# Sensor-Level Mosaic of Multistrip KOMPSAT-3 Level 1R Products
## 2 Methods
### Projection onto the Reference Image  
Except for the reference image, the other images, i.e., collateral images, are required to be projected onto the reference image space using the sensor modeling information. This study used RPCs instead of the physical model for compatibility with little difference in accuracy [15]. 1. Ground to image projection:  
Ground to image projection is called the forward projection, which equation is expressed as Equation (1). Given 3D ground coordinates \((\phi,\lambda,h)\), the corresponding image coordinates \((l,s)\) can be obtained based on the non-linear equation of 78 coefficients (RPCs) [16]. \[ \begin{array}{l}Y=\frac{Num_{l}(L,V,W)}{Den_{l}(U,V,W)}=\frac{q^{T}u}{b^{T}u} \\ X=\frac{Num_{l}(U,V,W)}{Den_{l}(U,V,W)}=\frac{c^{T}u}{d^{T}u}\end{array}  \tag{1}\]  
Figure 2: Flowchart of the proposed method for the sensor-level mosaic. with  
\[U=\frac{\phi-\phi_{O}}{\phi_{S}},V=\frac{\lambda-\lambda_{O}}{ \lambda_{S}},W=\frac{h-h_{O}}{h_{S}},Y=\frac{l-L_{O}}{L_{S}},X=\frac{s-S_{O}}{S_{ S}}\] \[u=\left[1\ V\ U\ W\ VU\ VW\ UW\ V^{2}\ \ \ U^{2}\ W^{2}\ UVW\ V^{3}\ VU^{2}\ VW^{2}\ V^{2}U\right.\] \[\left.U^{3}\ UW^{2}\ V^{2}W\ U^{2}W\ W^{3}\right]^{T}\] \[a=[a_{1}\ a_{2}\ \ \ldots\ a_{20}]^{T},\ b=[1\ b_{2}\ \ \cdots\ b_{20}]^{T},\ c=[c_{1}\ c_{2}\ \ \cdots\ c_{20}]^{T},\ d=[1\ d_{2}\ \ \cdots\ d_{20}]^{T}\]  
where \((\phi,\lambda,h)\) are the geodetic latitude, longitude, and ellipsoidal height. \((l,s)\) are the image row and column coordinates. \((X,Y)\) and \((U,V,W)\) are the normalized image and ground coordinates, respectively. \((\phi_{O},\lambda_{O},h_{O},S_{O},L_{O})\) and \((\phi_{S},\lambda_{S},h_{S},S_{S},L_{S})\) are the offset and scale factors, respectively for the latitude, longitude, height, column, and row. However, the major problem is that the target elevation must be given, and there is no closed solution for the ground elevation computation. Figure 3 depicts the iterative ground elevation search process is depicted. Given an image point, the first image to ground projection is performed to the reference elevation, such as the mean elevation of RPCs. The computed horizontal coordinates are used to look up the ground elevation in SRTM DEM. Next, the second image to ground projection is tried for the estimated ground elevation. This iterative process continues until the no changes in the computed horizontal coordinates. Image to ground projection is called the backward projection. Given an image coordinates \((l,s)\) with the ground elevation \((h)\), the horizontal ground coordinates \((\phi,\lambda)\) are computed using Equation (2). The backward projection is a non-linear equation that requires to be linearized as Equation (2). The linearized equation requires the initial horizontal ground coordinates \((\phi^{0},\lambda^{0})\) for \((U^{0},V^{0})\). The solution is obtained by iteration until \((dU,dV)\) it reaches near zero. \[\left[ \begin{array}{c}V\\ U\end{array} \right]=\left[ \begin{array}{c}V^{0}\\ U^{0}\end{array} \right]+\left[ \begin{array}{c}dV\\ dU\end{array} \right] \tag{2}\]  
\[\left[ \begin{array}{c}dV\\ dU\end{array} \right]=\left[ \begin{array}{cc}\frac{\partial Y}{\partial V} \big{|}_{V=V^{0}}&\frac{\partial Y}{\partial U}\big{|}_{U=U^{0}}\\ \frac{\partial X}{\partial V}\big{|}_{V=V^{0}}&\frac{\partial X}{\partial U} \big{|}_{U=U^{0}}\end{array} \right]^{-1}\left[ \begin{array}{c}Y-Y^{0}\\ X-X^{0}\end{array} \right]\]  
Figure 3: Iterative ground elevation search. where  
\[