Implementation

The observation equations are given by the polynomial model (y = A x):

$$\left[ \begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_N \\ \end{array... ...} \\ \alpha_{01} \\ \alpha_{20} \\ \vdots \\ \alpha_{0d} \\ \end{array} \right]$$ (T2)

Where:
y contains the observed offsets in a certain direction.
$l_i$ denotes the location (line number) of the observed offsets in a certain direction.
$p_i$ denotes the location (pixel number) of the observed offsets in a certain direction.
$\alpha_{lp}$ denotes the unknown coefficients of the polynomial.

The data is rescaled (to the interval [-2, 2], see Annex D) so the normalmatrix is rescaled. otherwise there could occur very high values for, e.g., $l^d=25000^5$. The least squares parameter solution is given by:

$$A^T Q_y^{-1} y = A^T Q_y^{-1} A x = N x$$ (T3)

Where:
$Q_y^{-1}$ is the (diagonal) covariance matrix of the observations. this matrix can be equal to identity or to the correlation values in version 1. (CPM_WEIGHT card).

The coefficients are estimated by factorization of the matrix N.

The inverse of matrix N is also computed to check the solution (stability) and to compute some statistics.

A check number is given ( ${\rm\max(abs(}N N^{-1} - I))$) that gives a hint on the stability of the solution.