# Comparison of Different Algorithms to Orthorectify WorldView-2 Satellite Imagery
## Materials and Methods
### Algorithms for Orthorectification  
In this section, several algorithms that are present in the software most commonly used for image orthorectification are described. According to [PERSON] [2], whatever the mathematical functions used, the geometric correction method and processing steps are: acquisition of image; acquisition of the GCPs and check points (CPs) with image coordinates and map coordinates X, Y, Z; computation of the unknown parameters of the mathematical functions used for the geometric correction model; and image rectification with or without digital elevation model (DEM). The two-dimensional (2D) polynomial model has been adopted for image rectification since the 1970s [35]. It relates images coordinates (X\({}^{\prime}\), Y\({}^{\prime}\)) to cartographic coordinates (X, Y) by the following equations:  
\[x^{\prime}=\sum_{i=0}^{m_{1}}\sum_{j=0}^{m_{2}}a_{ij}x^{i}y^{j},\;y^{\prime}= \sum_{i=0}^{m_{1}}\sum_{j=0}^{m_{2}}b_{ij}x^{i}y^{j} \tag{1}\]  
If n is the order of the equation, the following relations are valid:  
\[0\leq m_{1}\leq n;\quad 0\leq m_{2}\leq n;\quad m_{1}+m_{2}\leq n. \tag{2}\]  
The 2D polynomial function model of the first order (6 parameters) corrects the image by a rotation, a translation in x and y, scaling along both axes and an obliquitous transformation. The 2D first-order polynomial function is also called affine transformation. The 2D polynomial function model of second order (12 parameters) adds to the previous transformation's correction of the torsion and convexity along both axes. A cubic polynomial function (20 parameters) further increases the correction flexibility, in a way that does not necessarily correspond to any physical reality of the image acquisition system [36]. By substituting the coordinate values of GCPs in Equation (1), polynomial coefficients (\(a_{lij}\) and \(b_{lij}\)) can be calculated. Consequently, the new location of each pixel in the cartographic plane can be defined. The minimum number of GCPs needed is established by the order \(n\) of the polynomial functions according to the formula [37]:  
\[N_{\text{min}_{GCPs}}=\frac{(n+1)\cdot(n+2)}{2} \tag{3}\]  
To increase the positional accuracy of the resulting image, a greater number of GCPs is chosen, which are distributed regularly. Thus differences between map coordinates and corrected image coordinates for the same features tend to be reduced. To evaluate the quality of the results, errors not only for GCPs but also for other points, named check points (CPs), are considered [2; 38]. Using polynomial functions is a very uncomplicated method to orthorectify VHRSI: often considered outdated, they correct for basic planimetric distortion at the GCPs; because they do not take ground elevation into consideration, they are limited to small and flat areas [39]. 3D rational polynomial functions (RPFs) may be more useful. They define a relationship between the image coordinates (x\({}^{\prime}\), y\({}^{\prime}\)) and the 3D object coordinates (X, Y, Z) [23; 40]. Specifically:  
\[x^{\prime}=\frac{P_{1}^{n}\left(X,Y,Z\right)}{P_{4}^{n}\left(X,Y,Z\right)} \qquad y^{\prime}=\frac{P_{3}^{n}\left(X,Y,Z\right)}{P_{4}^{n}\left(X,Y,Z\right)} \tag{4}\]where \(P_{1}^{n},P_{2}^{n},P_{3}^{n},P_{4}^{n}\) are usually cubic polynomials. Each of these includes 20 coefficients and can be expressed as:  
\[P_{l}^{n}\left(X,Y,Z\right)=\sum_{i=0}^{m_{1}}\sum_{j=0}^{m_{2}}\sum_{k=0}^{m_{3} }a_{i_{ijk}}X^{i}Y^{j}Z^{k} \tag{5}\]  
where:  
\[ \begin{array}{l}n=3\\ l=1,2,3,4\\ 0\leq m_{1}\leq 3;\\ 0\leq m_{2}\leq 3;\\ 0\leq m_{3}\leq 3;\\ m_{1}+m_{2}+m_{3}\leq 3\end{array}  \tag{6}\]  
Equations in (4) are known in the literature as _Upward RFM_, as they enable the image coordinates to be obtained starting from the 3D coordinates of a ground point [41]. Substituting \(P\) in (4) with the polynomials in (5) and eliminating the first coefficient in the denominator, there are 39 RPF coefficients in each equation: 20 coefficients in the numerator and 19 (and the constant 1) in the denominator [42].