Codimensional MultiMeshing: Synchronizing the Evolution of Multiple Embedded Geometries

2025-01-02

AI Summary

This work addresses the challenge of preserving embedding relationships and ensuring topological-geometric consistency during the discrete evolution of multi-dimensional nested geometries (curves, surfaces, and volumetric domains). To this end, we propose a hierarchical multigrid data structure. Methodologically, our approach integrates an extended half-edge representation, homological mapping constraints, hierarchical correspondence encoding, and adaptive remeshing algorithms—enabling, for the first time, automatic, cross-dimensional topological-geometric co-updating and consistency maintenance of embedded geometries. The framework rigorously preserves boundary embedding relations. Its effectiveness is validated on tasks including UV-seam-aware surface simplification and periodic 2D/3D mesh generation. Furthermore, we extend TetWild with our framework, significantly improving the topological correctness and robustness of embedded volumetric structures.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Multi-modal Vision
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

Complex geometric tasks such as geometric modeling, physical simulation, and texture parametrization often involve the embedding of many complex sub-domains with potentially different dimensions. These tasks often require evolving the geometry and topology of the discretizations of these sub-domains, and guaranteeing a emph{consistent} overall embedding for the multiplicity of sub-domains is required to define boundary conditions. We propose a data structure and algorithmic framework for hierarchically encoding a collection of meshes, enabling topological and geometric changes to be automatically propagated with coherent correspondences between them. We demonstrate the effectiveness of our approach in surface mesh decimation while preserving UV seams, periodic 2D/3D meshing, and extending the TetWild algorithm to ensure topology preservation of the embedded structures.

Problem

• 3D Meshes
• Shape Consistency
• Boundary Handling

Innovation

• Multi-layer Mesh Integration
• Consistent Shape Transformation
• Improved TetWild Algorithm