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X-Ray
Tomography and Transport Theory
(Guillaume
Bal, Steve
McDowall, and Gunther
Uhlmann). The first
topic of this minicourse is the
study of the X-ray transform in
two dimensions (or Radon
transform) arising in medical
imaging, in particular in Computed
Tomography (CT), and many other
fields. X-ray tomography is one of
the basic inverse problems and
consists of determining the
density of tissue by measuring the
attenuation of X-rays passing
through the body. The measurements
are modeled by the X-ray transform
and the inverse problem is to
invert this transform.
The second topic
is the mathematical study of the
attenuated X-ray transform arising
in medical imaging in particular
in Single Positron Emission
Tomography (SPECT). A patient is
given a pharmaceutical labeled by
a radionuclide, which emits
photons. The goal is to recover
the function that gives the
distribution of the radiation
sources. The measurements are
modeled by the attenuated X-ray
transform and the inverse problem
is to invert this transform.
Both the X-ray
transform of CT and and the
attenuated X-ray transform of
SPECT are merely reflections of a
deeper mathematical object, the
so-called radiative transport
equation. This equation also
handles many other problems, for
instance optical tomography. This
inverse boundary problem consists
of reconstructing the absorption
and scattering coefficient of an
inhomogeneous medium by probing it
with diffuse light. The problem is
modeled by the linear Boltzmann
equation. The third and final
topic of the class will be to
study a direct problem and the
corresponding inverse problem
associated with this equation.
During the
afternoon the participants will
have problem sessions and Matlab
sessions on simulation and
inversion of X-ray transforms and
direct and inverse transport
theory.
Finite Volume
Methods and the Clawpack
Software (Randall
LeVeque and Donna
Calhoun) This
minicourse will provide a
concentrated introduction to the
theory and application of
hyperbolic partial differential
equations, a broad class of
equations that model wave
propagation problems arising in
nearly all fields of science and
engineering. Applications include
ultrasound, seismic waves, shock
waves, tsunamis, detonation waves,
and traffic jams. Solving inverse
problems in medical or seismic
imaging often requires accurate
techniques for solving the forward
wave propagation problem, often
defined by a hyperbolic system.
The minicourse will also provide a
hands-on introduction to the
software package Clawpack,
which implements a popular class
of numerical methods for solving
such problems, incorporating
adaptive mesh refinement for the
efficient solution of
multidimensional problems
Please
direct questions or comments about
the IPDE Summer School to
ipdemail@math.washington.edu.
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