In the simplest case, let us consider an object whose attenuation coefficient with respect
to *X*-rays at the point *x* is *f(x)*. We scan the cross section by a thin *X*-ray
beam *L* of unit intensity. The intensity past the object is

This intensity is measured, providing us with the line integral

The problem is to compute *f* from *g*.

In principle this problem has been solved by Radon (1917). Let *L* be the straight line where and .
Then, (1.1) can be written as

*R* is known as the Radon transform. Radon's inversion formula reads

where is the derivative of *g* with respect to *s* and . In principle, (1.3) solves our problem. So,
why do we write an article on tomography?

First, inversion formulas such as (1.3) do not exist in all cases. For instance, in emission tomography, the mathematical model involves weighted line integrals, which in general do not admit explicit inversion. Also, even if explicit inversion is possible, it is not obvious how to turn an inversion formula such as (1.3) into an efficient and accurate algorithm. Many problems concerning sampling and discretization arise. Often not all of the data in an explicit inversion formula can be measured. Finally, (1.1) is a prime example for many imaging techniques, and a proper understanding of the inversion of (1.1) is a necessary prerequisite for the understanding of more complicated problems.

Thu Sep 10 10:51:17 MET DST 1998