| 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.
Please direct questions to mthmaster@nosferatu.cas.usf.edu.
Last updated: 26-Oct-2001.
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