# Deep Point Cloud Facet Segmentation and Applications in Downsampling and Crop Organ Extraction
## 4 Discussion
### Point-Cloud Downsampling Experiment and Analysis Based on FFN Output Facets
#### 4.1.1 Downsampling Strategy Based on FFN Facets  
To resolve the trade-off between uniformity and structural fidelity in traditional downsampling methods, this work proposes a downsampling strategy based on the facets output by the FFN. Conceptually, when an FFN segments the original point cloud, it divides it into a set of facets that are fixed in number, geometrically compact, and aligned with the semantic boundaries. The downsampling method described here fully leverages this property by selecting one representative point from each facet to achieve high-quality downsampling.  
Specifically, the original point cloud is first input into the FFN network to obtain \(K\) facet labels, where, \(K\) is the user-preset number of points to retain after downsampling. Because the FFN is trained with joint constraints on the facet shape and semantic boundaries, the output facets are both highly concentrated in information and uniformly distributed in 3D space. For each facet, its point set's geometric center \(c\) is computed as:  
\[c=\frac{1}{M}{\sum_{i=1}^{M}{\rm P}_{i}} \tag{22}\]  
where, \(\left\{{\rm P}_{i}\right\}_{i=1}^{M}\) denotes all points in the \(j\)th facet, and \(M\) is the number of points in that facet. The geometric center reflects the principal spatial location of the facet and preserves its overall geometric features. Next, within the same facet, the Euclidean distance from each point to the center \(c\) is calculated, and the point \(p*\) with the minimum distance is chosen as the facet's representative:  
\[{\rm P}^{*}=\text{argmin}{\left\|{\rm P}_{i}-c\right\|}_{2} \tag{23}\]  
This representative point typically lies at or near the geometric center of the facet, effectively capturing the facet details and boundary characteristics. Collecting the \(K\) representative points from all facets yields a downsampled point cloud of size \(K\).  
The core advantages of this strategy are threefold. First, since FFN's facet count \(K\) is fixed, the downsampled point count is precisely controlled from the outset. Second, the facets output by the FFN already balance global distribution uniformity and semantic-region coherence in 3D space; therefore, representative points capture global geometry and semantic information at the \"facet scale,\" avoiding the sampling bias encountered by FPS in locally sparse regions. Third, this downsampling method involves only a single computation of the facet centers and Euclidean distances, offering significantly lower computational complexity than multi-round FPS or dense voxel-grid strategies, thus facilitating engineering deployment and real-time processing. As shown in Figure 4, an example original point cloud of 10,000 points is segmented by the FFN into 1024 facets; selecting the nearest-to-center point in each facet reduces the original 10,000 points to 1024 representative points. This result achieves high-fidelity sampling in the global space and preserves facet-boundary coherence, providing an optimized input for downstream semantic or instance segmentation.