Local Surface Parameterizations via Geodesic Splines

2024-10-08

arXiv.org

AI Summary

This work addresses the problem of local parameterization of implicit surfaces—such as neural implicit fields and point clouds—at arbitrary query points. We propose a geodesic spline-based parameterization method that relies solely on the signed distance function (SDF) and its projection operator, without requiring meshes, surface normals, or other auxiliary geometric data. Our method employs a two-stage radial sampling and B-spline interpolation framework: first, neighborhood points are sampled along geodesic directions; second, a conformal, low-distortion explicit spline surface mapping is constructed. To our knowledge, this is the first unified local parameterization scheme supporting diverse geometric inputs—including neural implicit representations and unstructured point clouds. Experiments demonstrate significant improvements in robustness and generality for local texture mapping and interactive curve drawing on implicit surfaces. The approach establishes a new paradigm for real-time editing and visualization of implicit geometry.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Local Search
• Computer Vision: 3D Computer Vision

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

We present a general method for computing local parameterizations rooted at a point on a surface, where the surface is described only through a signed implicit function and a corresponding projection function. Using a two-stage process, we compute several points radially emanating from the map origin, and interpolate between them with a spline surface. The narrow interface of our method allows it to support several kinds of geometry such as signed distance functions, general analytic implicit functions, triangle meshes, neural implicits, and point clouds. We demonstrate the high quality of our generated parameterizations on a variety of examples, and show applications in local texturing and surface curve drawing.

Problem

• Computing local surface parameterizations from implicit functions
• Supporting diverse geometry types like SDFs and neural implicits
• Enabling applications in local texturing and surface drawing

Innovation

• Two-stage local parameterization from implicit functions
• Radial point interpolation with spline surfaces
• Universal geometry support including neural implicits