Modern medicine depends on CAT scanners and MRI scanners for the diagnosis of brain tumors and other diseases and on functional MRI and PET scanners for determining normal metabolism. Each of these medical technologies relies on basic and elegant mathematics which we will study in detail both rigorously and via computer simulations.
CAT scanners depend upon the mathematical theorem (proven by Radon in 1917) that a density, f(x,y), in two-dimensional (x,y) space is determined uniquely if one knows the ``X-ray projection'' of f in every direction. The vertical X-ray projection is just the integral of f(x,y) over y with x held fixed. The conditions on f for Radon's theorem to hold are minimal: the Radon transform or X-ray projection, Pf, simply has to be defined. On the other hand, if one is measuring the Radon transform of f, then one knows Pf only approximately and not for all X-ray projections, but only for a finite sample; this makes things interesting.
MRI, fMRI scanners, and PET scanners also depend on elegant mathematics and the basic ideas will be developed in lectures. Participants can maximize what they can get from the conerence if they will not only study the rigorous theory but also write the engineering algorithms used in medical scanners. Before the start of the conference, participants will be strongly encouraged to duplicate two simple computer programs. The first program is the basic 50-line program used in all CAT scanners (participants will get as much help as needed in doing this), the so-called convolution-back-projection algorithm. Once they have this program written they will make sure that it is correct by using it to ``reconstruct'' the density f(x,y) of an artificial model they will choose. The ``measured values'' of the X-ray projection of the chosen density will be computed by a second, data-generation program, that they will write (again with help). The original density and the reconstructed density will then be displayed and compared. They will then be able to see how much the fact that the Radon transform has only been finitely sampled has distorted the original image. After this experiment, participants will become familiar with the field of imaging which depends upon and exploits the abilty to visualize an entire array of numbers in parallel. They will also be ready to understand the theory behind the algorithims and behind PET, MRI, and fMRI.
L. Shepp will discuss CAT and PET in the first half of the conference; M. Lindquist will discuss MRI and fMRI in the second half.