3D地圖探索是機器人技術中的關鍵技術之一。
然而，由於環境未知，尋找最佳的勘探路徑是一個挑戰。
這項研究提出了拓撲傅立葉稀疏集(TFSS)演算法，
使無人飛行器(UAV)能夠在理論上保證探索3D環境效能。
該演算法由Rips複合體和傅立葉稀疏集組成。
Rips複合體用於擴展子目標以進行地圖探索，
而傅立葉稀疏集用於學習子目標選擇的子模函數。
由於空間探索的目標函數被重新構造為路徑限制下的最大化子模函數，
貪婪算法可以達到(1/2)(1-e^(-1))的近似最佳值。
使用該演算法進行的實驗證明，無人機可以比基準方法探索更多環境。
此外，該演算法顯示了探索問題的持續同調性。

3D map exploration is one of key technologies in robotics.
However, nding an optimal exploration path is a challenge due to unknown environments.
This research proposed the Topological Fourier Sparse Set (TFSS) algorithm to enable an unmanned aerial vehicle (UAV) to explore 3D environments with theoretical guarantees.
The algorithm consists of the Rips complex and Fourier sparse set.
The Rips complex is to expand subgoals for map exploration while the Fourier sparse set is to learn submodular functions for subgoal selection.
Since the objective function of spatial exploration is reformulated as a maximizing submodular function with path constraints,
greedy algorithms can achieve (1/2)(1-e^(-1)) of the optimum.
Experiments conducted with this algorithm demonstrates that the UAV can explore the environments more than the benchmark approaches.
Furthermore, the algorithm shows the persistence homology of exploration problems.

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