Seeing it Whole
"We wanted to make the task of visualizing and reconstructing medical shapes easy
for doctors," says Ravi Malladi, who is in the Mathematics Department of the Lab's
Computing Sciences Directorate. "Of course trained physicians and medical technicians
can find the boundaries even in noisy images, and indeed hospitals hire interns to sit in
front of computers and painstakingly click out the edges on series of images. The
challenge is to create a program that can make these decisions in automatic fashion."
Malladi and James Sethian, also of Computing Sciences and a professor of mathematics at UC Berkeley, have developed a fast new way to compute threedimensional models of internal organs and other anatomical features, using data from flat images created by many different medicalimaging techniques. "Now all a physician has to do is click once or twice inside the region of interest and the program will build a model in a few seconds," Malladi says. Among many other models, Sethian and Malladi have made 3D images of organs with shapes as intricate as those of a human brain—and from the same data, the same person's skull; from sonograms they have modeled a fetus in the womb; they have made movies of a pumping heart, relating blood flow in and out of the chambers to the thickness of the heart walls. Their program incorporates two mathematical methods useful in recovering medical shapes, one called "Level Sets" and another known as "Fast Marching;" the underlying mathematical approach is an implicit representation of curves, a form of partialdifferential equations pioneered by Sethian which tracks boundaries as they evolve in space and time. The process begins when the physician uses the computer to plant a visual seed in the
image to be modeled. "Even a single point, represented by a computer mouseclick, can
be thought of as a very small circle or sphere," Malladi says. From that point an
increasingly complex shape starts to grow. Knowing when to stop, however, depends on the
algorithm's ability to recognize edges, which are often hard to read. The trick lies in mathematically adjusting the speed of the growing curve. Fast Marching is a method of approximating the position of curves and surfaces moving under a simple "speed law," which attracts the evolving curve to a boundary and closely relates it to the regularity of the emerging shape—for example by adding a little "viscosity" according to changes in curvature. As the curve advances, it encounters changes in the grayscale values of the pixels in the flat image. Where changes are small from one pixel to the next, the curve moves quickly—the algorithm assumes there is no nearby boundary—but where changes are large, the curve senses a boundary and slows down. Tooabrupt changes in curvature are also smoothed by the calculation. In this way a complex shape can be quickly and accurately bounded and filled in. Level Sets is a method of modeling curves and solids by incorporating an extra dimension—viewing the representation from above, as it were. It's easy to imagine adding a third dimension to a twodimensional curve; it's hard to picture a fourth spatial dimension associated with a volume. Yet mathematically it is no more difficult to use the Level Sets approach with solids. Since the mathematical view is always from one dimension "above" the shape being modeled, a complex 3D model can be quickly constructed, with the propagating boundary curve easily working its way around holes and voids. What's inside and outside a complex solid can be modeled separately—brain and skull, for example. Models constructed at different moments in time can be used to produce movies of 3D shapes in action, such as a beating heart. "One of the first things we set out to do was make cleaner images without destroying essential information," Malladi says, but they have achieved far more. The computer program developed by Sethian and Malladi makes medical images useful to doctors in real time, aiding fast, well informed decisions for effective treatment. Stills and movies of Malladi's and Sethian's anatomical models can be found on the web at http://www.lbl.gov/~malladi. A discussion of implicit representation of curves, also with movies, can be found on the web at http://math.berkeley.edu/~sethian/level_set.html. 

