Once the weighting factors are calculated, the reconstruction values are obtained by equations (3-3) with simple additions and multiplications. This classic algebraic method, known as Matrix Inversion, is rigorous and straightforward, but in practice it is employed only with small pictures because the weighting factors matrix contains n4 cells, and its dimensions becomes quickly prohibitive with increasing values of n (for a picture with 100x100 cells we would need a weighting factors matrix with 100⁴ = 10⁸ cells).

The theoretical limit

The Matrix Inversion method is not widely used in practice (because the dimensions of the weighting factors matrix are usually prohibitive), but from a theoretical point of view is extremely useful, because it allows us to calculate the minimum number of projections which are required for a complete reconstruction. If we have p projections of a structure, and each projection contains r rays, a reconstruction procedure amounts to solving a system of p·r equations in n² unknowns, and algebra tells us that a solution exists only if the number of linearly independent equations is equal to the number of the unknowns.
The condition that equations are linearly independent is easily understandable, because it amounts to saying that projections obtained at different angles must transport different information (if they didn’t, the total information of the projections would be inferior to that of the original picture and the reconstruction would be impossible).