Figure 3-1 During the projection of a three-dimensional structure on a two-dimensional plane, information is lost, and it is therefore necessary to collect a plurality of projections at different angles in order to reconstruct the original structure.

The algebraic method

The simplest case is the reconstruction of two-dimensional structures from one-dimensional projections (Figure 3-1). A digitized two-dimensional structure, for example a television picture, can be described as a n.n matrix [f_ij] of size D and cells (i,j) of size d=D/n (Figure 3-2). A projection of the picture at an angle ø is a set of parallel rays (ø,k) which totally cover the picture at the angle ø, and any projection ray can be represented by a n.n matrix (Figure 3-3) where each element a_ij^øk is the fraction of the cell (i,j) which is contained within the ray (ø,k).
The picture matrix and the ray-matrices are easily transformed into vectors (Figura 3-4). More precisely, the picture matrix [f_ij] is replaced by a column-vector [f_z], and the ray matrices [a_ij^øk] are described by row-vectors [a_z^øk] with the transformations: