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：过去二十年中，我们见证了各领域采集的不规则形状点云数量呈指数级增长。受点云实体建模重要性的驱动，我们基于四面体剖分上的多元样条提出一种新型高效平滑工具，用于提取点云中的潜在信号并构建三维实体模型。所提出的平滑方法能有效对点云进行去噪或去模糊处理，并提供实际信号的多尺度重建。此外，该方法可处理稀疏且非均匀分布的点云，并恢复潜在轨迹。所提出的平滑与插值方法还提供了一种自然的数量降维途径。进一步地，我们建立了该方法的理论保证：具体推导了估计量的收敛速度与渐近正态性，并证明其收敛速度达到了非参数最优收敛速度。通过大量模拟研究与实际数据案例，我们展示了该方法在估计精度与数据降维效率方面相较于传统平滑方法的优越性。