Los Angeles Times

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Tuesday, July 14, 1998

Math Whizzes Want Respect in Equation
Mathematicians are the behind-the-scenes workers for Hollywood, shuttle launches and crime scenes.
By K.C. COLE, Times Science Writer

When Tom Hanks was filmed walking out onto the tarmac in the movie "Apollo 13," the weather was cloudy. The director later decided he wanted the sky clear. Who made the sun shine?
When the creators of a Coke commercial for use in China wanted to show a giant panda sipping their product, who helped them make the brown liquid slosh realistically?
When the government needed to test nuclear weapons without violating the test ban treaty, who figured out a way to do it?
Who are you going to call when faced with any of these problems? And many more besides?
The surprising answer is: a mathematician.
But like plumbing and electric wiring that lies unheralded behind the walls in the house, mathematics stays well out of sight. And that makes mathematicians cross.
"Math never gets into the story," complains UCLA math department Chairman Tony Chan. "Everyone else gets the credit."
The hidden machinations of mathematicians drive virtually all aspects of modern society, from heavy industry to entertainment, from health care to sports. But you won't see them mentioned in the credits.
They are the behind-the-scenes fixer-uppers on Hollywood sets, shuttle launches, at crime scenes, for stock market transactions, real estate deals, medical diagnoses, cell phone conversations. But you won't get their bill.
"Even people in science don't know what we do," said Stanley Osher, another UCLA mathematician.
Chan heads a department known for its ability to solve hands-on problems, applying mathematical truths to the real world. They do everything from sharpening fuzzy images for police and astronomers to designing new materials from scratch.
Surprisingly, the mathematician accomplishes all these diverse tricks with a comparatively small set of tools--rather like a skilled carpenter who can create an elaborately carved dollhouse or rebuild a garage with hammer, saw, lathe, nails, sandpaper and shellac.
Whether the problem is simulating weather or seeing stars, mathematicians always begin by turning things into numbers and relationships into equations. And that requires a clear understanding of the situation at hand. Sometimes, they get help from unexpected places.

Life Around the Edges

Take the strange case of the rose tattoo that played a pivotal role in the Reginald Denny case. During the uprising that followed the Rodney G. King verdicts in Los Angeles, the beating of truck driver Denny was captured on videotape. The image revealed a speck on the arm of a man who threw a brick at Denny but not much more. Mathematical manipulation was able to resolve the speck into a rose tattoo, leading to the capture of the suspect.
How did the mathematicians do it?
UCLA's Osher at the time was a partner at Cognitech, a company that specializes in image enhancement. The key to successfully describing the problem in mathematical terms came when Osher recognized that it was exactly the same as another, already familiar, problem: edge detection.
More than anything, he realized, making out the details of fuzzy images requires pinning down the shape and location of edges--whether they be edges of petals, stems, clouds, lips, roofs, mountains or planets. An edge is a place where things change abruptly.
Physicists had already developed equations to deal with shock fronts--the "edge" between two things moving at different speeds. Waves pounding on beaches, airplanes breaking the sound barrier, and sudden slow-downs on freeways that lead to traffic tie-ups are all traveling shock fronts that can be described with the same equations.
"When you perceive images, you're really perceiving edges," Chan explained. So Osher borrowed familiar shock front equations and applied them to imaging. Such borrowing is common, explained Chan. "You use analogies from other areas to solve problems in your own area."
Once Osher sees the analogy, he said, "I can draw on 300 years of experience--ever since Newton invented calculus."
Doug Robles of Digital Domain, the Venice-based company that produced the simulations for "Titanic," "Apollo 13," "'Interview With the Vampire" and other films, also relies on ancient equations--such as when he made the Coke slosh around realistically in a bottle for the China commercial.
"I'm stealing from a hundred years of fluid dynamics equations, and I get paid for it," he said with a grin. Sometimes, however, coming up with the right mathematical description isn't always that simple. "Sometimes we borrow from Newton, but sometimes we have to invent it ourselves," Chan said. To create realistically moving monsters with realistic-looking hair, for example, requires the mathematicians to start from scratch.
Once they have the detailed mathematical description, called a "model," the next step is to get the model to tell them what they want to know. For that, they need to invent the right algorithm to transform the numbers and equations into useful solutions.
While "algorithm" seems like a term only a mathematician could love, it's actually something everyone uses every day. An algorithm is nothing more than a set of steps for doing a task. It's fair to say that nearly everyone over the age of 6 has learned the algorithm for tying one's shoes: First, cross the left lace over the right. Then make a loop in the right lace. Cross the loop behind the left lace and through the hole. Pull.
We use algorithms to pump gas, bake a cake, ride a bike, brush our teeth. And we invent new ones every time we figure out a better way to accomplish a task.
An algorithm in a computer tells the electronics what steps to take to transform the input into something useful.
"I write the algorithms that make the computer sing," said Osher. "I'm the Barry Manilow of mathematics. The algorithm takes you from the equations to the solution."
Osher's algorithm, based on Newton's equations, told the computer to take the information in the fuzzy dot (the tattoo) and focus on places of stark contrast between bright and dark--that is, edges. Presto! A rose appeared out of the fuzz, and the brick thrower was eventually identified and convicted.
The brain uses similar algorithms to transform the information from waves of compressed air crashing on our eardrums into voices and music. The eye turns photons of light into people and things.
"Physiology has evolved over millions of years, so it's found the best way to handle a lot of things," said UCLA mathematician Russel Caflisch. "The right way to understand sound is with waves, and Fourier analysis [a way of breaking up waves into their mathematical components]. That's what the ear does. The best way to understand images is edge detection. And that's what the eye does."

Reducing Everything to Numbers

Other algorithms can help a movie director--or the Defense Department--simulate an explosion. Mathematicians start with the simple Newtonian equations that explain how things fall and move in space. Then they factor in externals, such as wind and air pressure. They approximate how the motion changes by tiny, discrete steps, because that's how digital computers work.
All these steps transform the numbers that stand for velocity, position, temperature, stock price, location of an edge, brightness, whatever. After the numbers are graphed on a computer, the numbers are made to correspond to images on a screen--images that can be rotated, shrunk, enlarged and altered in myriad shape-shifting forms.
The process is similar to the way the human eye and brain change discrete, abstract photons of light into different versions of the same scene: the street light that becomes, on closer inspection, the moon, or the shadow in the corner that turns out to be your cat.
And just as the brain can be fooled, so can the mathematician. Sometimes, they discover that their solution doesn't work. "The model can be wrong," Chan said. "The computer can fool you."
"The most innocent looking equation has hidden complexities," Osher said. "[If you're doing a climate simulation], you can predict snowflakes in the Sahara."

The Days of 'Deconvolution'

Once they have a plausible solution, the question is: Is it right? Does it accurately represent whatever it is they're trying to represent?
Often, that's a matter of taste, or judgment. Do the vampire teeth look realistic? Does the CAT scan image look like a brain or tumor ought to look? Does the starlike object look fuzzy because it's out of focus? Or is it really a fuzzy object?
Chan runs into this problem constantly. One of his specialties is what mathematicians call "deconvolution." Basically, it means taking a blurry image and putting it into focus using clever algorithms. Taking out the blur is a different mathematical challenge than taking out the "noise"--say, ferreting out a faint signal in a loud background.
Human beings are quite good at eliminating noise using their own built-in perceptual apparatus. Most people can maintain a phone conversation even when other people are talking around them.
But blurring is harder to fix. "Humans can denoise things very well, but we can't deblur," Chan said. "That's why we wear glasses."
Or hire mathematicians.
Chan calls up a computer image depicting the planets Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune. Before deblurring, only the last four--the gas giants--are visible. Saturn has no clear ring and Jupiter has no distinctive atmospheric stripes. In the "deconvoluted" image, however, the small rocky planets suddenly appear in the image, and the gas giants take their more familiar form.
Of course, when the image involves something of unknown form--say some never before seen object in space--how does one know what it's supposed to look like?
"You cannot decide without a priori information," Chan said. "The astronomer has to decide [what the object really looks like]. The mathematician gives you different versions to choose from."
Sometimes, the ultimate test of the mathematics is reality itself. Mathematicians know the model and algorithms work when the solutions they produce are useful in real life.
For example, Caflisch is trying to create a virtual computer world that mimics high-tech materials used in communication satellites. It's critical that these materials be as perfect as possible because small imperfections can garble communications the way a bad phone connection garbles words. For the information highway to transport signals efficiently, it has to be smooth and free of potholes. The materials Caflisch is trying to perfect is the "asphalt" that goes into making those roads.
Why bring in a mathematician to make materials?
Since it's not possible to actively monitor the making of the material, the next best thing is to build a replica inside a computer.
Engineers can't directly monitor the quality of the materials as they are made the way inspectors can monitor the quality of cars coming off an assembly line. For one thing, the crystals are too small to see with the naked eye--only 50 atoms thick. For another, they are created under extreme conditions--high vacuums and searing temperatures. Even if an instrument could watch the crystals as they develop, the microscope itself would disrupt the growth process.
"There comes a point where you just can't do the experiments you need to get the data you want," Caflisch said. "But with mathematics, you can work in places where you can't get real experimental results."
(It's the same approach the U.S. Defense Department has taken to test nuclear weapons without violating the test ban treaty. They set off "virtual" nuclear bombs inside the computers that tell them as much as the real thing.)
Once the model is working inside the computer, Caflisch can adjust the "temperature" and other manufacturing variables to get perfect "pothole-free" crystals.
"Instead of doing it in a lab, he can sit there at a computer and try doing it different ways, and check in a microsecond which works best," said Mark Greene, another UCLA mathematician.

Making Movie Magic

So far, Caflisch and his collaborators at Hughes Research Laboratories have received almost $10 million from the National Science Foundation and the Defense Advanced Research Projects Agency for the work, which could have broad applications for satellites, cell phones and almost any communications device that relies on precision materials.
But the idea of simulating what you can't get at directly is much broader than that. CAT scans and other methods allow physicians to "see" the insides of patients. Mathematical tools, says Greene, allow them to look at heart valves "without ripping out my heart."
In contrast, Doug Robles' work in movies involves almost pure fantasy. "What we're trying to do is make things look real," he said. "And that takes physics and math."
The mathematicians, programmers and artists at Digital Domain work together to figure out how to remove or put things into scenes (the stars shining over the Titanic at night, for example), or how to create realistic sequences from scratch (most of the launch sequence on "Apollo 13" was created inside a computer). "That allows the director the ultimate freedom," he said.
Recently, Robles showed off a sequence from "Dante's Peak" for the mathematicians at UCLA. "See that mountain?" he said. "It's fake, fake, fake."
His biggest challenge is creating realistic hair, clothes, muscles, gas, fire, water. "Explosions and floods are expensive and dangerous," he said. Better to do it with math--linear algebra, calculus, fractals, wavelet technology.
"We want to be able to have a creature stomp into a puddle of water and have it look right," he said.
After his talk on the mathematics in the movies, Robles turned to his audience of mathematicians and made his final pitch--as usual, asking the all-purpose handy guys for help. It wasn't the kind of plea that people normally expect to hear in Hollywood:
"If you guys have good fast differential equation solvers," he said plaintively, "I'd sure love to have them."

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