Homogeneity Pursuit for 3D Point Clouds

This paper proposes a novel classification learning approach to identify the latent group structures in 3D point clouds collected with irregular shapes, which allows 3D point clouds to be classified into different groups with the number of groups and the group membership unknown. 3D point clouds within a group have a common mean function, while mean function heterogeneity is allowed across groups. We first obtain preliminary estimates of the individual mean functions by penalized trivariate splines over triangulation. Based on these preliminary estimates, we develop an agglomerative clustering algorithm to estimate the unknown group structures and propose a ratio criterion to determine the number of groups. We establish the consistency of the estimated number of groups and group membership. Furthermore, we investigate a post-grouping trivariate penalized spline estimation procedure for group-specific mean functions and derive the corresponding asymptotic theory. The proposed method is validated through simulation studies and showcased with an application to the imaging data from the Alzheimer's Disease Neuroimaging Initiative database.