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 an algorithm that establishes a connection between this representation and the corresponding mapping. Experimental results showcase the algorithm's effectiveness in eliminating foldings in 3D mappings, as well as in mapping reconstruction and generation. 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.

翻译：多维数据中映射关系及其畸变的精确分析是当前研究中的重要挑战。虽然Beltrami系数能够精确描述二维映射中的畸变特征，但目前的三维空间分析工具尚不具备这种能力。本文提出了一种创新方法：建立能捕获三维映射局部膨胀特征的3D拟共形表示，并设计了连接该表示与对应映射的算法。实验结果表明，该算法能有效消除三维映射中的折叠现象，并在映射重建与生成中展现出显著效果。上述特性与二维线性Beltrami求解器技术具有相似性。本文工作为三维数据及其映射的畸变精确分析与调控提供了具有前景的解决方案。