The question is absolutely natural because the vortices appear to have a precise, and often even obvious, meaning. If a negative (or a positive) density value keeps reappearing in the same point for T consecutive times, it is clear that in the original structure that point must be a minimum (or a maximum). But if this is true, it is clearly useless to keep treating that point as an unknown, and we can therefore erase it from the list of the unknowns. The advantage of this operation is obvious: while the number of equations (p·r) remains constant, the number of the unknowns (n²) is decreasing.
If this is confermed, the problem of reconstructing structures from incomplete projections could be solved. The key obstacle, in this problem, is precisely the fact that the number of equations is smaller than the number of unknowns, but if the unknowns are continuously reduced eventually they would reach the same number of the equations, and at that point a complete reconstruction would be guaranteed. As anyone can see, the production of “illegal” density values – which was looking like a structural defect of the algorithm – opens the way to unexpected developments.

Density Modulation

The first algorithm to use memory matrices was presented at Brookhaven’s first international workshop on reconstruction techniques with the name of Density Modulation (Barbieri, 1974). This method recognizes the vortices with formulae (3-7) and (3-8), and then subtracts them from the list of the unknowns. By indicating with N^O_øk and N^M_øk the number of negative and positive vortices that fall in the ray (ø,k), the values of the reconstructed matrix at iteration q+1 are obtained with the following instructions