Ingrid Daubechies won the 2023 Wolf Prize in Mathematics last February. She is the James B. Duke Distinguished Professor of Mathematics and Electrical and Computer Engineering at Duke University.

Crop from 2021 New York Times profile by Siobhan Roberts, photo by Jeremy Lange

Today, Mother’s Day in the US, we congratulate her on this award and note some non-theoretical applications of her work.

The award cites “her work in the creation and development of wavelet theory and modern time-frequency analysis.” It goes on to say:

Her discovery of smooth, compactly supported wavelets, and the development of biorthogonal wavelets transformed image and signal processing and filtering. Her work is of tremendous importance in image compression, medical imaging, remote sensing, and digital photography. Daubechies has also made unparalleled contributions to developing real-world applications of harmonic analysis, introducing sophisticated image-processing techniques to fields ranging from art to evolutionary biology and beyond.

Daubechies started as a physicist and wrote a PhD thesis on quantum mechanics. Although she is the first woman to win a Wolf Prize in Mathematics, the Physics Wolf Prizes began with a female winner right away, in 1978.

That was Chien-Shiung Wu, who won in 1978 for her work on weak interactions and execution of the first experiment that demonstrated the non-conservation of parity. Wu’s prize partially righted a now universally-acknowledged wrong of not including her in the 1957 Nobel Prize of Tsung Dao Lee and Chen Ning Yang. There were only four female winners, all in Medicine, until Ada Yonath in Chemistry in 2006/2007, but women had exact parity in the twelve prizes in 2022.

The French word for wavelet, which is ondelette, may have been first employed in a scientific context by the French-American physicist Alexander Grossmann around the time he was one of Daubechies’s two PhD advisors. Without going into detail, we venture a highly compressed evocation of wavelets.

An abstract statement of the problem for which wavelets have often provided effective solutions is:

Given a generator of elements in a high-dimensional Hilbert space, what is the best choice of basis to minimize the expected complexity of representing items drawn from over that basis, either exactly or optimizing a combined score of complexity and approximation?

An example of a high-dimensional vector space is the set of possible monomials of degree (up to) in variables. Working over the standard basis is like assembling a polynomial term-by-term, which can be poky. There are other bases composed of mutually orthogonal polynomials for which small sums may be closer to typical outputs from certain generators .

When represents a static configuration or a process that changes uniformly and slowly, there might not be much improvement on the standard basis. But when is dominated by a few transient events, bases aligned with such events perform better. Getting very rough in our description, many events of interest are like throwing rocks into a pond. The resulting ripples are well described over a wavelet basis. Whereas, using a standard basis—which is fine when the pond is still—will leave blocky artifacts when low resolution fails to discriminate the ripples.

Wikipedia “Daubechies wavelet” source

The origin story of how Daubechies realized the wider applicability of wavelets, as told here, is that she noticed blocky artifacts in grass while watching a soccer match. The movement of grass blades is more concisely explained as a response to transient wind and game events than a uniform and locally independent process. This has proved universal in practice, as the story says: “Anytime you go to a movie theater, or watch live sports on ESPN, each frame has been compressed using Daubechies’ wavelet-based method.”

Art and Originality

My wording with “explained” hints at a yet-wider world of application in data-science inference. I would love to claim in a grand sweep that an artwork such as a painting is best represented as the sequence of conceptual events that governed its creation. The events should be describable at a level where the exact details of execution can be inferred while saving many bits over an exhaustive “standard-basis” representation. I am not going as far as the low-complexity art of Jürgen Schmidhuber, but share the motivation that art may be better described in bases that align with succinct formulations and salient impulses.

On firmer ground is the use of descriptive techniques to identify events that have happened to artwork since its creation. In a wonderful 2016 article by Daubechies for Quanta on art restoration, she describes two kinds of events affecting centuries-old altarpieces:

• Natural cracking of the wood panels
• Previous efforts at restoration.

The latter included past restorers imposing a lattice support on the wood that confused with natural cracks. A third kind of event they were able to distinguish fits the creation type in my first paragraph: strokes of fine text delicately painted by the original artists. Daubechies’s methods were able to highlight those just enough for professional antiquarians to identify the religious text that was intended.

Excerpt from large image in Daubechies article

In earlier work at Princeton described in this 2008 article, Daubechies and team applied wavelets to distinguish authentic paintings by Vincent van Gogh from forgeries. To do this, they “created a statistical portrait of van Gogh’s style using wavelets.” What emerged from their analysis was that the forgeries could not so easily be described in their van Gogh basis:

The forgeries and copies … had more tiny wavelets in the image, which [quoting team member Shannon Hughes] “we think are tiny fluctuations and wobbles. … You can imagine that if someone is painting really slowly trying to copy another work, they are hesitating a little bit.”

This reminds me also of telltale hesitations that enable online chess platforms to infer when a human player is concentrating on something else besides playing the game. A final quote from Hughes in the article tends toward my grander thought about artistic creation:

[Wavelet technology may help figure out] “what the artist was thinking and what the artist was doing and why they were doing it.”

Daubechies has been involved in some art creation of her own: the art installation Mathemalchemy.

Open Problems

What inferences will wavelets enable about the creative process? As someone who fancies composing but has no ability to play or transcribe music faster than a snail’s pace, I can vouch that music is not created in the standard basis.