Ingrid Daubechies is James B. Duke Professor of Mathematics at Duke University. She is the author of fundamental work on wavelets analysis and of innumerable applications of the theory; she gave important contributions in a wide range of areas in science. She has been the recipient of many prestigious prizes, the last of which has been William Benter Prize in Applied Mathematics awarded in June 2018. In this interview we ‘surf’  from wavelets to equal opportunities in science,  and much more.
Interview by Anna Maria Cherubini and Francesca Arici.

Your work encompasses many areas of research. Which problems are you most interested in at the moment?

I am working on a number of different topics. One involves differential and computational geometry. This project is in collaboration with morphological biologists who study whole organisms (rather than at the molecular or cellular level). They are looking for algorithmically feasible ways that have biological relevance to compare and quantify similarities and dissimilarities between surfaces like bones or teeth. We found approaches to compute automatically mappings from one surface to another that make biological sense to our collaborators. We do this by considering not just pairs, but whole collections of surfaces to construct our mappings; this corresponds nicely to what biologists do by “hand”, using coordinates of landmark tags that they pick themselves.

The framework we use in this project is that of a fibre bundle: for instance, when you compare second molars of primates, you can view the whole collection as a kind of “molar manifold”, in which every individual tooth of the collection is a different point on the manifold, but each tooth is also a 2-dimensional surface. So you really have a fibre bundle (in which each fibre is a 2-dimensional surface) and the mappings from tooth surfaces define a connection on that fibre bundle.

At the start of our construction, we have a “noisy connection”: the initial correspondences are defined pairwise, which exhibit large mistakes in many cases, but because we have the whole collection, we can de-noise the collection of mappings, using consistency arguments. This has shown to be a very interesting, productive and useful approach. We are now at the stage where we are producing output that the biologists find very helpful.

Another project, about time-frequency analysis, is in collaboration with a former student, Hau-Tieng Wu, who was already trained as a radiologist before doing a PhD in applied math. We mostly look at medical signals, but our approach can also be applied to other types of signals, like bird songs or other animal vocalizations. The signals we consider look periodic at first sight, but are in fact not quite periodic – they typically exhibit slight variations in period and amplitude. These slight variations are what we really want to keep track of. An example would be an electrocardiogram, in which the time between peaks changes constantly, following (but not in perfect sync with) the subject’s respiration; other changes might indicate occasional heart malfunction.

The first reflex in signal analysis, when encountering a periodic signal, is to use a Fourier transform, a beautiful and well-understood mathematical tool that exploits periodicity in an essential way. For our not-quite periodic signals, Fourier analysis would, however, give a very confused, blurry picture, so we have to use a more adaptive procedure, based on harmonic analysis and nonlinear operations. As a toy example, one could imagine a function that depends in a truly periodic way on an argument that is not exactly proportional to time t, as t progresses – instead of being a linear function of t, its slope varies continuously. As a function of t, this will be “not quite periodic”; typically we are interested in keeping track of the non-linearly varying “instantaneous frequency”.

Even when the time-warping varies slowly, and a windowed Fourier transform (i.e. Fourier transforms carried out on localized pieces of the signal) manages to do a reasonable job of tracking the changes in frequency, it typically still introduces “artifacts” in its analysis. More precisely, if the decomposition of the original function into cosines and sines has several components, then each of them will be present as well in the windowed Fourier transform of the warped function, with the frequency of each following a curve dictated by the time warping. These are the “harmonics” we see in spectrograms of musical sounds or spoken language. If a signal consists of a superposition of several functions of this type, each with their own warping, then the harmonics of each can interfere with each other (with e.g. the 3rd harmonic of one “crossing” the 2nd harmonic of another), impeding the legibility of the representation. But, in a very real sense, the harmonics observed are an artifact of the analysis method: they appear only because Fourier analysis “insists” on decomposing into sines and cosines. Our adaptive method tries to identify the shape of the periodic function(s) that is (are) time-warped, and use this in the time-frequency decomposition.

I am also still working on applications of wavelets in image processing. Some of these are research questions, but for other projects I am working with teams of undergraduates who use wavelets and other image processing techniques for the analysis of high-resolution pictures of art. It is a lot of fun to work with art conservators and curators.

Since you mentioned wavelets, one of the areas of research for which you are most famous: how would you explain to a nonspecialist what ‘wavelets’ are, and what their applications are to real-world issues?

When explaining wavelets to art conservators I tell them they first have to accept that we can digitalise images, i.e. we can make tables of numbers (one number for each pixel) from which we reconstruct a good approximation to the image. To see that the digital image contains information at many different scales, you imagine just blurring the image a bit; this will make you lose some information; the blurred digital image also consists of numbers, and so you could subtract the numbers of the blurred version from those of the original, pixel by pixel. Those differences give you an idea of the information lost by blurring.

You can repeat the operation with the blurred version: blurring it a bit more, and again looking at the difference image obtained by subtracting the even more blurred image from the first blur. And so on. In the end, you get an extremely blurry version; you will also have a whole series of successive difference images – if add these, one after the other, to the very blurry image, you gradually build up again all the detail that you lost, to reconstruct the original. Effectively, you have constructed a wavelet decomposition of the original. In those successive difference layers (the wavelet layers) one can recognize sharp edges in the original image (they show up in every single difference), but also other features, that may “exist” at some scales and not at others.

Wavelet decompositions provide you with a succinct description of the important features ate each scale; the succinctness of this description explains why wavelets are good for image compression. This potential for image compression is interesting for many applications, for instance for digital cinema.

For our work on art paintings, compression is not as interesting as the decomposition itself, which helps us in analyzing and manipulating images. In one project we collaborated with a museum to carry out a virtual restoration of a 14th-century Italian altarpiece. We did automatic detection and virtual in-painting of the cracks, followed by a remapping of the colors, from the aged and faded pigments to what they must have looked like when they were fresh. It ended up being very interesting. The undergrads who worked on it with us learned a lot working on much more interesting images than the standard image processing stalwarts. The project became part of an exhibit at the museum; we are continuing this collaboration and there are many more questions we want to answer.

Now that our approaches have shown they can be useful, art historians come with more questions – questions about image processing that we might not come up with naturally ourselves. Some of these questions require new approaches, and so they become research questions; for some others we already developed tools that we just need to adapt – those become project for undergraduates.

Can you state the authenticity of a work of art?

I try to stay very much away from any art authentication questions. I have no standing in any such question. What we do is provide an extra tool that experts can use, if they wish. I sometimes do get contacted by people who own a painting that they would like to see authenticated, and who offer to pay me for a study of their painting. I never accept – it would jeopardize my credibility with respect to the museums with which I want to work. Authenticity, to be honest, is not the most interesting question for me. I find it more interesting to help art conservators get a better insight into the painting process, or get a better view of layers hidden by the top painting, which may show a different painting, completely overpainted. Or tear apart the X-ray view of a panel painted on both sides (such as the doors of a polyptych) into the contributions that came from each panel separately. This turns out to be a hard problem, that. I wouldn’t have thought of dreaming up myself, but it’s a problem that occurs in reality

A side question: what are surfing wavelets?

That was just the title of a talk, as in ‘surfing over the subject’

Can you tell us something about your personal pathway to your profession? When did you decide to be a scientist and why?

I have always been interested in why things work the way they do. I actually studied physics — I originally intended to become an engineer, but then I decided that I wanted more to understand the physical nature of the world. I went into theoretical physics, where you have to learn a lot of maths and I liked that very much. For my PhD. I worked on quantum mechanics and on semiclassical approximations; the tools you use there have a lot to do with Fourier analysis. I found that many of those tools were useful also for signal analysis and I started to talk to engineers. Some of the tools I had studied gave me a different point of view on signal analysis, and I wrote a few papers on the subject using these other ways of looking at signals. Then all of a sudden people considered me an applied mathematician. So I feel I ‘surfed’ the opportunity.

Can you remember any particular difficulty along your way or any particular help and support?

I was always kind of stubborn. I went to public schools and in Belgium at that time they were gender segregated. So I didn’t really encounter the attitude that because I was a girl I was supposed to be less good at science. I came across that at university, but then I thought, well, if somebody thinks that, they are just stupid. I always wanted to understand things and I would ask lots of questions. At university all professors were male, but we had one assistant professor who was female and whom I liked a lot and was very supportive. Later, when I started doing research, I didn’t see many women mathematicians; the first few I met had no children and I always thought I would have family and children. Then I met Cathleen Morawetz and she had four children, so I didn’t worry anymore.

I do believe in role models. I was involved in the Program for Women in Mathematics that Karen Uhlenbeck started (still going strong after 25 years!) at the Institute for Advanced Studies in Princeton. It involves junior and senior women, so everybody sees women in all career stages and everybody can be mentor as well as mentee. I think that is great. Many of the women who meet at that program stay in touch for their whole career. I believe in associations like EWM; recently the IMU created its Committee for Women in Mathematics, which is very important for women in developing countries. In some developing countries there are more opportunities for women than in the West, because unfortunately university positions pay very little and ambitious young men look elsewhere. They leave the field to the women and … we can do it!

In your opinion which are the main challenges for women in science?

There’s still an incredible variation, across different countries in Europe, in the number of women. Women are underrepresented in academics in mathematics everywhere, but in some countries more than others. I hope that the mobility that comes with European funds will help in pulling up the numbers of women in countries where they are exceptionally low in numbers. These things change so incredibly slowly. We have to work to make it possible for those who have the talent and the wish to be a researcher to achieve their dream. That can depend on very small things, so it’s crucial to identify what the factors are.

Making quality childcare easily available and affordable, is probably a big factor. And whenever there are negative factors, really really try to act against them. For example, only now do people bring out in the open to what extent sexual harassment in academia discourages some young women, who then go to different professions. It’s essential to make sure that it doesn’t happen and there is not a tolerant attitude towards this kind of behavior. I don’t know a single woman my age who has not had to deal with some jerks of this type. In general, we evaded it, or made a joke about it and avoided that person in the future. But it is something that could be very discouraging – I think that we still have to watch out for the younger ones. Unfortunately, I don’t have a silver bullet, if I did I would use it.

There are many actions in place regarding women and science: what works? What more could or should be done?

We could actually study different subfields in mathematics, because some subfields have many more women than others, even in the same country. I think it would really be worthwhile to perform a sociological study of these communities and identify differences in culture and behavior. I really believe that community atmospheres are different in different subfields and that some fields are more attractive than others for women because there is a different atmosphere – more congenial and less competitive, maybe.

Is there any particular problem you dream of solving now?

There are people who spend their whole life around a particular problem but I’m not that way. I’m very opportunistic in that respect, I would never have spent my life on the Riemann hypothesis. I will be 64 this year, it will be probably my last power of two, and I think the legacy you leave is not the problems you solve but the community you help to build and sustain and pass on. My students wanted to organize a conference for my birthday but I don’t want a conference in which everybody talks about what a fantastic person I am. So we will do a much more communal thing, still with speakers, but there will be students, and students of students, as an extended family. On the first day everybody will get three minutes, and they will talk on something they work on, so people who get interested in something will get in contact with the speaker who mentioned that topic, even if they didn’t know him or her beforehand.

What are your passions, apart from mathematics?

I like to cook and I’m a very fervent gardener. I like doing things with my hands, so I started doing ceramics again, after many years of not doing it while the children were growing. And I’m a voracious reader.

And what are you reading at the moment?

I’m reading the last crime novel by Elizabeth George, and ‘Hidden figures’, on the Afro-American women ‘computers’ at NASA. And a book by Neil Gaiman about Norse mythology. And a book on how to help elderly parents.

Photo: Les Todd, Duke Photography