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Topic: Mathematicians and Mathematics Behind the Scenes
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Jerry P. Becker

Posts: 13,803
Registered: 12/3/04
Mathematicians and Mathematics Behind the Scenes
Posted: Jan 22, 2000 2:34 PM
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*************************************************
From the Los Angeles Times -- Tuesday, July 14, 1998, Part A Section/Home
Edition
*************************************************

COLUMN ONE Math Whizzes Want Respect in Equation

Mathematicians are the behind-the-scenes workers for Hollywood, shuttle
launches and crime scenes.

By K.C. Cole

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."
---------------------------
K.C. Cole is a Los Angeles Times Science Writer
***************************************************************

Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618) 453-4244
Phone: (618) 453-4241 (office)
(618) 457-8903 (home)
E-mail: jbecker@siu.edu

mailto://jbecker@siu.edu





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