Volume 1, Issue 3
Boundaries Unbounded
James Sethian is the author of Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (Cambridge University Press, 1999).
What do inkjet printing, MRI brain scans, and microchip manufacturing have in common? They've all been improved by UC Berkeley professor James Sethian's pioneering work on the mathematics of boundaries.
"The world is filled with dynamic boundaries or interfaces where one thing is separated from another," says Sethian, who is also the head of the Applied and Computational Mathematics Department at Lawrence Berkeley National Laboratory (LBNL). "Oil and water is the most obvious example. The etching process in semiconductor manufacturing is less obvious. I'm in the business of creating mathematical tools that can help formulate and solve all sorts of problems involving dynamic interfaces."
Indeed, Sethian has built a career on looking at the subtle and striking motion of myriad interfaces, from flames dancing in a fireplace to a robot weaving around obstacles. Earlier this year, he was awarded the prestigious Norbert Wiener Prize in Applied Mathematics in honor of his "seminal work on the computer representation of the motion of curves, surfaces, interfaces, and wave fronts, and for his brilliant applications of mathematical and computational ideas to problems in science and engineering."
From the petroleum industry to Hollywood's computer animation studios, Sethian is known for his mathematical approach to battling the boundary problem. Algorithms based on his methodologies already aid oil companies in checking layers of rock for pockets of oil and inform engineers how to design more precise inkjet printers.
Time sequence of a three-dimensional axisymmetric inkjet printhead. The ink bath (at bottom) undergoes a rapid pressure change which forces ink out of the nozzle; the moving ink separates into satellite droplets as it progresses. (courtesy the researchers)
Most numerical techniques for solving "evolving interface problems" use a collection of marker points placed at various locations on the shifting boundary line. Imagine that a handful of buoys are connected by ropes on the surface of a lake. Mathematically representing those boundaries becomes incredibly difficult if the buoys cross, a sharp angle forms between two of the buoys, or multiple strings of buoys combine. Tracking this over time is even more complicated.
On the other hand, Sethian's Fast Marching Methods and Level Set Methods, the latter developed with UCLA professor Stanley Osher, work by overlaying a grid, or coordinate system, on top of the area being studied. In this higher-dimensional perspective, the value of each coordinate changes depending on its distance to the shifting boundary.
"If this were basketball, buoy methods would resemble man-to-man coverage and level set methods provide a zone defense," Sethian says.
With the basic algorithmic research already under his belt, Sethian is now focused on a growing number of applications for Level Sets and Fast Marching Methods. For several years, he has collaborated with semiconductor companies to improve the efficiency and accuracy of their manufacturing. Level Set Methods, he explains, are well-suited to model the etching and deposition processes used to carve out channels in integrated circuits and deposit insulators.
"It's art moved down to the nanometer scale," Sethian says. "It's difficult to dig long trenches, make smooth surfaces, and lay down insulators. This is coupled to equations off the surface of the wafer, in the plasma chamber, for instance, where the rate of the etching is controlled. But it's all moving boundaries."
The human body is also filled with moving boundaries. Some, like tumors, must be carefully tracked if they're to be beaten. Sethian is developing ways that his mathematical techniques can be used to automatically extract "news you can use" from the raw data of medical scans such as magnetic resonance imaging (MRI) and computer-aided tomography (CAT).
This reconstruction of the brain's cortical structure was created by applying the edge-detecting capabilities of Level Set Methods to MRI data. (courtesy the researchers)
"Right now, a physician identifies a tumor in a CATscan of the brain by drawing a line around it on the computer," Sethian says. "It's hard though to detect the noise so the doctor can throw it away and find something meaningful, like if a tumor is getting bigger or smaller."
Sethian, in collaboration with postdoctoral fellow Thomas Deschamps and LBL mathematician Ravi Malladi, have devised an automatic approach for edge detection in medical images. Previous techniques extract the outline, say, of the colon by looking for differences in intensity between neighboring pixels. However, determining the intensity value that signifies one side of a boundary versus the other is tricky. Sethian and his colleagues instead start with a small shape placed on the image in a location that's obviously within the boundaries of the colon. The shape then evolves outwards. The Fast Marching and Level Set algorithms cause the shape to locally slow its expansion at the places where individual pixel intensities begin to change. When it comes to a halt, the boundary has been defined and the colon is "filled in from the inside."
"We can then run a robotic algorithm to figure out the best way for an endoscopic probe to go through without puncturing the sides," Sethian says.
So far, the researchers have demonstrated the potential of their technique for virtual colonoscopies, vascular reconstruction, and brain scans. It's not perfect, Sethian points out, but it's fast, automated, and gets around the pockets of noise. As with all of his research, the key to improving the tools' effectiveness is continued collaboration with the people they're intended for.
"I get excited by teaching this bag of tricks to people who know their particular scientific fields much better than I do," Sethian says. "Hopefully they can then benefit from applying these general mathematical ideas to their own specialized problems."