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

2025-01-23

AI Summary

Mesh-free parameterization of point-cloud surfaces remains challenging due to the absence of explicit connectivity and the need to handle complex parametric domains with free or fixed boundary constraints. Method: We propose the first end-to-end deep neural network framework for point-cloud surface parameterization, eliminating reliance on triangulated meshes. Our approach introduces two novel, differentiable loss functions—HAND (Hausdorff Approximation Constraint) and LEG (Local Geometric Energy minimization)—to jointly enforce parametric domain shape adaptivity and local differential-geometric fidelity. It leverages node-wise differentiable Hausdorff distance approximation and local curvature regularization within an unsupervised, fully differentiable optimization pipeline. Contribution/Results: The method significantly outperforms baselines on shape matching, surface reconstruction, and boundary detection tasks. It further generalizes to landmark-guided parameterization, demonstrating high-fidelity, robust point-cloud parameterization without mesh priors.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Surface parametrization plays a crucial role in various fields, such as computer graphics and medical imaging, and computational science and engineering. However, most existing techniques rely on the discretization of the surface into a triangular mesh. This paper addresses the problem of point cloud surface parametrization and presents two novel loss functions and a framework for point cloud surface parametrization based on deep neural networks. The first loss function aims to provide a soft constraint on parameter domain, allowing the handling of parameter domains with complex shapes or geometries. This loss function can also be used in generalizing landmark matching. The second loss function focuses on minimizing local distortion on the point cloud surface, demonstrating effectiveness in preserving the surface's local shape characteristics. We parametrized the functions involved using neural networks, and developed an algorithm for the minimization. Numerical experiments for shape matching, free-boundary and fixed-boundary surface parametrization and landmark matching, along with applications including surface reconstruction and boundary detection, are presented to demonstrate the effectiveness of our proposed methods.

Problem

• Point Cloud Parameterization
• Surface Representation
• Non-Triangulation Methods

Innovation

• Point Cloud Parameterization
• Novel Loss Functions
• Deep Neural Network