Colloquia — Fall 2001
Friday, November 2, 2000
| Title |
Backprojections in X-Ray Tomography, Spherical Functions and
Addition Formulas: a Challenge |
| Speaker |
Professor F. Alberto Grünbaum
Department of Mathematics
University of California, Berkeley |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Nagle Lecture Committee |
Abstract
One of the most numerically efficient algorithms in parallel beam X-ray tomography
(with arbitrary directions) depends crucially on certain properties of special
functions like the Gegenbauer polynomials. I will give an abinitio discussion
of this material and show how the crucial property here is exactly the definition
of spherical functions for a symmetric space G/K where
G is a Lie group and K a compact subgroup of it. This
property of the Gegenbauer polynomials (which depend on a continuous parameter)
holds even when there is no group around. The Gegenbauer polynomials are the
spherical functions when G = SO(n + 1) and
K = SO(n). In this case G/K
is the usual n dimensional sphere.
The property in question is a consequence of what is called an “addition formula,”
i.e., an extension of the property exp(x + y) = exp(x)
exp(y). It would be nice to find a use for this formula in tomography,
and this remains as a challenge.
I will discuss the case of “fan beam tomography”, the one found in present
day hospital machines and discuss the (apparent) failure of a similar mathematical
treatment in this case.