# 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
### Basic Coordinate Transformation of Target Positioning Algorithm
\begin{array}{l}N_{0}=R_{e}\\ h_{0}=\left[(x_{T})^{2}+(y_{T})^{2}+(z_{T})^{2}\right]^{1/2}-(R_{e}R_{p})^{1/2} \\ \varphi_{0}=\arctan\biggl{\{}\frac{z_{T}[(1-e^{2})N_{0}+h_{0}]}{[(x_{T})^{2} +(y_{T})^{2}]^{1/2}(N_{0}+h_{0})}\biggr{\}}\\ \\ \left\{\begin{array}{l}N_{i}=\frac{R_{E}}{\left(1-e^{2}\sin^{2}\varphi_{i-1 }\right)^{1/2}}\\ h_{i}=\frac{\left[(x_{T})^{2}+(y_{T})^{2}\right]^{1/2}}{\cos\varphi_{i-1}}-N_ {i-1}\\ \varphi_{i}=\arctan\biggl{\{}\frac{z_{T}[(1-e^{2})N_{i-1}+h_{i-1}]}{[(x_{T})^ {2}+(y_{T})^{2}]^{1/2}(N_{i-1}+h_{i-1})}\biggr{\}}\end{array}\right.\end{array} \right. \tag{6}\]  
\(N_{0},H_{0},\varphi_{0}\) are the initial parameter of iteration. Longitude can be calculated directly from the earth's coordinates. The conversion equation and positive-negative relationship are shown in Equation (8). \[\left(\lambda_{P}\right)_{0}=\arctan(\frac{x_{P}^{E}}{y_{P}^{E}}),\;\lambda_{ P}= \begin{cases}\left(\lambda_{P}\right)_{0}&x_{T}>0\\ \left(\lambda_{P}\right)_{0}+\pi&x_{T}<0,\;\lambda_{0}<0\\ \left(\lambda_{P}\right)_{0}-\pi&x_{T}<0,\;\lambda_{0}>0\end{cases}  \tag{8}\]  
The origin of the geographic coordinate system is situated at the local positioning station (aircraft or airborne camera), and the positioning station is regarded as a mass point \(A\) when establishing the geographic coordinate system. Depending on the direction of the coordinate axis, the coordinate system type can be classified into the north-east-down (NED) coordinate system and the east-north-up (ENU) coordinate system. In the NED Coordinate System, with the aircraft as the origin \(A\), the \(A-N\) axis and \(A-E\) axis point to due north and due east, respectively, and the \(A-D\) axis points to the center of the earth along the ellipsoid normal. The relationship between the geographic coordinate system and the earth coordinate system is revealed in Figure 3, and the conversion relationship is presented in Equation (9). \[C_{ECEF}^{NED}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&R_{n}+h\\ 0&0&0&1\end{bmatrix} \times \begin{bmatrix}-\sin\varphi&0&\cos\varphi&0\\ 0&1&0&0\\ 0&0&-\sin\varphi&0\\ 0&0&0&1\end{bmatrix} \times \begin{bmatrix}\cos\lambda&\sin\lambda&0&0\\ -\sin\lambda&\cos\lambda&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \times \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&R_{n}e^{2}\sin\varphi\\ 0&0&0&1\end{bmatrix}  \tag{9}\]The origin of the aircraft coordinate system (A) is the same as the origin of the geographic coordinate system. The \(X\)-axis points to the nose, the \(Y\)-axis points to the right wing, and the \(Z\)-axis is vertical to the plane of the carrier plane. During the flight, the three-axis attitude is roll (\(\varphi\)), pitch (\(\theta\)), and yaw (\(\psi\)), respectively. When the attitude angles of the three axes are all zero, the coordinate system of the carrier aircraft completely coincides with the geographic coordinate system. The relationship between the carrier coordinate system and the geographic coordinate system is shown in Figure 4, where LOS represents the direction of the camera's visual axis (line of sight). The transformation matrix between the carrier coordinate system and the geographic coordinate system is indicated in Equation (10) [64]. \[C_{NED}^{AC}= \begin{bmatrix}1&0&0&0\\ 0&\cos\varphi&\sin\varphi&0\\ 0&-\sin\varphi&\cos\varphi&0\\ 0&0&0&1\end{bmatrix} \times \begin{bmatrix}\cos\theta&0&-\sin\theta&0\\ 0&1&0&0\\ \sin\theta&0&\cos\theta&0\\ 0&0&0&1\end{bmatrix} \times \begin{bmatrix}\cos\psi&\sin\psi&0&0\\ -\sin\psi&\cos\psi&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{10}\]  
The origin of the camera coordinate system (C) mostly is the projection center of the camera optical system \(S_{O}\). The \(Z_{C}\) axis is the main optical axis and \(S_{O}Z_{C}\) points to the detection target when the target is imaged at the center of the detector. Modern aerial cameras are usually installed in a two-axis frame consisting of two universal shafts.