## 3 Method  
### Basic Idea  
The geometric principle underlying the VTW involves determining whether the projection region intersects with or separates from the target. Since the VTW is only a small portion of the global calculation interval, it is evident from the process that boundary discretization and angle calculation are also required during the separation times. However, the angle calculation during the separation phase can be considered as unnecessary computation. This part of the computation would increase the amount of computation time. Assuming the accuracy of the orbit propagation model, if it is possible to predict the time of the separation and intersection based on the orbit position information, excluding ineffective time, then boundary discretization, angle calculation, and calculations are necessary only within the vicinity of the window. This approach ensures an improvement in computational speed while maintaining calculation accuracy. Therefore, this paper proposes an orbit filtering method based on an extended region, as illustrated in Figure 4. In the figure, the orange circular part D represents the payload's field of view projection, S is the sub-point, and \(m\) is the radius of the projection. The irregular blue region is the ground target region A, the rectangular line region B is obtained by expanding A based on the four extreme points range, and the rectangular dark blue region C is obtained by expanding region B with the radius \(m\) of the field of view projection D. Thus, it is necessary to determine only the relationship between the sub-point S and region D, transforming the positional relationship judgment between the regions into a judgment between a point and a region. In Figure 4a-c, regions C and D are shown in different scenarios as follows:  
1. In case (a), regions C and D are separate, with region C not containing point S;
2. In case (b), regions C and D intersect, with region C not containing point S;
3. In case (c), regions C and D intersect, with region C containing point S.  
In situations (a) and (b), regions D and A are not intersecting, while only in situation (c) is there a possibility of intersection between region D and A. Therefore, this approach  
Figure 4: **(a)** Regions C and D are separate, and region C does not contain point S; **(b)** Regions C and D intersect, and region C does not contain point S; **(c)** Regions C and D intersect, and region C contains point S.  
can effectively exclude cases where there is no intersection, specifically by determining the positional relationship between the sub-point S and the expanded region C.  
Building upon the extended region, the paper introduces the TTIHS. As depicted in Figure 5, gray represents the time intervals, golden arrows indicate the interval lengths, red, blue, and green, respectively, represent the orbit filtering, large-step angle calculation, and small-interval binary search strategies adopted in the global time interval \(\Delta t_{1}\), the coarse calculation time interval \(\Delta t_{2}\), and the precise calculation time interval \(\Delta t_{3}\). The time intervals form a pyramid structure from bottom to top, indicating the gradual shortening of the calculation time intervals. The first step involves employing the concept of an expanded rectangle within the global time interval [\(t_{1s}\), \(t_{1e}\)] (\(\Delta t_{1}\)) to implement an orbit filtering strategy. Utilizing the longitude and latitude of rectangle C and the orbit model, the times where point S is not within rectangle C are excluded. This phase requires only the calculation of the orbit position, effectively eliminating a large range of non-visible time intervals to reduce computation. This results in the coarse calculation time interval [\(t_{2s}\), \(t_{2e}\)] (\(\Delta t_{2}\)). Within \(\Delta t_{2}\), a large-step angle calculation strategy is employed, calculating the angle relationship between the sub-points and boundary discretization points.