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.
