If exact inversion formulas are not available, iterative methods are the algorithms of choice. But even if exact inversion is possible, iterative methods may be preferable due to their simplicity, versatility and ability to handle constraints and noise.

Iterative methods are usually applied to discrete versions of the reconstruction problem.
These discrete versions are obtained either by starting out from discrete models - as in the
EM algorithm below - or by a projection method - in the tomographic community ``series expansion method'', Censor (1981). This means that the unknown function *f* is written as

with certain basis functions . With the *i*-th measurement, the measurement process
being linear, we obtain the linear system

for the expansion coefficients , the matrix elements being the *i*-th measurement for the Basis function . In tomography we always have . Also, the matrix is typically sparse. Often is the characteristic
function of pixels or voxels. Recently smooth radially symmetric functions with small support
(the ``blobs'' of Lewitt (1992), Marabini et al. (1998)) have been used. Blobs have several advantages over pixel or voxel based functions. Due to the radial symmetriy it is easier
to apply the Radon transform (or any of the other integral transforms) to , making it easier to set up the linear system (4.1). The smoothness of the prevents the ``checkerboard'' effect (i.e. the visual appearance of the pixels or voxels in the
reconstruction) and does part of the necessary filtering a smoothing.

The linear system (4.1) may be overdetermined *(M>N)* or underdetermined *(M<N)*, consistent or inconsistent. Useful iterative methods must be able to handle all these cases.

- ART (algebraic reconstruction technique)
- EM (expectation maximation)
- MART (multiplicative algebraic reconstruction technique)

Thu Sep 10 10:51:17 MET DST 1998