IEEE Circuits and Systems Magazine - Q4 2022 - 4

Question 2: You received a B.S. degree in physics, and
later a Ph.D. in quantum mechanics. Six years later after
your Ph.D., you made your best-known discovery on the
wavelet theory. Can you elaborate on how these diverse
backgrounds connect and impact your research and
career?
Prof. Daubechies: The work on quantum mechanics I
had been done during my Ph.D. was very mathematical.
I worked in a field called mathematical physics,
which essentially means working on physical models,
but with rigorous mathematics. Mathematical physics
results can be useful elsewhere: for instance, the initial
work on the Ising model is now widely used in machine
learning. When you do signal analysis, you really are
looking at the properties of functions locally in time
and in frequency. This is very similar to what you do
in quantum mechanics where you want to understand
the wave function as it depends on space and momentum
- the dual variable. In this sense, signal analysis
and quantum mechanics are both related to the Fourier
transform and their mathematical techniques are
very similar. So it's not such a big jump to go from one
to the other. My background in physics was very useful
to me and I had learned some techniques different
from what electrical engineers learn. Also, I had
learned much more to look at time and frequency as
two aspects of one reality - having a different point of
view really helps.
Question 3: How did you come up with the idea of
" Daubechies wavelets " ? How to explain it to average
people? How does the concept of wavelets compare to
the currently widely used machine learning approaches
in signal processing?
Prof. Daubechies: When I started working on wavelets,
I built on research I had done in time and frequency.
And then, when the first wavelet basis was constructed,
it linked into harmonic analysis that I didn't know, and I
started studying that. It was a beautiful theory, but for the
practical application to signals analysis, it would have to
be mutilated, and in this process lose some of its beautiful
properties. Mathematicians think that is natural because
when you work on practical applications rather than mathematical
models you have to take aspects into account
that don't fit into the model. I wanted to see if it was really
true that I had to give up all the beautiful features in
order to do the applications. To answer this question, you
must think about it differently. You need to find out what
the constraints are imposed by the applications, whether
those constraints can be added to the ones of the model,
and whether the whole construction can be rebuilt. All
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IEEE CIRCUITS AND SYSTEMS MAGAZINE
these questions inspired the construction of " Daubechies
wavelets " . This way of looking at the problem was particular
to me and can be attributed to my mixed background. I
solved the problem in a roundabout way, otherwise I don't
think people would have solved it so easily.
In old-fashioned machine learning, before neural networks
came along, algorithms were essentially trying to
learn a representation in which the class of things that
you were studying could be represented simply. A natural
way of doing so is using orthogonal polynomials. Normally,
the first few terms are going to be easy to compute
and it becomes harder to find more terms. The situation
is different when you can compute easily a large number,
say a million of terms. In this case it may still be useful to
retain only some small collection of these terms, but maybe
not necessarily the first few terms, which becomes a
completely different mathematical approach. So that's
where wavelets come in - instead of looking at the original
representation, you can make a basis transformation
on the signal and characterize function spaces by only
focusing on a few of the coefficients you are interested in.
Furthermore, we can identify the types of functions that
can provide sparse expansion for certain tasks, making
it possible for compression. Neural networks, in a sense,
also make it possible for learning compressed representations,
though we don't well understand - at least not the
level that I call understanding- the theory behind them. I
think the next decade is going to see a lot of theory work
that helps us to better understand neural networks. We
will be able to get a better idea of how to construct neural
nets, and their computation will be less wasteful than
they are today.
Question 4: Can you talk about your recent works on
applying mathematical skills to fine art? What projects
have you been working on?
Prof. Daubechies: Art conservation often involves the
use of very high-resolution images and has completely
different tasks about these images from standard image
analysis. Taking one of my recent projects as an example,
which is X-ray image separation for artworks with
concealed designs. X-ray images play a vital role in conservation
and preservation of artworks, but many X-ray
images of paintings have concealed sub-surface designs
deriving from the reuse of the painting support or revision
of a composition by the artist. In practice, it is very
difficult to separate the contributions from both the
surface painting and the concealed features. The best
algorithm we have now is one that uses neural network,
but there are still many open questions, which I believe
in turn will lead to a lot of interesting research. For me
and my students, working with these images (such as
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IEEE Circuits and Systems Magazine - Q4 2022

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