the entire point cloud. All reconstructed areas are concatenated and then passed through the FC layer by $$\bm{P} = \bm{W} [\bm{A}^{'}_1 \oplus \bm{A}^{'}_2 \oplus \cdots \oplus \bm{A}^{'}_T] + b.$$ Here, $\oplus$ represents the concatenation operation.

Loss Function
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We propose a new loss function to train the network in an end-to-end fashion. There are two parts in the loss function, local scale reconstruction and global point cloud reconstruction, respectively. As mentioned earlier, we should encourage accurate reconstruction of local areas and the global point cloud at the same time. Suppose $\bm{A}_t$ is the $t^{th}$ scale area in the multi-scale establishment subsection, then, the local reconstruction error for $\bm{A}^{'}_t$ is measured by the well-known Chamfer distance, $$\begin{split}
L_{local} = d_{CH}(\bm{A}_t, \bm{A}^{'}_t) = \sum_{t=1}^T  (\frac{1}{|\bm{A}_t|}\sum_{p_i \in \bm{A}_t} \min_{p^{'}_i \in \bm{A}^{'}_t} \lVert p_i - p^{'}_i \rVert_2  \\ +  \frac{1}{|\bm{A}^{'}_t|}\sum_{p^{'}_i \in \bm{A}^{'}_t} \min_{p_i \in \bm{A}_t} \lVert p_i - p^{'}_i \rVert_2),
\end{split}$$ Similarly, let the input point set be $\bm{P}$ and the reconstructed point set be $\bm{P}^{'}$. The global reconstruction error can be denoted by $$\begin{split}
     L_{global} = d_{CH}(\bm{P}, \bm{P}^{'}) = \frac{1}{|\bm{P}|}\sum_{p_i \in \bm{P}} \min_{p^{'}_i \in \bm{P}^{'}} \lVert p_i - p^{'}_i \rVert_2 \\
     + \frac{1}{|\bm{P}^{'}|}\sum_{p^{'}_i \in \bm{P}^{'}} \min_{p_i \in \bm{P}} \lVert p_i - p^{'}_i \rVert_2.
