Spline Smoothing of 3-D Geometric Data

• Activity Number: 494 - Advanced Developments in Methods and Algorithms for Modern Complex Imaging Data
• Type: Invited
• Date/Time: Thursday, August 11, 2022 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Shan Yu* and Xinyi Li and Yueying Wang and Guannan Wang and Lily Wang
• Companies: University of Virginia and Clemson University and Columbia University and College of William and Mary and George Mason University
• Keywords: Complex domain; Nonparametric smoothing; Penalized splines; Trivariate splines; Tetrahedra partitions

Abstract

Over the past two decades, we have seen an increased demand for 3D visualization and simulation software in medicine, architectural design, engineering, and many other areas, which have boosted the investigation of geometric data analysis and raised the demand for further advancement in statistical analytic approaches. In this paper, we propose a class of spline smoothers appropriates for approximating geometric data over 3D complex domains, which can be represented in terms of a linear combination of spline basis functions with some smoothness constraints. We start with introducing the tetrahedral partitions, Barycentric coordinates, Bernstein basis polynomials, and trivariate spline on tetrahedra. Then, we propose a penalized spline smoothing method for identifying the underlying signal in a complex 3D domain from potential noisy observations. Furthermore, the convergence rate and asymptotic normality of the proposed estimator are established, where the convergence rate achieves the optimal nonparametric convergence rate, and the asymptotic normality holds uniformly. Simulation studies are conducted to compare the proposed method with traditional smoothing methods on 3D complex d