Optimal Level of Wavelet Decomposition for Denoising

Question

If a certain mother wavelet is defined and the optimal threshold shall be defined via Donoho, how can I define the optimal decomposition level to achieve the best denoising results.

Answer 1

Best denoising should be related to certain quality measures, often requiring the clean signal reference, which you do not have in general. Or you could rely on some reference-free measures.

To the best of my knowledge, many SURE wavelet methods allow to derive risk estimators without a reference, on a given wavelet decomposition (with given levels) in a scalar fashion, coefficient by coefficient, before reconstructing with the inverse transform.

There are a few references that address a selection of appropriate levels:

• Using Stein’s Unbiased Risk Estimate (SURE) to Optimize Level of Decomposition in Stationary Wavelet Transform Denoising
• Optimal determination of wavelet threshold and decomposition level via heuristic learning for noise reduction
• Selection of Optimal Decomposition Level Based on Entropy for Speech Denoising Using Wavelet Packet

but they seem relatively ad-hoc to me (based on a minimum of the Stein risk, entropy, genetic algorithms). It seems quite difficult to design a model of what happens across subbands.

Indeed, scalar thresholding on orthogonal wavelets often leads to relatively poor results. One generally should invest in (slightly) redundant (and oriented for images) wavelet decompositions to improve the signal/noise separation (at the cost of correlation), block thresholding to base threshold on groups of coefficients, inter- and intra-scale, and to coefficient weighting. The paper A Nonlinear Stein Based Estimator for Multichannel Image Denoising reviews previous models and also exploits coefficients from different channels, but the block/redundancy pair still applies for mono-channel data.

[20181218: Additional notes] one should be aware that the level of decomposition combined with wavelet filter length generates globally long convolutions, that should be restricted with respect to the signal length. Side effects, like data extension (periodization) and aliasing caused by thresholding are important factors as well.

Answer 2

I'd say it depends on the noise properties and of course the image itself.

What you can think is that most Denoise Filters can handle only the High Frequencies of the noise.
Hence the decomposition process moves Low Frequency of the noise to the High Frequency part for the spectrum.

So if your noise is white you need to go down as you can. If it colored and most of its energy is in the High Frequency you can do only few levels.