# Comparison of Different Algorithms to Orthorectify WorldView-2 Satellite Imagery
## Materials and Methods
### Algorithms for Orthorectification
In order to solve the 78 coefficients, at least 39 GCPs are required [43]. The 78 coefficients are calculated by the data provider considering the position of the satellite at the time of image acquisition. They are then included in the RPC (rational polynomial coefficient) file. However, they can also be calculated using GCPs [43], as in the case above for polynomial functions (PFs). At least 39 of them are needed. In fact, if the camera model is not available, the ground control points need to be selected in a conventional way; that is, through collimation of the homologous points on the cartography or DEM (digital elevation model) or through specific GPS (Global Positioning System) survey campaigns [44]. Usually the Equations in (4) are resolved using a least squares iterative process, on the basis of the measurement of a large number of GCPs. The accuracy of the results depends on the number and the distribution of GCPs. Several GCPs (more than 39) with a regular planimetric and altimetric distribution contribute to the high quality of the results [17; 45; 46]. A DEM of the whole area is also required for RPFs. By applying PFs as well as RPFs to the primary image, a matrix of \"empty\" cells is computed. To calculate the radiometric value to be assigned to each pixel, a resampling method, such as nearest neighbor, bilinear interpolation, or cubic convolution, is used [47; 48]. The nearest-neighbor algorithm is used for this application because it preserves the original radiometric values, which is fundamental for further image processes (calculation of vegetation indexes, classifications, etc.). Several algorithms are described in the literature for VHRSI orthorectification using rigorous physical models. Based on the collinearity equations, they reconstruct the geometry of the scene during the image acquisition through the knowledge of parameters concerning both the platform and the sensor. The traditional photogrammetric method of transforming from object space to image space using collinearity equations is highly suited to frame cameras, but not to pushbroom sensor products [23]. In fact, unlike the traditional frame-based aerial photos, each line of the linear array image is collected in a pushbroom fashion at a different instant of time; therefore, the perspective geometry is only valid for each line whereas it is close to a parallel projection in along-track direction; in addition, for each line, there is a different set of (time-dependent) values for the exterior orientation elements [49]. For 1-meter GSD pushbroom sensors such as IKONOS, fully parametrized camera models are extremely complex and difficult to implement; this is highlighted in the IKONOS System Geometric and Mathematical Model document that consists of 183 pages [50]. OrthoEngine--PCI geomatics software (PCI, 2016) supplies [PERSON]'s rigorous model [51; 52]. Originally developed for SPOT, the model was also later extended for Landsat and satelliteswith synchronous imaging acquisition i.e., IKONOS II, QuickBird [53], WV-2, Komposat-2, and OrbView [54]. Other examples of rigorous models are mentioned in the literature, i.e., the model implemented in SISAR, which is a software for high-resolution satellite images at the Area di Geodesia e Geomatica--Sapienza Universita di Roma. The approximate values of the physical parameters can be computed thanks to the information contained in the metadata file provided with each image and corrected by a least squares estimation process based on an appropriate number of GCPs [55]. In this work orthorectification processes were carried out on the WV-2 panchromatic image using PCI Geomatica OrthoEngine Version 2015. The following algorithms were applied:  
* 2D PFs;
* 3D RPFs with original RPCs;
* 3D RPFs without original RPCs;
* [PERSON]'s rigorous model;
* A combination of 3D RPFs and 2D PFs. Both GCPs and CPs were homogeneously distributed over the study area and located on well-defined features.