Gordon and Herman have also proposed a variety of formulae which allow one to compute the distance between original picture and reconstructed matrix, and therefore to evaluate the efficiency of a reconstruction algorithm. The ART method, in conclusion, is simple, fast and versatile, which explains why it has become an ideal starting point for the research of a new class of reconstruction algorithms.

The memory matrix

In reconstructions performed with iterative algorithms we find, at each iteration, values that are inferior to the minimum and superior to the maximum, but we have already seen that it is always possible to bring these “illegal” values within the legitimate range.
Let us assume, however, that we want to discover something else about those irregular values, a part from the fact that they do exist. It could be interesting, for example, to find out if their distribution in space is totally random or is following some king of regularity.
In order to answer this kind of questions, we can perform reconstructions by using not only the structure matrix [f_ij] but also an additional matrix [m_ij], of the same size, where we “memorize” the illegal values which appear at each iteration. This allows us to conserve a “memory” of them even when they have been erased from the structure matrix, and for this reason their matrix has been called memory matrix.
The construction of the memory matrix is performed by taking as a starting point a totally “blank” matrix [m_ij^o = 0], and by applying the following operations:

where ß is a parameter which is chosen to represent the presence of an “illegality” in any convenient way.