A New Geometric Representation for 3D Bijective Mappings and Applications

TL;DR

This paper introduces a novel geometric representation called 3D quasiconformality (3DQC) for bijective 3D mappings, enabling efficient manipulation and interpolation while preserving geometric properties.

Contribution

The work extends 2D Beltrami coefficients to 3D, deriving a PDE and discretization method for efficient computation of bijective 3D mappings.

Findings

Effective numerical algorithms for 3D mapping interpolation

Robustness demonstrated through extensive experiments

Efficient solution via conjugate gradient method

Abstract

Three-dimensional (3D) mappings are fundamental in various scientific and engineering applications, including computer-aided engineering (CAE), computer graphics, and medical imaging. They are typically represented and stored as three-dimensional coordinates to which each vertex is mapped. With this representation, manipulating 3D mappings while preserving desired properties becomes challenging. In this work, we present a novel geometric representation for 3D bijective mappings, termed 3D quasiconformality (3DQC), which generalizes the concept of Beltrami coefficients from 2D to 3D spaces. This geometric representation facilitates the scientific computation of 3D mapping problems by capturing local geometric properties in 3D mappings. We derive a partial differential equation (PDE) that links the 3DQC to its corresponding mapping. This PDE is discretized into a symmetric positive-definite linear system, which can be efficiently solved using the conjugate gradient method. 3DQC offers a powerful tool for manipulating 3D mappings while maintaining their desired geometric properties. Leveraging 3DQC, we develop numerical algorithms for sparse modeling and numerical interpolation of bijective 3D mappings, facilitating the efficient processing, storage, and manipulation of complex 3D mappings while ensuring bijectivity. Extensive numerical experiments validate the effectiveness and robustness of our proposed methods.