With the advancement in 3D scanning technology, there has been a surge of interest in the use of point clouds in science and engineering. To facilitate the computations and analyses of point clouds, prior works have considered parameterizing them onto some simple planar domains with a fixed boundary shape such as a unit circle or a rectangle. However, the geometry of the fixed shape may lead to some undesirable distortion in the parameterization. It is therefore more natural to consider free-boundary conformal parameterizations of point clouds, which minimize the local geometric distortion of the mapping without constraining the overall shape. In this work, we propose a novel approximation scheme of the Laplace--Beltrami operator on point clouds and utilize it for developing a free-boundary conformal parameterization method for disk-type point clouds. With the aid of the free-boundary conformal parameterization, high-quality point cloud meshing can be easily achieved. Furthermore, we show that using the idea of conformal welding in complex analysis, the point cloud conformal parameterization can be computed in a divide-and-conquer manner. Experimental results are presented to demonstrate the effectiveness of the proposed method.

中文翻译：

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点云的自由边界共形参数化

随着 3D 扫描技术的进步，人们对在科学和工程中使用点云的兴趣激增。为了促进点云的计算和分析，先前的工作已经考虑将它们参数化到一些具有固定边界形状的简单平面域上，例如单位圆或矩形。然而，固定形状的几何形状可能会导致参数化中出现一些不希望的失真。因此，更自然地考虑点云的自由边界共形参数化，它可以在不限制整体形状的情况下最小化映射的局部几何失真。在这项工作中，我们提出了一种新的拉普拉斯-贝尔特拉米算子在点云上的近似方案，并利用它来开发一种用于盘型点云的自由边界保形参数化方法。借助自由边界保形参数化，可以轻松实现高质量的点云网格划分。此外，我们表明，在复杂分析中使用保形焊接的思想，可以以分而治之的方式计算点云保形参数化。实验结果证明了所提出方法的有效性。点云共形参数化可以以分而治之的方式计算。实验结果证明了所提出方法的有效性。点云共形参数化可以以分而治之的方式计算。实验结果证明了所提出方法的有效性。