Plastic surgery to the'nines'

July 22, 2002

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.

Did Cleopatra?

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.

'Perfect 10'

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.
-end-


Washington University in St. Louis

Related Mathematics Articles from Brightsurf:

A new method for boosting the learning of mathematics
How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

Could mathematics help to better treat cancer?
Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

People can see beauty in complex mathematics, study shows
Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Improving geothermal HVAC systems with mathematics
Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

How the power of mathematics can help assess lung function
Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Mathematics pushes innovation in 4-D printing
New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

More democracy through mathematics
For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics
Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

US educators awarded for exemplary teaching in mathematics
Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored
Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Read More: Mathematics News and Mathematics Current Events
Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.