# ILRS Reference Point Determination Using Close Range Photogrammetry
## 2 Mathematical Background
### Bundle Adjustment  
The functional model of the bundle adjustment relates the planar image coordinates \(( \begin{array}{cc}x_{i}^{\prime}&y_{i}^{\prime}\end{array} )^{\rm T}\) to the spatial object point coordinates \({\bf P}_{i}=( \begin{array}{cc}X_{i}&Y_{i}&Z_{i}\end{array} )^{\rm T}\) using the well-known collinearity equations ([16] p. 354), i.e.,  
\[\left( \begin{array}{c}x_{i}^{\prime}\\ y_{i}^{\prime}\end{array} \right)_{j}=\left( \begin{array}{c}x_{0}^{\prime}\\ y_{0}^{\prime}\end{array} \right)+\left( \begin{array}{c}x_{i}^{*}\\ y_{i}^{*}\end{array} \right)_{j}+\left( \begin{array}{c}\Delta x_{i}^{\prime} \\ \Delta y_{i}^{\prime}\end{array} \right)_{j}, \tag{1}\]  
where  
\[x_{i}^{*} =-c\frac{r_{11}(X_{i}-X_{0}^{\prime})+r_{21}(Y_{i}-Y_{0}^{ \prime})+r_{31}(Z_{i}-Z_{0}^{\prime})}{r_{13}(X_{i}-X_{0}^{\prime})+r_{23}(Y_ {i}-Y_{0}^{\prime})+r_{33}(Z_{i}-Z_{0}^{\prime})}, \tag{2}\] \[y_{i}^{*} =-c\frac{r_{12}(X_{i}-X_{0}^{\prime})+r_{22}(Y_{i}-Y_{0}^{\prime} )+r_{32}(Z_{i}-Z_{0}^{\prime})}{r_{13}(X_{i}-X_{0}^{\prime})+r_{23}(Y_{i}-Y_{0 }^{\prime})+r_{33}(Z_{i}-Z_{0}^{\prime})}. \tag{3}\]  
The principal distance is \(c\), and \(x_{0}^{\prime},y_{0}^{\prime}\) are the coordinates of the principal point. The distortion parameters are denoted by \(\Delta x^{\prime}\), \(\Delta y^{\prime}\). This set of parameters is referred to as interior orientation parameters. The so-called exterior orientation parameters are the \(j\)th spatial position \(( \begin{array}{cc}X_{0}^{\prime}&Y_{0}^{\prime}&Z_{0}^{\prime}\end{array}  )_{j}^{\rm T}\) and orientation \({\bf R}_{j}\) of the camera. The matrix  
\[{\bf R}=\left( \begin{array}{ccc}r_{11}&r_{21}&r_{31}\\ r_{12}&r_{22}&r_{32}\\ r_{13}&r_{23}&r_{33}\end{array} \right)\]  
is an orthogonal rotation matrix, i.e., \(\det{\bf R}=+1\) and \({\bf R}{\bf R}^{\rm T}={\bf R}^{\rm T}{\bf R}={\bf I}\), and describes the spatial rotational sequence from the image frame to the object frame ([16] p. 281). The distortion parameters \(\Delta x_{i}^{\prime}\), \(\Delta y_{i}^{\prime}\) compensate for the radial-symmetric lens distortion and the decentring distortion. Beside noncausality models, which compensate for the effects without specifying the physical causes, causal models are used in photogrammetric applications, which model effects due to physical interactions ([17] p. 505). The [PERSON] [18,19] approach is a causal model frequently applied in close range photogrammetry and, therefore, is used in this investigation. The radial-symmetric lens distortion is characterized by the polynomial function  
\[\Delta r_{\rm sym,i}^{\prime}=A_{1}r_{i}^{\prime 3}+A_{2}r_{i}^{\prime 5}+A_{3}r_{i}^{ \prime 7},\]  
which is proportionally applied to the coordinate components, i.e.,  
\[\Delta x_{\rm sym,i}^{\prime} =x_{i}^{*}\frac{\Delta r_{\rm sym,i}^{\prime}}{r_{i}^{\prime}}, \tag{4}\] \[\Delta y_{\rm sym,i}^{\prime} =y_{i}^{*}\frac{\Delta r_{\rm sym,i}^{\prime}}{r_{i}^{\prime}}, \tag{5}\]where \(A_{1}\), \(A_{2}\), and \(A_{3}\) are the polynomial coefficients, and \(r_{i}^{\prime}=\sqrt{{x_{i}^{*}}^{2}+{y_{i}^{*}}^{2}}\) denotes the \(i\)th radial distance [19]. The decentring distortion with parameters \(B_{1}\), \(B_{2}\) is given by [18; 19]  
\[\Delta x_{\text{asy},i}^{\prime} = B_{1}\left(r_{i}^{\prime 2}+2{x_{i}}^{2}\right)+2B_{2}{x_{i}^{*}} _{i}^{y}, \tag{6}\] \[\Delta y_{\text{asy},i}^{\prime} = B_{2}\left(r_{i}^{\prime 2}+2{y_{i}}^{2}\right)+2B_{1}{x_{i}^{*}} _{i}^{y}. \tag{7}\]  
Moreover, further parameters \(C_{1}\) and \(C_{2}\) concerning the affinity and shear components of the image plane can be taken into account ([16] p. 179f), i.e.,  
\[\Delta x_{\text{aff},i}^{\prime} = C_{1}x_{i}^{*}+C_{2}y_{i}^{*}, \tag{8}\] \[\Delta y_{\text{aff},i}^{\prime} = 0. \tag{9}\]  
However, for modern cameras, such parameters are rarely significant and can be neglected without losing accuracy [20]. Substituting Equations (4)-(9) into Equation (1) yields the functional model of the bundle adjustment, where the total correction term reads ([16] p. 180)  
\[\left(