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

TL;DR

VecKM introduces a linear-time, linear-space local point-cloud geometry encoder that vectorizes a Gaussian kernel mixture over each neighborhood. The method provides theoretical guarantees of reconstructiveness and similarity preservation, leveraging a factorizable dense variant to avoid explicit neighborhood graphs. Empirically, VecKM delivers massive speedups (up to 100x) and accuracy gains across normal estimation, classification, and segmentation, while integrating smoothly with PointNet++ and transformer-based architectures. This approach enables scalable, noise-robust preprocessing for large 3D point clouds with practical impact on downstream 3D understanding tasks, by using formulas like E_A(N(x0)) ≈ (1/n) sum exp(i (x_k - x0) A) and Gaussian-kernel-based representations.

Abstract

We propose VecKM, a local point cloud geometry encoder that is descriptive and efficient to compute. VecKM leverages a unique approach by vectorizing a kernel mixture to represent the local point cloud. Such representation's descriptiveness is supported by two theorems that validate its ability to reconstruct and preserve the similarity of the local shape. Unlike existing encoders downsampling the local point cloud, VecKM constructs the local geometry encoding using all neighboring points, producing a more descriptive encoding. Moreover, VecKM is efficient to compute and scalable to large point cloud inputs: VecKM reduces the memory cost from $(n^2+nKd)$ to $(nd+np)$; and reduces the major runtime cost from computing $nK$ MLPs to $n$ MLPs, where $n$ is the size of the point cloud, $K$ is the neighborhood size, $d$ is the encoding dimension, and $p$ is a marginal factor. The efficiency is due to VecKM's unique factorizable property that eliminates the need of explicitly grouping points into neighbors. In the normal estimation task, VecKM demonstrates not only 100x faster inference speed but also highest accuracy and strongest robustness. 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 10 times.

Figures (9)

• Figure 1: Our VecKM encoding is descriptive, robust to noise, and efficient in runtime and memory cost. Upper Left: Raw VecKM encodings, without any training, already capture rich geometric features such as orientations and shapes. Lower Left: Under varying levels of noise, VecKM encodings remain highly consistent. Upper Right: Existing encoders face memory costs of $(n^2+nKd)$, while VecKM costs only $(nd+np)$ memory. Existing encoders compute $nK$ MLPs, whereas VecKM only computes $n$ MLPs. Lower Right: VecKM is 10x$\sim$100x faster than existing encoders in wall-clock time and scalable to large point cloud inputs.
• Figure 2: Theoretical outline of VecKM illustrated by 2d shapes. A point cloud, sampled from a shape distribution function, is associated with a Gaussian kernel mixture and a corresponding VecKM encoding, where the VecKM encoding is proved to be reconstructive and isometric to the Gaussian kernel mixture. Since the Gaussian kernel mixture can approximate the shape function, the VecKM encoding yields a good representation of the shape.
• Figure 3: Visualization of Gaussian kernel $\mathcal{G}$ and its approximation with Lemma \ref{['thm:lemma1']}.
• Figure 4: VecKM can be seamlessly integrated into deep point cloud architectures, improving both accuracy and efficiency.
• Figure 5: VecKM's robustness to data corruptions. VecKM can reconstruct the local shape under corrupted inputs. The VecKM encodings remain highly similar under data corruptions.

Theorems & Definitions (9)

• Theorem 1: Pointwise Local Geometry Encoding
• Proposition 1: Reconstruction
• Proposition 2: Similarity Preservation
• Theorem 2: Dense Local Geometry Encoding
• Lemma 1: VecKM embodies a Gaussian kernel
• Lemma 2: Reconstruction
• Lemma 3: Similarity Preservation
• Lemma 1: VecKM embodies a Gaussian kernel
• proof