点云的自由边界共形参数化
Free-boundary conformal parameterization of point clouds
摘要：
随着3D扫描技术的进步，在科学和工程中使用点云的兴趣激增。为了便于点云的计算和分析，先前的工作已经考虑将它们参数化到具有固定边界形状（诸如单位圆或矩形）的一些简单平面域上。然而，固定形状的几何结构可能导致参数化中的一些不期望的失真。因此，考虑点云的自由边界共形参数化是更自然的，其在不约束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，我们开发了一个自由边界共形参数化方法的磁盘类型的点云，其中涉及到一种新的近似方案的点云拉普拉斯与累积余切权重，以及在边界点的特殊处理。借助自由边界保形参数化，可以方便地实现高质量的点云网格划分。此外，我们表明，使用保形焊接在复杂的分析的想法，点云保形参数化可以计算在一个分而治之的方式。实验结果证明了该方法的有效性。
英文摘要：
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 develop a free-boundary conformal parameterization method for disk-type point clouds, which involves a novel approximation scheme of the point cloud Laplacian with accumulated cotangent weights together with a special treatment at the boundary points. 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.
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