# Surface Reconstruction Planning with High-Quality Satellite Stereo Pairs Searching
## 4 High-Quality DSM with Stereo Pairs Searching  
### Quality-Driven Planning Model of Stereo Sensing  
To ensure the acquisition of proper images for high-quality 3D surface reconstruction, scientific planning of Earth-imaging tasks is essential, by comprehensively considering satellite orbit dynamics, the pairing of imaging time windows, and the collaborative use of multiple satellites, etc. This is especially important when there are a large number of stereo imaging tasks to be fulfilled by a constellation. In this case, the imaging resources are usually limited and the planning for efficient collaboration of satellites can avoid overlapping tasks and redundant use of resources while fulfilling the tasks as soon as possible.  
The parameters and variables used in the proposed planning model are summarized in Table 1. The prediction model of reconstruction quality as trained in Section 3.2 is denoted as a function of imaging pair as \(DSM(i,j)\) in the table, to estimate the DSM error for planning model optimization.  
This paper characterizes the planning with a multi-objective problem including the objective functions of task quantity, reconstruction DSM error, and task timeliness as follows:  
\[\begin{split} Obj1:\underset{i,j}{\sum}x_{ij}\\ Obj2:\underset{i}{\sum}\underset{j}{\sum}DSM(i,j)\cdot x_{ij}\\ Obj3:\underset{i}{\sum}\underset{j}{\sum}\max(te_{i},te_{j}) \cdot x_{ij}\end{split} \tag{5}\]  
where \(\max(te_{i},te_{j})\) is the latest end time for a matched imaging time window pair \((i,j)\). According to the above formalized objectives, the \(Obj1\) (number of tasks to be fulfilled) needs to be maximized while \(Obj2\) (DSM error) and \(Obj3\) (total end time) need to be minimized. The overall objective can be calculated as the weighted sum of above objectives, to be maximized as  
\[Obj=\omega 1\cdot Obj1+\omega 2\cdot(-Obj2)+\omega 3\cdot(-Obj3) \tag{6}\]  
In calculating Equation (6), the objectives are normalized to the same scale to avoid the dominance of any objective value, where \(\omega 1\), \(\omega 2\), and \(\omega 3\) are weights for these objectives to control their importance in the model. To provide feasible plans for stereo imaging, the following constraints need to be satisfied when optimizing the above objective function.  
Task time span constraint: The observation time window of each task must be within the entire task period.  
\[t_{start}\leq ts_{i}\leq te_{i}\leq t_{end},\forall i \tag{7}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(ts_{i}/te_{i}\) & Start/end time of imaging time window \(i\) \\ \(t_{start}/t_{end}\) & Start/end time of entire tasks \\ \(mt_{k}\) & Maneuvering time needed for satellite \(k\) \\ \(\alpha_{i}\) & Side-swing angle of imaging time window \(i\) \\ \(\alpha_{\max}\) & Maximum side-swing angle of satellites \\ \(DSM(i,j)\) & DSM error of stereo imaging pair \((i,j)\) \\ \(\theta_{ij}\) & Intersection angle of imaging pair \((i,j)\) \\ \(\theta_{\min}/\theta_{\max}\) & Minimum/maximum intersection angle \\ \(x_{ij}\) & Decision variable indicating whether to select stereo imaging pair \((i,j)\). \\  & \(x_{ij}=1\) if \((i,j)\) is selected; otherwise \(x_{ij}=0\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: quality-driven planning parameters and variables.  
Maneuvering time constraints: This is to ensure that the satellite has sufficient maneuvering time to adjust its attitude for two consecutive observations.  
\[ts_{i+1}-te_{i}\geq mt_{k_{x}},\forall i,k \tag{8}\]  
Swing angle constraint: Satellite swing angle has to be less than the maximum range.  
\[|\alpha_{i}|\leq\alpha_{\max},\forall i \tag{9}\]  
Intersection angle constraint: The intersection angle value of the stereo imaging pair \((i,j)\) ( \(x_{ij}=1\)) needs to be within the valid range.  
\[\theta_{\min}\leq\theta_{ij}\leq\theta_{\max},\forall i,j \tag{10}\]