Volumetric Parameterization for 3-Dimensional Simply-Connected Manifolds

TL;DR

This paper introduces novel volumetric parameterization methods for 3D simply-connected manifolds, enabling balanced control over geometric and density distortions, with demonstrated effectiveness in manifold remeshing.

Contribution

It presents new algorithms for 3D manifold parameterization that balance multiple properties, addressing limitations of previous approaches.

Findings

Methods effectively preserve geometric structure.

Achieve density equalization in mappings.

Demonstrated accuracy in manifold remeshing applications.

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.