# Additive white Gaussian noise and undecimated DWT

One of the benefits of DWT is that it is an orthonormal transform.

There are statements that the energy of noise component mainly concentrates on the high-frequency (detail) part and distributes homogeneously. The energy of noise component is included in more wavelet coefficients with smaller amplitudes, while the energy of the useful signal concentrates on fewer wavelet coefficients with bigger amplitudes. (Yu et al., 2010)

As far as I understand, that is due to the orthogonality property of DWT.

Does this behavior of white Gaussian noise also hold true:

• for undecimated DWT (like SWT or MODWT).
• for less redundant forms of DWT (comparing to undecimated version) - DTCWT, for example.

As far as I understand both of them aren't orthogonal transforms.

One of the benefits of DWT is that it is an orthonormal transform

Well, not quite. Some standard DWT are orthonormal, but not all of them. The others used in practice are biorthogonal. Which makes computations more difficult. However, for close-enough-to-orthogonal wavelet transforms, the application of orthogonal results to non-strictly-orthogonal transforms tends to work in practice. Real-world noise are rarely exactly white. But let us start from the noise.

Undecimated DWT or DTCWT belong to frames, a set of generating vectors subject to some bounds: for all $x$ (I am skipping technical conditions) transformed into coefficients $X$, there are $A>0$ and $B<\infty$ such that:

$$A\|x\| ^2 \le \|X\| ^2 \le B\|x\| ^2$$

of which $\|X\| ^2 = \|x\| ^2$ (orthonormality) is a special case. The case $A=B$ corresponds to tight frames, the closest "redundant" equivalent to orthonormality. In this (close-to) tight frame case, things are generally manageable. So for the noise part, the noise coefficients are not white in general, as some correlation appears with redundancy or non-orthogonality.

However, not all is lost with the noise:

• wavelet frames generally keep some noise whitening properties,
• sometimes, a SWT can be implemented as a union of orthogonal bases, which can be processed separately, then recombined (eg with cycle-spinning), but this is a bit suboptimal, as happens with some others redundant transforms: scalar thresholding is common, but suboptimal,
• some technical results still can be obtained with the SWT, see for instance The redundant discrete wavelet transform and additive noise, 2005, J. Fowler
• for DTCWT, you are much less redundant, and can even get a tight frame. The good news is that, due to special features of the Hilbert transform on primal/dual wavelets (and their cross-correlation), you can express the noise covariance very precisely, see for instance Noise covariance properties in Dual-Tree Wavelet Decomposition, C. Chaux et al., 2007. This helps in designed good block thresholding algorithms, like in A Nonlinear Stein Based Estimator for Multichannel Image Denoising, C. Chaux et al., 2008.

So more or less, indeed,

the energy of noise component is included in more wavelet coefficients with smaller amplitudes

Now, let us focus in the signal. Orthogonality lays a lot on constraints in a basis: first vector has $N$ degrees of freedom, the second $N-1$, etc. Thus, orthobasis vectors may be less prone to nicely match, hence concentrated, structured signals or images. If one relaxes orthogonality, one enhances the diversity of projection vectors, and tends to have an increased sparsity, so more or less:

the energy of the useful signal concentrates on fewer wavelet coefficients with bigger amplitudes

But wait, in the transformed domain only, which can be redundant, and correlates noise.

However, all in all, with a little well-managed redundancy (tight or almost tight-frame), and clever thresholding, non-critical wavelet transforms are often beneficial with respect to critically sampled DWT. This also happens with more generic filter banks, see for instance Optimization of Synthesis Oversampled Complex Filter Banks, 2009, J. Gauthier et al.

• Hi, Do you have any experience with Bandlets? – Royi Feb 27 '17 at 6:48
• A little one only, honestly, on the theory and the use as a demo. Why are you asking? – Laurent Duval Feb 27 '17 at 7:14
• Just thought about playing with them and wanted to know how do they compare to other methods in, for instance, denoising image? – Royi Feb 27 '17 at 7:28
• Their power is quite related to the flow regularity of contours and textures in your data, this may depend on the type of your images – Laurent Duval Feb 27 '17 at 7:51

One property of Orthogonal Transformations is that White Noise stays White (Uncorrelated) under Orthogonal Transformations (One could say it's a property of White Noise).

Usually this property assist with deriving properties of operators (Things are simpler).

Yet this property doesn't not hold for Transformations in general which might "Color" the noise in the new base.