Interesting Open Questions In Wavelet Theory for UG Research?

Question

Just as background, I am going into my senior year of undergrad and I have a pretty light load this coming semester but I want to fill this time with something that will really make my grad school resume sparkle. As you might guess from my username, I am fascinated by wavelets (and harmonic analysis in general) and I have decided to try to at least attempt some research over the next year to close out my UG studies. I spoke with a professor about this and he said he'd be more than happy to help me as much as he can and get me credit for it but I'd need to come up with the topic and expect to do the vast majority of it on my own (I actually prefer working more independently).

Anyways, I have been trying to find a problem that interests me enough that I can see myself working on 5-6 hours a day but I haven't had any luck. While I feel confident in my knowledge of wavelet theory, I don't really know which way is up when it comes to the actual research end of things. I know the major players like Daubechies, Mallat, Morlet, etc. but I am having a hard time figuring out what constitutes "cutting edge".

Obviously, I expect that any research is going to be more on the applied end of things but I'd prefer it to be as "pure" (i.e. not computational) as possible just because I am limited in how much computing power I have access to (I've had my computer crash many many times from trying to do wavelet projects in MATLAB) and I am skeptical that the department will let me anywhere near the big guns. My knowledge base is sufficient that I can understand most of Mallat's "A Wavelet Tour of Signal Processing: The Sparse Way", though I find it to be quite a slog just because of the notation and lay out. I am particularly interested in some potential interplay between stochastic processes (sufficiently nice ones at least) and wavelet analysis. I do not have a great background in stochastic integration quite yet but I learn fast enough that I feel confident it wouldn't pose too much of barrier.

I would be indebted to anyone who could point me in the right direction.

Answer 1

As you understand most of Mallat's A Wavelet Tour of Signal Processing, I guess you are familiar with Bessel sequences and frames.

The following is an open problem regarding the extension of wavelet systems to wavelet frames:

Let the wavelet system $\{D_j T_k g \}_{j, k \in \mathbb{Z}} \subset L^2 (\mathbb{R})$ be a Bessel system in $L^2 (\mathbb{R})$. Is there a wavelet system $\{D_j T_k h \}_{j, k \in \mathbb{Z}} \subset L^2 (\mathbb{R})$ such that the union $$ \{D_j T_k g \}_{j, k \in \mathbb{Z}} \cup \{D_j T_k g \}_{j, k \in \mathbb{Z}} $$ forms a wavelet frame for $L^2 (\mathbb{R})$.

There exists a similar question regarding dual wavelet frames.

In case you are interested in this problem, I suggest you to have a look in the recently published book An Introduction to Frames and Riesz Bases (Second Edition) by Ole Christensen.

Answer 2

Another route you can go is to focus on practicalities and implementation and then Wim Sweldens did a polynomial factoring scheme called the lifting scheme back in the 90s (I think). It may be rather old nowadays, but sure can give some practical speed-ups as well as opens up new ways to design wavelets and filter banks and factoring wavelets and adapting algorithms on the fly.

A short matrix schematic of one scale of the factorization with $N$ lifting steps:

$$\left[\begin{array}{c}\bf l\\\bf h\end{array}\right]=\underset{\text {Save to multiply one-by-one}}{\underbrace{\left(\prod_{i=1}^N \left[\begin{array}{cl}{\bf I}&\bf 0\\{\bf U}_i&\bf I\end{array}\right]\left[\begin{array}{lc}\bf I&{-\bf P}_i\\\bf 0&\bf I\end{array}\right] \right) }} \underset{\text {Multiply together}}{\underbrace{\left( \left[\begin{array}{cc}\bf S&\bf 0\\\bf 0&\bf S\end{array}\right]\left[\begin{array}{c}\bf I\\\bf L\end{array}\right]\right)}} {\bf v}$$ Where

• $\bf I$ does nothing - just copies.
• $\bf L$ is the lazy filter which delays samples by 1.
• $\bf S$ is subsample a factor 2
• ${\bf U}_i$ are the update filters.
• ${\bf P}_i$ are the predict filters.
• $\bf v$ is column vector with the signal samples.
• $\bf l,h$ are low-pass and high-pass filtered results from current scale.

Then the practical part is that the dyadic subdivision has already been done by the block matrix construction in-place, so we just need to iterate on the correct half of the ${\bf v}$ with no pesky permutations to keep track of.