# Review of Target Geo-Location Algorithms for Aerial Remote Sensing Cameras without Control Points
## 2 Bibliometric Study and Typical Aerial Camera-to-Ground Target Localization Algorithm
### Target Positioning Algorithm Based on Earth Ellipsoid Model  
Early target geo-location algorithms assume the target area as a complete plane, ignoring the influence caused by geodetic elevation and the curvature of the earth to the localization results [12]. To tackle this challenge, a method is given in the following. As shown in Figure 6, when the aerial camera acquires the target image, it will be projected on the focal plane of the detector. The detector of an airborne camera in most cases is made of charge-coupled devices (CCD). If the size of a single pixel of the CCD is \(a\), and the focal length of the camera optical system is \(f\), then the projected position of the target point in the camera coordinate system can be described by Equation (12).  
\[T_{s}{}^{\prime}=[m\times a,n\times a,-f,1]^{T} \tag{12}\]  
Figure 5: Schematic diagram of camera coordinate system.  
Figure 6: Target projection point position.  
Based on auxiliary coordinate systems described above, the coordinates of the target projection point in the earth coordinate system \({T_{E}}^{\prime}\) can be expressed as Equation (13) according to the coordinate rotation transformation theory [13; 14].  
\[{T_{E}}^{\prime}=\begin{bmatrix}x_{T}{}^{\prime}\\ y_{T}{}^{\prime}\\ z_{T}\\ 1\end{bmatrix}=C_{NED}^{ECEF}\times C_{AC}^{NED}\times C_{S}^{AC}\times{T_{S}}^{\prime} \tag{13}\]  
If the origin of the camera coordinate system is \(O_{S}=[x_{S},y_{S},z_{S}]^{T}\), the coordinates of the target in the earth coordinate system \([x_{T},y_{T},z_{T}]^{T}\) should be located on the line Equation (14) established between the camera system center and the target projection point.  
\[\frac{x_{T}-x_{S}}{x_{T}^{\prime}-x_{S}}=\frac{y_{T}-y_{S}}{y_{T}^{\prime}-y_{S }}=\frac{z_{T}-z_{S}}{z_{T}^{\prime}-z_{S}} \tag{14}\]  
If the altitude of the target is \(h_{T}\), the earth ellipsoid equation at this area can be written as  
\[\frac{{x_{T}}^{2}}{(R_{e}+h_{T})^{2}}+\frac{{y_{T}}^{2}}{(R_{e}+h_{T})^{2}}+ \frac{{z_{T}}^{2}}{(R_{e}+h_{T})\times(1-e^{2})^{1/2}}=1 \tag{15}\]  
The coordinates of the target in the earth coordinate system can be calculated by combining Equations (14) and (15). Obviously, there should be two sets of analytical solutions to the equation, and the set of solutions closer to the camera is the true position of the target [15; 16].  
This kind of algorithm has been widely used in the 1990s and the beginning of the 21 st century. It realizes the positioning only by the angle, solves the influence of the curvature of the earth on the target positioning results, and is the basis of the follow-up research in the field of target positioning. The equipment with the highest positioning accuracy using this method is the Global Hawk UAV. Depending on the high-precision angle sensor, Global Hawk can control the positioning error within 20 m (imaging distance 18 km) by using the target positioning algorithm based on the earth model [13].  
The target geo-location algorithm based on the earth ellipsoid model is simple to calculate and can be directly used for location when the accuracy requirement is not high enough. The algorithm is extremely affected by the elevation error because the algorithm needs the average elevation information in advance and regards it as the reference height of the target; this leads to large position errors in areas with huge topographic relief.