In this section we give a detailed description of the most important algorithm in 2D tomography. The discrete implementation depends on the scanning geometry, i.e. the way the data is sampled.

This algorithm is essentially a numerical implementation of the Radon inversion formula (1.3). However, a different approach avoiding singular integrals is simpler.
We describe this approach for the *n*D Radon transform

where *f* is a function in and , . Let

be the backprojection operator and let *V*, *v* be functions such that .
Mathematically, is simply the Hilbert space adjoint of the Radon transform *R*.
It is easy to see that

where the convolution on the left hand side is in , while the convolution on the right hand side is a 1D convolution with respect to the second variable:

The idea is to choose *V* as an approximation to Dirac's -function. Then,
is close to *f*. The interrelation between *V*, *v* is easily described in terms of the Fourier transform. Denoting with the same symbol `` '' the *n*D
Fourier transform

and the 1D Fourier transform

we have

see Natterer (1986). By we mean the euclidean length of .

The choice of *V* determines the spatial resolution of the reconstruction algorithm. We use the notion of resolution from sampling theory, see Jerry (1977).
We give a short account of some basic facts of sampling theory. A function *f* in
is said to be band-limited with bandwidth , or simply -band-limited,
if for . An example for *n = 1* is the sinc function

which has bandwidth 1. Obiously, sinc has bandwidth . -band-limited functions are capable of representing details of size but no smaller ones. This becomes clear simply by looking at the graph of sinc.

In tomography the functions we are dealing with are usually of compact support. Such functions can't be strictly band-limited, unless being identically zero. Hence we require the functions only to be essentially -band-limited, meaning that for in an appropriate sense, see Natterer (1986).

A reconstruction method in tomography is said to have resolution if it reconstructs reliably essentially -band-limited functions.

For strictly -band-limited functions we have the following propositions, which hold also, with very good accuracy, for essentially -band-limited functions.

**1.**- If
*f*is -band-limited and (Nyquist-condition), then*f*is uniquely determined by the values*f(hk)*, , and **2.**- If
*f*is -band-limited and , then **3.**- If , are -band-limited and , then

where is a filter with the property

This follows from the formula . This means that for the
filter function *v* we must have

Examples are the ideal low-pass

the filter

and the filter

which has been introduced in tomography by Shepp and Logan (1974). The corresponding
functions *v* for *n = 2* are

for the ideal low pass,

with the same function *u*, and

for the Shepp-Logan filter. More filters can be found in Chang and Herman (1980).

The integral on the right hand side of (2.2) has to be approximated by a
quadrature rule. We have to distinguish between several ways of sampling *g = Rf*.

- Parallel geometry in the plane
- The interlaced parallel geometry
- Standard fan beam geometry
- Linear fan beam geometry
- Fast backprojection

Thu Sep 10 10:51:17 MET DST 1998