Over the past two decades, we have seen an exponentially increased amount of point clouds collected with irregular shapes in various areas. Motivated by the importance of solid modeling for point clouds, we develop a novel and efficient smoothing tool based on multivariate splines over the tetrahedral partitions to extract the underlying signal and build up a 3D solid model from the point cloud. The proposed smoothing method can denoise or deblur the point cloud effectively and provide a multi-resolution reconstruction of the actual signal. In addition, it can handle sparse and irregularly distributed point clouds and recover the underlying trajectory. The proposed smoothing and interpolation method also provides a natural way of numerosity data reduction. Furthermore, we establish the theoretical guarantees of the proposed method. Specifically, we derive the convergence rate and asymptotic normality of the proposed estimator and illustrate that the convergence rate achieves the optimal nonparametric convergence rate. Through extensive simulation studies and a real data example, we demonstrate the superiority of the proposed method over traditional smoothing methods in terms of estimation accuracy and efficiency of data reduction.
翻译:在过去20年中,我们看到以不同区域非正常形状收集的点云数量急剧增加,由于对点云进行固态建模的重要性,我们根据四面形分区的多变量浮点线,开发了一种新的高效平滑工具,以提取基本信号,并从点云中建立3D固态模型;拟议的平滑方法可以有效地淡化或淡化点云,并提供对实际信号的多分辨率重建;此外,它能够处理分散和不定期分布的点云,并恢复基本轨迹;拟议的平滑和内插方法也为减少数字数据提供了自然的方式;此外,我们建立了拟议方法的理论保障;具体地说,我们得出了拟议估计值的趋同率和零乐观的正常度,并表明合并率达到了最佳的非对称汇率。通过广泛的模拟研究和一个真实数据实例,我们展示了拟议方法在估计数据减少的准确性和效率方面优于传统平滑方法。