Reality in wavelets
|
|
|
|
|
Ingrid Daubechies (Photo
by Denise Applewhite)
|
Making sense of the world is something professors (and most people, for that matter) spend lifetimes doing.
But for Ingrid Daubechies, math professor and director of the Program in Applied and Computational Mathematics, this is a somewhat more literal calling.
Daubechies studies ways to represent reality more clearly by using "wavelets," a type of mathematical tool used to compress data and then to unbundle it, usually via computer, so that the information sent is reproduced as accurately as possible. Applications include encryption, viewing photos from the Internet and medical diagnostic imaging.
In some respects, what Daubechies is doing is bringing to electronics what the human brain already does pretty well. People are constantly fine-tuning the world around them to focus on a task while blocking out extraneous stimuli, or noise. We do this unconsciously. Reading, for instance, not only takes some degree of concentration but also involves the process of screening out background noise--people moving or talking, air ventilation systems humming, telephones ringing.
While the brain performs screening tasks well because of tens of thousands of years of evolution, signal processing via electronics is a relatively new endeavor, and scientists like Daubechies are working to make the process more efficient and more accurate.
Images have noise
The field of wavelets underwent a revolution about 14 years ago, coinciding with the rise of the capabilities of computers for processing huge amounts of information, including the data embedded in visual images such as photos being sent over the Internet or a pulsing heart visualized by MRI.
"There are a whole lot of visual images, and these images have noise," says Daubechies. "Nothing is perfect, nothing is completely sharp."
She approaches the challenge from a mathematical point of view, talking to people in various fields where data transmission technology is applied, such as medicine, neuroscience and engineering. "I talk to them, get a sense of what is important to them and translate that into mathematical tools."
Daubechies' tools are often undulating waveforms, which can be added together or subtracted to identify bundles of data. Her goal is to communicate data--say a photograph of breast tissue being evaluated for a possible tumor--in the simplest possible way, forming the bundle of information into maximum compression with a minimum of noise.
Ways to reduce a data stream
Reducing data to as few building blocks as possible increases the likelihood that the data will appear on the other end of transmission closely resembling the original. Think of a game of telephone, where a sentence
is whispered from person to person, or a document that is to be copied by hand: the fewest words often make for the greatest accuracy.
In practice, this is done by "decomposition," which is reducing the scale of the information. For example, if there was a photo with a large block of white, rather than transmit data about every point of white, including nuances in shading, the data would be decomposed to wavelets which would describe a simplified white block. Through reductions via successive averages, similar portions would register as "0" rather than longer numerical descriptions. Having all these 0s would make it easier to compress the entire data bundle.
"You would have a representation that is close to the original but described with much less memory in the computer," says Daubechies.
In theory, the approach is simple: find ways to reduce a data stream while maintaining the integrity of the information being transmitted. In practice, it is very difficult, requiring highly refined mathematical approaches.
Most fun, most frustrating
If it were simple, it wouldn't hold much interest for Daubechies.
"Figuring out problems," she says, "is the most fun and also most frustrating thing for me. It's baffling. You feel there is something there, feel it in your gut, but don't know where it is.
So you must take it apart, and when you really understand it, it is beautiful. Usually that feeling lasts a couple of days, and then you feel really stupid for not having seen it sooner."
The approach--whether for processing the digital sound of CDs into sound waves or converting the imaging of breast tissue via x-ray to computer screen in a mammogram--requires both elegant mathematics and an understanding of the application in order to get the job done.
"I spend time talking and finding out why they use a particular system," says Daubechies. "Then I think about it and find different mathematical properties."
After understanding and building the math involved with the processing, she feeds it back into the application. "I like to live between what is important for these applications and the mathematics."
Math from physics
It was this desire to inhabit two worlds--both pure and applied mathematics--that brought Daubechies to where she is today. While studying physics at the Free University in Brussels, where she received both her 1975 bachelor's degree and her 1980 doctorate,
she gravitated towards quantum mechanics, trying to abstract the math from physics.
In 1987 she threw herself into the fulltime study of application-oriented problems when she joined the technical staff at AT&T Bell Labs, where she worked for seven years. She still kept one foot in academia during those years, taking leave to spend six months at the University of Michigan and two years at Rutgers as professor of mathematics. In 1993 she joined the faculty at Princeton, becoming the first woman appointed to a tenured position in math.
Daubechies' enthusiasm for sharpening the way we see things extends to students, especially those who wouldn't normally explore mathematics. In 1997 she introduced Math Alive, a course aimed at nonmath and nonscience majors. Her goal was to teach how mathematicians think and how that thinking has been applied to making everyday things work, such as playing CDs and using the Internet.
Based on student interest and feedback, the course has been extremely successful. There are two sessions offered this spring, one taught by Daubechies, the other by Philip Holmes, professor of mechanical and aerospace engineering.
"I enjoyed developing the course," she says, "but the proof of concept is someone else teaching it successfully."
Daubechies also wants time to develop another course, she says, for juniors with a math or engineering background. The course, which she describes as "Math Alive for the Math Aware," will go into the math several layers deeper, sharpening the focus yet again.
top