royi Oct 4 '17 at 21:40