Reduced normal matrix for solving fast photogrammetric bundle adjustment

Reduced normatrix for photogrammetric bundle adjustment based on the collinearity equations

Note: There are $m_1$ photos and $m_2$ object points, and the number of EOPs is 6. If 6*$m_1$ is less than 3*$m_2$, solve unknowns related with EOPs. If not, solve object points coordinates, first. This post shows only the first case, 3*$m_2$ > 6*$m_1$.

[Fig. Normal matrix sample]

$$ \begin{pmatrix} N_1 & N_{12} \\ N_{12}^T & N_2 \end{pmatrix}\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} = \begin{pmatrix} C_1 \\ C_2 \end{pmatrix} $$ $$ N_1 X_1 + N_{12} X_2 = C_1$$ $$ N_{12}^T X_1 + N_2 X_2 = C_2$$ In case $6m_1<3m_2$, let's slove $X_1$ first.
$$ X_2 = N_2^{-1} C_2 - N_2^{-1} N_{12}^T X_1$$, therefore
$$ (N_1 - N_{12} N_2^{-1} N_{12}^T) X_1 = C_1 - N_{12} N_2^{-1} C_2$$ $$ X_1 = (N_1 - N_{12} N_2^{-1} N_{12}^T)^{-1}(C_1 - N_{12} N_2^{-1} C_2)$$ And
$$ X_2 = N_2^{-1}(C_2 - N_{12}^T X_1) $$ For $X_2$, we can avoid solving the inverse matrix of $3m \times 3m$ .
For $i_{th}$ point,
$$ X_{2_i} = N_{2_{i}}^{-1}(C_{2_i} - N_{12_i}^T X_1) $$