## 2 Data and Methods
Standard correction equations include a matrix of plan \(A\), a vector of unknowns \(x\), and a measurement vector of \(I\):  
\[\mathbf{v}=\mathbf{Ax}-I \tag{1}\]  
As all unknowns in the equations are in a linear form, there is no need to linearize the equations. The derivatives of transformation equations according to individual variables appear in the matrix \(A\). For the affine transformation of one map sheet we have a total of six unknowns, for the polynomial transformation of the second degree we have 12 unknowns. The equations of conditions must be added to these equations. In our case, the conditions define the identity of the corresponding edges. In the case of an affine transformation, the identity of the end points of the edges after the adjustment is ensured, in the case of a polynomial transformation, the identity of the points in the middle of the edges after the adjustment must also be used:  
\[\mathbf{Bx}=\mathbf{0}. \tag{2}\]  
Figure 1: Example of a map sheet of the First Military Mapping. The solution then has the form:  
\[\left( \begin{array}{cc}\mathbf{A}^{T}\mathbf{A}&\mathbf{B}\\ \mathbf{B}^{T}&\mathbf{0}\end{array} \right)\left( \begin{array}{c}\mathbf{x}\\ \mathbf{k}\end{array} \right)=-\left( \begin{array}{c}\mathbf{A}^{T}\mathbf{I}\\ \mathbf{b}\end{array} \right). \tag{3}\]  
For the total calculation of the joint adjustment it is necessary to prepare the matrices \(\mathbf{A}\), \(\mathbf{B}\) and the vector \(\mathbf{I}\). In the joint adjustment of 250 map sheets by means of an affine transformation, there will be a total of 1500 unknowns. The matrix \(\mathbf{A}\) will have 1500 columns. The number of rows in the matrix will correspond to the number of measurements, i.e., the number of GCPs multiplied by two. In our case, it will be 13,698 lines. The number of conditions, and thus the number of rows in the matrix \(\mathbf{B}\), is given by the layout of the map sheets in the mosaic. It is important that the conditions are independent of each other. The calculation would collapse if one of the conditions were a linear combination of the others. In our case, the layout of the map sheets is obvious from Figure 2. For a polynomial transformation of the second degree we will have a total of 3000 unknowns, the number of measurements remains the same, the number of conditions increases due to the identity of the center points of the edges. The result of the adjustment are transformation parameters for individual map sheets. After applying them to the image data, the adjacent edges fit exactly together. The drawing behind the map frames remains a problem. This must be masked within the mosaic using defined masks. After the affine transformation, the masks are parallelograms, after the polynomial transformation of the second degree, they are formations composed of four parabolic arcs. In order to increase the reliability of our results, it is also possible to use the robust IRLS method. It is based on the repeated calculation of the adjustment, with the weights being reassigned to individual measurements in the calculation on the basis of an estimate of their remoteness. To adjust the weights, it is crucial to choose the correct score function. The parabolic function is used for LSM. To obtain a robust estimate, we use [PERSON]'s solution, where the edge parts of the parabola are replaced by straight lines. Then, it is possible to derive an influence function and a weight function that assigns specific weights. Figure 2: Mosaic of map sheets with marked position of GCPs. As a parameter \(c\) for [PERSON]'s estimate, we chose \(c=1.5\), i.e., the case where the expected contamination of measurements is 4%. More about this statistical approach can be found in [25]. After each iteration, it is possible to calculate the ratio of the resulting mean standard errors (RMSE). The end of the calculation is set with a difference of less than 0.1%. The whole calculation is quite demanding and, therefore, we decided to create the MultiGeoref software that solves this problem [28].