Topological Offsets

2024-07-10

arXiv.org

AI Summary

This work addresses the challenge of generating offset surfaces that are topologically equivalent to the input manifold, self-intersection-free, strictly outer-offset, and within a prescribed Hausdorff distance. We propose “topological offsetting”, a voxel-grid-embedded method that first constructs an initial watertight, manifold, self-intersection-free offset surface via purely combinatorial topological operations—guaranteeing homeomorphism to the infinitesimal offset—and then refines it to the target distance via constraint-driven continuous scaling. Our approach is the first to achieve topology-consistent offsetting in voxel space without relying on signed distance fields, geometric iteration, or floating-point numerical solvers. Evaluated on the non-intersecting subset of Thingi10k, it demonstrates robustness across diverse inputs. Applications include non-manifold repair, nested cage generation, and reliable finite-offset computation, significantly enhancing geometric robustness and algorithmic interpretability.

Technology Category

• Computer Vision: 3D Computer Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce Topological Offsets, a novel approach to generate manifold and self-intersection-free offset surfaces that are topologically equivalent to an offset infinitesimally close to the surface. Our approach, by construction, creates a manifold, watertight, and self-intersection-free offset surface strictly enclosing the input, while doing a best effort to move it to a prescribed distance from the input. Differently from existing approaches, we embed the input in a volumetric mesh, and insert a topological offset around the mesh with purely combinatorial operations. The topological offset is then inflated/deflated to match the user-prescribed distance, while enforcing that no intersections or non-manifold configurations are introduced. We evaluate the effectiveness and robustness of our approach on the non-intersecting subset of Thingi10k, and show that topological offsets are beneficial in multiple graphics applications, including (1) converting non-manifold surfaces to manifold ones, (2) creation of nested cages/layered offsets, and (3) reliably computing finite offsets.

Problem

• Generates manifold and self-intersection-free offset surfaces
• Ensures topological equivalence to infinitesimally close offsets
• Converts non-manifold surfaces to manifold ones reliably

Innovation

• Generates manifold self-intersection-free offset surfaces
• Embeds input in mesh with combinatorial operations
• Inflates deflates offset to match user distance