Colloquia — Fall 2001
Friday, November 2, 2001
Backprojections in X-Ray Tomography, Spherical Functions and Addition Formulas: a Challenge
F. Alberto Grünbaum
Department of Mathematics
University of California, Berkeley
Nagle Lecture Committee
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.