Ingrid Daubechies' work on wavelets was a revolution — a tour de force — in the way the information in images and other signals are stored compactly in computers.
Image compression can be dramatic: an image could be stored at a minuscule fraction of its original number of bits and visually look the same as the uncompressed image. How is that possible?
Imagine an image of a sky with clouds and balloons. There are slight color variations of gray and white within the pixels representing the clouds, slight variations in color within each balloon, and, similarly, various shades of blue in the sky. If we slightly alter these pixels — in a way that is so subtle that the human eye will not notice — so that patterns in the shades of color are slightly more repetitive, we can store the image much more easily.
For instance, if a similar variation in pixels occurs to create the color shading within the different balloons, we can use that same pattern multiple times. If we compress to the extreme, the image could be stored similarly to a cartoon, with edges separating objects of different colors, where one color is used for each balloon. In those cases, one needs only to store the edges and the colors of each object.
In the years before Ingrid's work, it was not known how to store images efficiently, because images have lots of edges, and previous techniques like Fourier analysis have a terrible time handling them. These techniques prefer smooth changes between colors. But, as we discussed above, images have lots of edges.
Ingrid invented a really elegant way of storing the important information within images that preserves edges and allows compression, but also allows almost perfect reconstruction of the image, even from highly compressed versions of it.
What makes this magical is the math that makes it all work.
Consider two principles: (1) Natural images of the world are "scale invariant," meaning that an image taken from far away has similar properties (edges, colors) to images taken closer to the subject. And (2) Images have local information (such as a balloon in one location).
If one translates the previous two sentences into mathematical expressions and works out several pages of math, they could derive the Daubechies classes of wavelets, where a wavelet means "little wave" (named by the French mathematicians that Ingrid was working with). In one dimension, the most famous of these wavelets looks like a craggly upside-down mountain followed by a craggly right-side up mountain.
This symbol, of the wave shaped like the two craggly mountains, has become iconic among people working with images. Anyone who has seen this math cannot help but be stunned by its elegance, and the derivation boosted Ingrid into legendary status among the mathematical world as well as among electrical engineers.
When I went to a conference with Ingrid many years ago, the other students and I had to step aside while a line of people — mostly women — formed to introduce themselves to Ingrid and tell them what admirers they were of her work (even though they all butchered the pronunciation of her last name! It's pronounced "Dohb-shee," by the way). Ingrid is a deity among people working in image processing, and her wavelets are used often for many different image tasks, including compression and denoising. In recent years, Ingrid has even used it for art restoration and understanding.
I learned many important lessons from Ingrid when I was her Ph.D. student. Let me write down the "lessons" of Ingrid, though of course she never said any of these outright:
Trinity College of Arts & Sciences
