BijectiveRemesh: Maintaining Bijective Mappings for Data Transfer Across Remeshed Manifolds

2026-05-28

AI Summary

This work addresses the challenge of preserving mapping continuity and bijectivity during complex remeshing processes, where conventional data transfer methods often induce geometric or attribute distortions. The authors propose a composite mapping framework based on local bijective atlases, enhanced by a Shared Scaffold structure that guarantees global bijectivity. The approach is generalized to support a variety of remeshing operations and, for the first time, enables the construction of bijective mappings on 3D tetrahedral remeshings by innovatively integrating Steinitz’s theorem with Maxwell–Cremona lifting theory. This framework facilitates precise tracking of geometric entities—including points, curves, and surfaces—across remeshing sequences, significantly improving fidelity in high-precision applications such as texture transfer and volumetric simulation.

Abstract

We introduce BijectiveRemesh, a robust algorithm for maintaining a continuous, bijective mapping across complex remeshing sequences on both 2D triangle surfaces and 3D tetrahedral meshes. Unlike traditional data transfer methods that rely on interpolation or projection, our approach constructs a mathematically rigorous composite map from the input mesh to the output mesh by chaining local bijective atlases defined for each primitive remeshing operation. Our framework represents the overall mapping as a composition of local bijective atlases, one per remeshing operation. Building upon successive self-parameterization, we introduce a Shared Scaffold structure for 2D triangle meshes that enforces global bijectivity through local orientation preservation. We extend this approach to handle edge splits, edge swaps, and vertex smoothing beyond the original edge collapses. For 3D tetrahedral meshes, we generalize the local atlas construction using Steinitz's Theorem and Maxwell-Cremona lifting to ensure valid embeddings. This enables exact tracking of geometric entities, including points, curves, and surfaces, across remeshing, with applications from texture transfer to volumetric simulations.

Problem

• bijective mapping
• remeshing
• data transfer
• manifold
• mesh processing

Innovation

• bijective mapping
• remeshing
• data transfer
• mesh parameterization
• geometric correspondence