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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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