Beauty may well be in the eye of the beholder. But in the future it could be in the realm of mathematics. Steven G. Krantz, Ph.D., professor and chair of mathematics at Washington University in St. Louis, is working with collaborators on the West Coat to create a system that will make the practice of reconstructive surgery analytical.
The system consists of a new type of three-dimensional scanner and sophisticated graphics software that uses wavelet analysis to analyze a face, recommend procedures, assess the difficulty of those procedures, and evaluate the outcome of the surgery.
Wavelet analysis is a sophisticated kind of harmonic analysis that is integral in analyzing and compressing data -- video, sound, or photographic, for instance --for a wide range of applications.
Krantz is working on the software package, Thomas Lu, Ph.D., an electrical engineer from Petaluma, Calif., developed the scanner. Michael Cedars, M.D., a plastic surgeon from Berkeley, Calif., is the philosophical guiding light of the project and is going to implement the system with his patients.
The system is not intended to replace plastic surgeons; rather, its purpose is to aid the plastic surgeon in making more precise evaluations and procedures in performing plastic surgery. It will help him see his problems in new ways, and to more accurately predict outcomes.
Plastic surgery has been practiced since the days of ancient Egypt, some 2,500 years ago. Many strides have been made in the field since the end of World War II, and especially in the past 25 years. For instance, it's now possible for plastic surgeons to perform a procedure to change the distance between your eyes. In spite of this sophistication, the plastic surgeon essentially relies on his wits when performing a breast lift or nose job.
"The basic elements of plastic surgery are rather primitive," said Krantz. "The primary conceptual tools of the surgeon are experience and intuitive aesthetics. If you read the technical papers of the very best plastic surgeons it's like reading about art appreciation; there is a great emphasis on balance and perspective.
"But the face is a geometric surface and can be analyzed as such . Roughly speaking, that's what we're trying to do."
This software could have many applications outside the field of reconstructive surgery. For example, Krantz and his collaborators have taken a group of adults and a child, scanned in the faces, and matched the child to his parents.
"We hope to be able to predict the attributes of a child of two given parents, to match identical twins, and perform other 'facial recognition' functions," he said.
The capability of the system will be useful in security applications, in anthropology, in aesthetics, and in other as yet unforeseen areas.
Krantz discussed the research in a keynote address given at the Discrete Geometry Conference, held May 16-19, 2002, in Tallahassee, Fla. The conference was sponsored by the National Science Foundation. One of the main themes of the conference was brain mapping, but there were contributors who were interested in all aspects of physiological scanning and modeling.
The mathematics of beauty
Here's the mathematics behind beauty. A patient comes to a plastic surgeon's office for surgery to make him look like a young Paul Newman. His face is scanned with safe white light from the 3-D scanner ; instantaneously it turns the face into a very fine grid that looks like multiple triangles pieced together, representing the shape of the face with all its curves and creases. The grid is compared to a library of idealized faces and one is chosen. Upon this grid, the wavelet mathematics and differential geometric calculations go to work in measuring the nose, cheeks, eye sockets and calculating the difficulty in making changes as well as the global effects of surgery -- how changes in one part of the face, say, the forehead, will impact the shape beneath the eyes, for instance.
The wavelet functions can handle every different feature to any degree of accuracy, said Krantz. It is a geometry designed for each reconstructive surgery problem, allowing the surgeon to model a face by breaking the different parts of the face into units and custom build a new face from those units. Krantz and his colleagues, using wavelets, can actually take two faces and construct a 'difference' face.
"It's like subtracting one face from another to get a new one," Krantz said. "This gives the plastic surgeon a new way of looking at things.
"The hardest part of plastic surgery is trying to predict the outcome. It really takes a year of healing to be able to determine the final result. We think the system can make the predictive powers more accurate."
Beyond plastic surgery, the system could be used someday in security cameras to analyze the faces of burglars and compare them with a data base of known offenders, dispensing with the old store height chart and the manual perusing of hundreds of mugshots at the precinct.
A more provocative application?
"One of our long-term goals is to quantify beauty," Krantz said. "This is unsettling to some people. It is an age-old problem to turn 'beauty' into a quantifiable or analyzable attribute. Our goal is to attach a matrix or a family of numbers to a face and measure how beautiful it is. Certainly symmetry and balance would be parameters. But how do you establish a rubric that resonates with commonly held values?"
Krantz noted that there have been a number of interesting studies performed to categorize beauty. A well-known one took ten exquisitely beautiful faces and mathematically averaged them. The result was one 'plain Jane.'
Thus, the route to a perfect "10" is strewn with some tricky math.
"One would think symmetry would be the dominating parameter, " Krantz said. "But if you consider any number of classic beauties through history, from Greta Garbo, to Sophia Loren to Michelle Pfeiffer, it is often the asymmetries or irregularities that make them beautiful."
Think of a certain facial mole, a missing tooth.
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