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Mathematics & Statistics

# Colloquia — Fall 2001

## Friday, November 2, 2001

Title
Speaker

Time
Place

Backprojections in X-Ray Tomography, Spherical Functions and Addition Formulas: a Challenge
F. Alberto Grünbaum
Department of Mathematics
University of California, Berkeley
3:00pm-4:00pm
PHY 109
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.