A Linear Time and Space Local Point Cloud Geometry Encoder via Vectorized Kernel Mixture

We propose VecKM, a novel local point cloud geometry encoder that is descriptive, efficient and robust to noise. VecKM leverages a unique approach by vectorizing a kernel mixture to represent the local point clouds. Such representation is descriptive and robust to noise, which is supported by two theorems that confirm its ability to reconstruct and preserve the similarity of the local shape. Moreover, VecKM is the first successful attempt to reduce the computation and memory costs from $O(n^2+nKd)$ to $O(nd)$ by sacrificing a marginal constant factor, where $n$ is the size of the point cloud and $K$ is neighborhood size. The efficiency is primarily due to VecKM's unique factorizable property that eliminates the need of explicitly grouping points into neighborhoods. In the normal estimation task, VecKM demonstrates not only 100x faster inference speed but also strongest descriptiveness and robustness compared with existing popular encoders. In classification and segmentation tasks, integrating VecKM as a preprocessing module achieves consistently better performance than the PointNet, PointNet++, and point transformer baselines, and runs consistently faster by up to 10x.