Shape Representation using Gaussian Process mixture models

Abstract

Traditional explicit 3D representations, such as point clouds and meshes, demand significant storage to capture fine geometric details and require complex indexing systems for surface lookups, making functional representations an efficient, compact, and continuous alternative. In this work, we propose a novel, object-specific functional shape representation that models surface geometry with Gaussian Process (GP) mixture models. Rather than relying on computationally heavy neural architectures, our method is lightweight, leveraging GPs to learn continuous directional distance fields from sparsely sampled point clouds. We capture complex topologies by anchoring local GP priors at strategic reference points, which can be flexibly extracted using any structural decomposition method (e.g. skeletonization, distance-based clustering). Extensive evaluations on the ShapeNetCore and IndustryShapes datasets demonstrate that our method can efficiently and accurately represent complex geometries.

Figures (8)

• Figure 1: We partition a sparse point cloud using distance-based clustering and extract reference points (Sec. \ref{['ssect:lightweighht_shape_templates']}). Each 3D point $P_i$ is parameterized as a bearing vector in spherical coordinates $U_i$ relative to its reference point $C_i$. The directions, with distances as targets, serve as inputs to a Gaussian Process ($GP_i$) (Sec. \ref{['ssect:GP_prior_for_distance']}). The GP mixture model forms our shape representation.
• Figure 2: Spherical directional distance field (blue rays) centered at reference point $\pmb{C}$ (red).
• Figure 3: Ray-casting to the objects's surface (blue) from reference points (red) by retaining the first intersection. Denser coverage achieved with increasing number of reference points (left-to-right).
• Figure 4: Assignment of training points to clusters. (a)-(b) Ground truth - reconstructed without overlap. (c)-(d) Ground truth - reconstructed with overlap (blue).
• Figure 5: Qualitative comparison of shape representation on the ShapeNetCore dataset. The heatmap quantifies the distance from the ground truth point-cloud to the reconstructed; superimposed into the ground truth point cloud. We demonstrate two examples from the airplanes, chairs and sofas categories with each method. From left to right ours, DeepSDF park2019deepsdf, DeepSDF (10k), NKSR huang2023nksr