  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.
