# A Method for Efficient Quality Control and Enhancement of Mobile Laser Scanning Data
## Appendix B Solution of a Constraint Gauss-Helmert Model  
Our Jacobian matrix \(\mathbf{A}\) contains derivatives with respect to the unknown parameters. These include three translation and four rotation parameters. The number of rows in \(\mathbf{B}\) are three times the number of points. We compute derivatives with respect to all three coordinate components, arrange them in the diagonal, and append the partial matrices horizontally. The matrix grows quadratically, which is why it is necessary to treat \(\mathbf{B}\) as a sparse matrix for large point clouds. Matrix \(\mathbf{C}\) is a one row matrix, and its entries are derivatives of the constraint with respect to the unknown parameters.  
\[\mathbf{A}_{[n\times 7]}=\left[\begin{array}{cccccccc}\frac{ \partial f}{\partial\partial_{0}}&\frac{\partial f}{\partial\partial_{1}}& \frac{\partial f}{\partial\partial_{2}}&\frac{\partial f}{\partial\partial_{3}} &\frac{\partial f}{\partial\partial_{x}}&\frac{\partial f}{\partial\partial_{ y}}&\frac{\partial f}{\partial\partial_{z}}\end{array}\right]\Big{|}_{x_{0},\,v0} \tag{10}\] \[\mathbf{B}_{[3n\times 3n]}=\left[\begin{array}{cccccccc}\frac{ \partial f}{\partial v_{1}}&\dots&0&\frac{\partial f}{\partial v_{1}}&\dots&0& \frac{\partial f}{\partial v_{1z}}&\dots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&\dots&\frac{\partial f}{\partial v_{m}}&0&\dots&\frac{\partial f}{\partial v _{m}}&0&\dots&\frac{\partial f}{\partial v_{m}}&\end{array}\right]\right|_{x_{ 0},\,v0}\] \[\mathbf{C}_{[1\times 7]}=\left[\begin{array}{cccccccc}\frac{ \partial c}{\partial\partial_{0}}&\frac{\partial c}{\partial\partial_{1}}& \frac{\partial c}{\partial\partial_{2}}&\frac{\partial c}{\partial\partial_{3}} &\frac{\partial c}{\partial\partial_{x}}&\frac{\partial c}{\partial t_{y}}& \frac{\partial c}{\partial t_{z}}\end{array}\right]\Big{|}_{x_{0}}\]  
We find the solution to the constraint GHM model by solving the linear equation system  
\[\left[\begin{array}{cc}-\left(\mathbf{A}^{T}\big{(}\mathbf{B}\mathbf{Q}_{l} \mathbf{B}^{T}\big{)}^{-1}\mathbf{A}\right)&\mathbf{C}^{T}\\ \mathbf{C}&\mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ \mathbf{k}_{c}\end{array}\right]=\left[\begin{array}{c}-\mathbf{A}^{T}\big{(} \mathbf{B}\mathbf{Q}_{l}\mathbf{B}^{T}\big{)}^{-1}\mathbf{w}_{m}\\ \mathbf{w}_{c}\end{array}\right] \tag{11}\]  
with the misclosure vector and scalar being  
\[\mathbf{w}_{m} = -\mathbf{B}\,\mathbf{v}_{0}+\mathbf{f} \tag{12}\] \[\mathbf{w}_{c} = 1-c\;.\]  
Based on the mathematical model and the observations, we compute the vector of residuals  
\[\mathbf{v}=\mathbf{Q}_{l}\mathbf{B}^{T}\left(\mathbf{B}\mathbf{Q}_{l}\mathbf{B }^{T}\right)^{-1}\!\left(\mathbf{w}_{m}-\mathbf{A}\,\mathbf{x}\right)\,. \tag{13}\]  
The variance factor  
\[\hat{\sigma}_{0}^{2}=\frac{\mathbf{v}^{T}\mathbf{Q}_{l}^{-1}\mathbf{v}+\mathbf{ x}^{T}\mathbf{Q}_{\mathbf{x}}{}^{-1}\mathbf{x}}{n-7+1} \tag{14}\]  
is required to derive the variances and covariances of the seven transformation parameters by \(\boldsymbol{\sum}_{x}=\hat{\sigma}_{0}^{2}\,\mathbf{Q}_{\mathbf{x}}\). For more details and the mathematical derivation, the reader is referred to [40; 41].