Considering an imaging operator (for instance ), each row of corresponds to an output bin and each column corresponds to a data trace. Due to the irregular coverage, the columns and rows of are badly scaled and the matrix is ill-conditioned. Its condition can be improved by column scaling ().
Based on a similar approach, we propose two formulations for row and column normalization which we refer to as image normalization and data normalization. They involve pre- and post-multiplying the operator by a diagonal matrix whose diagonal entries are the inverse of the sum of the rows or columns of .
Since Kirchhoff operators are associated with matrices that contain
no negative elements, it is safe to use the sum of the elements.
In case of negative entry values, we can sum the absolute values
of the elements or compute the norms of the rows or columns.
Similarly for the case of complex values, we should use an L2 norm to
compute the diagonal entries of the normalization operator; i.e.,
the square root of the sum of elements squared.