It is instructive to compare RCM with the Algebraic Reconstruction
Method (ART) [9] traditionally used for many tomographic
reconstruction applications. Both ART and RCM are iterative
algorithms that can be used to solve the matrix equation
(1). Both ART and RCM break the problem into smaller, more
easily solved pieces to obtain an approximate solution, then iterate
to improve the approximation. The difference between ART and RCM is
in the way in which the matrix A is broken into pieces. In the
following discussion, the notation 
will be used for
vectors to emphasize the difference between scalars and vectors.
If A is divided into rows
and b is divided into elements
then the ART algorithm is obtained. The ART algorithm can be written as
where the i in 
and 
is interpreted mod N.
On the other hand, if A is divided into columns
and x is divided into elements
then the RCM algorithm is obtained. This version of RCM is one in which each submatrix of A is a single column of A. The RCM algorithm can be written as
where the i in 
and 
is interpreted mod M.
Equations 25 and 28 are very similar in form, but quite different in practice. Equation 25 is a vector equation, whereas equation 28 is a scalar equation. On the other hand, the quantity in parenthesis in 25 is a scalar quantity, whereas the quantity in parenthesis in 28 is a vector quantity. For ART, the previous solution is projected onto a hyperplane defined by a row of A. For RCM, the previous residual vector is projected onto a column of A. When the system matrix is sparse, ART takes advantage of the sparsity of the system matrix. When the system matrix is not sparse, but contains some column oriented structure, RCM takes advantage of the structure of the system matrix.
