Approximation of ${\bf A^{'}}{\bf A}$ by the sum of its columns

 We now analyze the properties of the approximate inverse that is defined by the substitution of ${\bf A^{'}}{\bf A}$ with a diagonal matrix $\widetilde{{\bf A^{'}}{\bf A}}$,when the diagonal elements of $\widetilde{{\bf A^{'}}{\bf A}}$ are equal to the sum of the corresponding rows of ${\bf A^{'}}{\bf A}$.Notice that ${\bf A^{'}}{\bf A}$ is symmetric, and thus summing over the columns would be equivalent to summing over the rows.

It can be noticed that the diagonal elements of $\widetilde{{\bf A^{'}}{\bf A}}$ are equal to the sum of the columns of the original operator ${\bf A}$as defined in equation (1); i.e., they are equal to the fold computed for each corresponding model trace.

The weights a_lj are interpolator weights, and they fulfill the constraint $\Sigma_{j} \; a_{lj} =1$ expressed in equation (1). It immediately follows that  

$$^{\Sigma}w_{i}^{m} = \left( \Sigma_{l} \; \; a_{li} \right)^{-1},$$ (10)

which is equivalent to equation (6). In summary, we just derived, for the model-space inverse, an approximation that is easy to compute, and it is equal to the fold normalization that was defined by heuristic considerations. However, the definition of the approximate inverse can be applied when more complex imaging operators are involved, and for which we do not have a simple heuristic to define the weights that improve the results obtained by application of the adjoint operator.