Led by Annie Cuyt, a professor at the
University of Antwerp and a member of the Royal Flemish Academy of
Belgium, the project combines a broad range of research and training
activities using multi-exponential analysis with applications in
industry. The EU grant of over $900,000 (in U.S. dollars) covers
research from 2021 to 2025.
Multi-exponential analysis has been described as ubiquitous in
mathematics, as almost all functions use the exponential function ez as a
building block. In that function, “e” is a constant known to
mathematicians as the Euler number, and z is a complex number. When z is
restricted to be the real numbers x, the function e-rx is said to
decrease or decay exponentially at rate r. In many applications, the
goal is to identify the exponential decay rates hidden in the given
data, which is usually a weighted sum of exponential functions with
various decay rates.
In simple terms, Ou said, multi-exponential analysis can be used to
produce a curve from a given set of data points and to find the
increase/decay rates hidden in the curve. For example, to show the
change in air temperature over a day, a person could take a thermometer
reading every hour, plot the numbers on a graph and draw lines
connecting the points. But even if the temperature is taken every
minute—or every second—the graph will still consist of a series of
straight connecting lines, even if each one is very short. To produce a
smooth curve, multi-exponential analysis can be applied to the data set,
in a procedure called the Prony’s interpolation.
In Ou’s own research on osteoporosis, she has used this type of
mathematical analysis to study how the deterioration of the
microstructure of porous materials, such as the osteoporotic bones, can
be quantified using ultrasound data.
After several years of work, she found the mathematical link between
the bone microstructure and the equations modeling ultrasound wave
propagation.
“There is something very complicated about your bones, so that waves
sent through the wrist bone present a difficult problem to solve,” she
said. “To understand the progression of osteoporosis, you can’t just
eyeball the x-ray images of bones. You need a mathematical way to look
at it.”
She has also been working with colleagues at NIH to monitor the
progression of Alzheimer’s disease by applying multi-exponential
analysis to the nuclear magnetic resonance (NMR) data.
“Multi-exponential analysis might sound remote, but it touches our
daily lives in many surprising ways, even if most people are unaware of
how important it is,” the EXPOWER project says on its website. “For
example, a substantial amount of effort in signal processing and time
series analysis is essentially dedicated to the analysis of
multi-exponential functions. Multi-exponential analysis is also
fundamental to several research fields and applications … [including]
remote sensing, antenna design, digital imaging, testing and metrology,
all impacting some major societal or industrial challenges such as
energy, transportation, space research, health and telecommunications.”
Ou and other mathematicians working with exponential analysis often
conferred over the years, particularly because the field and its
research problems are so interdisciplinary.
“We all work in different areas, but when we began coming together
for conferences and meetings, we realized this was a big topic and a
growing area of interest,” she said. “That’s when we decided to apply
for the grant.”
A meeting is scheduled for May 2022 at ASTRON, the Netherlands
Institute for Radio Astronomy, with participants hoping not only to
collaborate but also to bring new young researchers into the field.
“This work takes many people and many different tools,” Ou said. “We need as many different perspectives as we can get.”
More information about the project and the grant can be found at the EXPOWER website.
Article by Ann Manser
Published Feb. 10, 2022
