we simultaneously reconstruct local scale areas and global point clouds by hierarchical feature decoding. The output of L2G-AE is the reconstructed local areas $\bm{A}^{'}_t$ and the reconstructed $\bm{P}^{'}$ with same number of points to $\bm{P}$.

![A multi-scale example inside a local region of an airplane point cloud, where there are four scales areas $[\bm{A}_1, \bm{A}_2, \bm{A}_3, \bm{A}_4]$ with different colors around the centroid point (red).[]{data-label="fig:multiscale"}](ms.pdf){height="3.5cm" width="5cm"}

Multi-scale Establishment
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To capture fine-grained local patterns of point clouds, we first establish multi-scale areas in each local region, which is similar to PointNet++ [@NIPS2017_7095] and Point2Sequence [@liu2019point2sequence]. Firstly, a subset $\{p_{i_1}, p_{i_2},\\ \cdots, p_{i_M}\}$ of the input points is selected as the centroid of local regions by iterative farthest point sampling (FPS). The latest point $p_{i_j}$ is always the farthest one from the rest points $\{p_{i_1}, p_{i_2}, \cdots, p_{i_{j-1}}\}$. Compared to other sampling method, such as random sampling, FPS can achieve a better coverage of the entire point cloud with the given same number of centroids. As shown in Figure \[fig:multiscale\], around each sampled centroid, $T$ different scale local areas are established continuously by kNN searching with $\{K_1, K_2, \cdots, K_T\}$ nearest points, respectively. An alternative searching method is ball query [@NIPS2017_7095] 