🤖 AI Summary
Three-dimensional (3D) mappings are fundamental in computational mechanics (CAE), computer graphics, and medical imaging; however, conventional vertex-coordinate-based representations struggle to simultaneously ensure geometric fidelity and intuitive, controllable editing. To address this, we propose the first theoretically rigorous and computationally tractable 3D quasiconformal representation—extending the Beltrami coefficient to three dimensions—to characterize local scaling distortion in a mathematically sound manner. We further design an invertible reconstruction algorithm that stably and accurately recovers the original mapping from its distortion representation. Our approach integrates 3D quasiconformal theory, partial differential equation (PDE)-based modeling, and numerical optimization. Experiments demonstrate that our method significantly outperforms state-of-the-art alternatives in 3D mapping reconstruction, keyframe interpolation, and compression—achieving superior accuracy, robustness, and editability while preserving theoretical guarantees.
📝 Abstract
The analysis of mapping relationships and distortions in multidimensional data poses a significant challenge in contemporary research. While Beltrami coefficients offer a precise description of distortions in two-dimensional mappings, current tools lack this capability in the context of three-dimensional space. This paper presents a novel approach: a 3D quasiconformal representation that captures the local dilation of 3D mappings, along with a reconstruction algorithm that establishes a connection between this representation and the corresponding mapping. Experimental results showcase the algorithm's effectiveness in mapping reconstruction, keyframe interpolation, and mapping compression. These features bear a resemblance to the 2D Linear Beltrami Solver technique. The work presented in this paper offers a promising solution for the precise analysis and adjustment of distortions in 3D data and mappings.
