Point Cloud Surface Parametrization with HAND and LEG: Hausdorff Approximation from Node-wise Distances and Localized Energy for Geometry

TL;DR

The paper tackles point cloud surface parametrization without relying on mesh connectivity by learning a map $f:\mathcal{M}\to\Omega\subset\mathbb{R}^2$ via neural networks. It introduces HAND, a differentiable Hausdorff-distance surrogate, and LEG, a localized energy that encodes local geometric distortion, enabling domain alignment and geometry preservation directly on point clouds. The approach is underpinned by theoretical analyses linking HAND to the Hausdorff distance and LEG to angle distortion, and an SGD-based optimization with a staged scheme and an inverse-$\lambda$ network to control distortion. Applications include free-boundary and domain-constrained parametrizations, landmark matching, boundary detection, and surface reconstruction, demonstrating practical utility in graphics and geometric computing.

Abstract

Surface parametrization is a crucial part in various fields, having applications in computer graphic, medical imaging, scientific computing and computational engineering. The majority of surface parametrization approaches are performed on triangular meshes. On the contrary, the theories and methods of point cloud surface parametrization are less researched, despite its rising significance. In this work, we compute surface parametrization in an optimization approach using neural networks, with novel loss functions introduced without extrinsic information, together with theoretical analyses. Based on the theory, we develop an optimization algorithm to improve the parametrization quality. Using our methods, general open surfaces can be parametrized in either free-boundary manner or with arbitrary domain constraints. Landmark matching can also be enforced under our framework. Numerical experiments are conducted and presented, along with applications including surface reconstruction and boundary detection.

Figures (15)

• Figure 1: A figure illustrating the deformation of a triangle $T$ under mapping $f$.
• Figure 1: Architecture of the neural networks used in experiments. The arrows indicate the forward process. The bars indicate the latent variables, with dimensions stated under them.
• Figure 1: The mappings obtained by minimizing HAND with $\alpha = 2, 5, 10, 25, 50, 100$ on a 2D point cloud and a plot of their Hausdorff distance to a square, with boundary drawn in red.
• Figure 1: Result of boundary detection on a human face (top) and a car shell (bottom). Left to right: ground truth boundaries, convex hulls, and two detected boundaries with different thresholds, the detected boundaries with finer thresholds on the original point clouds.
• Figure 2: An illustration explaining how the mesh structure affects the factors.

Theorems & Definitions (11)

• Definition 4.1
• Definition 4.2
• Definition 4.3
• Lemma 4.4
• Definition 4.5
• Theorem 5.1
• Proof 1
• Theorem 5.2
• Proof 2
• Theorem 5.3