# Deep Point Cloud Facet Segmentation and Applications in Downsampling and Crop Organ Extraction
## 1 Introduction
Early methods converted unordered point clouds into regular structures, for example, by projecting point clouds onto multiple 2D views, performing 2D convolutional segmentation, and then back-projecting [14; 15], or discretizing point clouds into uniform voxel grids and using 3D convolutional networks (e.g., VoxelNet, VoxNet) for semantic segmentation, although computational and memory costs increase sharply as voxel resolution increases [16; 17]. To address the challenges posed by point cloud sparsity and disorder, PointNet was the first to directly take the point set as input and use symmetric functions to ensure output invariance to point order, achieving end-to-end learning [18]. PointNet++ was built by applying PointNet to local neighborhoods, enhancing local feature representation [19]. Based on these advances, some studies have combined facet over-segmentation with deep learning. For example, [PERSON] et al. transformed over-segmentation into local deep feature learning and graph partitioning, adapting to various geometric variations [20]; [PERSON] et al. used a PointNet-like network to extract facet features, then performed sampling and clustering to complete segmentation [21]; [PERSON] et al. jointly learned to generate a weighted soft affinity graph to improve facet boundary prediction accuracy [22]; and [PERSON] et al. transferred the SSN principle from 2D to 3D, requiring only the replacement of the front-end network to produce facet segmentation results for point clouds [9]. Although deep learning methods have achieved results in facet segmentation, two challenges remain: first, a high proportion of empty facets is generated by the network, leading to uncertainty in the facet count and affecting subsequent applications [23]; second, most methods focus on point- or voxel-level segmentation accuracy, with few systematic studies exploiting mid-scale facet features to optimize downsampling and instance segmentation [24]. To address these challenges, this study proposes a deep network capable of segmenting a fixed number of facets on 3D point clouds, called the Fixed-number Facets Network (FFN). Unlike traditional over-segmentation methods, the FFN explicitly incorporates a facet count constraint into its architecture, enabling it to generate a preset number of valid facets regardless of the input point cloud size. Specifically, FFN's front end employs a multiscale geometric feature encoder that fuses spatial position and normal information to integrate the local and global features of the point cloud, producing high-dimensional feature representations. The back end uses Differentiable SLIC for soft clustering and incorporates both facet count constraints and empty-facet suppression terms in the loss function, ensuring that each facet contains at least one real point. To maintain a constant count during testing, FFN also introduces the Seed Point Simple Linear Iterative Clustering (SPSLIC) algorithm: each point searches for its nearest neighbor among all seed points to compute a soft affinity matrix; when updating seed points, priority is given to the nearest real point, and each real point is restricted to correspond to only one facet, thereby avoiding empty facets and ensuring a consistent count. This design preserves the geometric coherence of the facets and significantly reduces the count uncertainty caused by empty facets, thereby improving the segmentation stability. Its core advantages are as follows: (1) by fixing the output facet count, facet-based downsampling and instance segmentation under a uniform facet budget maintain consistent computational complexity and evaluation criteria, facilitating fair algorithm comparison and practical deployment; (2) by combining multiscale geometric feature encoding with differentiable clustering, the spatial geometric consistency of the point cloud is preserved while semantic discriminability is also considered, ensuring that the output facets are both compactly coherent and semantically informative.