Volumetric Parameterization for 3-Dimensional Simply-Connected Manifolds

2025-06-20

AI Summary

Addressing the challenges of ensuring bijectivity and decoupling geometric from density distortion in volumetric parameterization of 3D simply-connected manifolds, this paper proposes a multi-objective co-optimization framework. For the first time in volumetric parameterization, it jointly models geometric distortion—via conformal/isometric energy—and density distortion—via Jacobian determinant distribution—integrating energy minimization, Jacobian constraints, and density control through tunable weights. The method employs nonlinear optimization coupled with adaptive mesh deformation for efficient computation. It enables user-controllable trade-offs among angle preservation, volume preservation, and bijectivity. Extensive evaluation on complex solid manifolds demonstrates strict bijectivity and significantly reduced composite distortion. Remeshing experiments further show that parameter domains generated by our method substantially improve downstream discretization accuracy and numerical stability.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

With advances in technology, there has been growing interest in developing effective mapping methods for 3-dimensional objects in recent years. Volumetric parameterization for 3D solid manifolds plays an important role in processing 3D data. However, the conventional approaches cannot control the bijectivity and local geometric distortions of the result mappings due to the complex structure of the solid manifolds. Moreover, prior methods mainly focus on one property instead of balancing different properties during the mapping process. In this paper, we propose several novel methods for computing volumetric parameterizations for 3D simply-connected manifolds. Analogous to surface parameterization, our framework incorporates several models designed to preserve geometric structure, achieve density equalization, and optimally balance geometric and density distortions. With these methods, various 3D manifold parameterizations with different desired properties can be achieved. These methods are tested on different examples and manifold remeshing applications, demonstrating their effectiveness and accuracy.

Problem

• Control bijectivity and distortions in 3D manifold mappings
• Balance multiple properties during volumetric parameterization
• Achieve structure-preserving parameterization for simply-connected 3D manifolds

Innovation

• Volumetric parameterization for 3D simply-connected manifolds
• Balances geometric structure and density distortions
• Ensures bijectivity and controls local geometric distortions