I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and so on. As far as I know most existing wavelets tend to focus on one or a few of the following properties, compact support, orthogonality, symmetry and smoothness of wavelet function ("capturing" as many local differentiations as possible). In this question we limit ourselves to biorthogonal discrete wavelets which have the following constraints for the scaling sequences: $$\sum_{n\in\mathbb{N}}a_n{\tilde a}_{n+2m} = 2 \delta_{m,0}$$ and the wavelet sequences can then be expressed or determined as: $$b_n = (-1)^n{\tilde a}_{M-n-1}\\ {\tilde b}_n = (-1)^n{a}_{M-n-1}\\s.t\\ n\in\{0,\cdots,N-1\}$$
Now to my question, would it be possible to create constraints to construct noise wavelets or highly irregular wavelets where the goal is to capture as much (additive) "unsmoothness" / "noise" as possible at every level. If possible, could they be useful? Any hints or references to such works are welcome as I am aware the question may be a bit too broad to answer completely.
Own work: By a crude and un-elegant optimization I have managed to get a few 4/4 tap wavelets with varying degrees of "smoothness". Here are two exampels.
Typical smooth 4/4 wavelet, a bit resemblant to Daubechies 4 tap in looking kind of fractal-like, is it not?
Example of one particularly more "random" or un-smooth wavelet, once again a 4/4 biorthogonal. These are the kinds I am curious about, since they almost never appear in the literature. How to build them, analyze them and potential uses.