TSSS: a novel triangulated spherical spline smoothing for surface-based data

Surface-based data are prevalent across diverse practical applications in various fields. This paper introduces a novel nonparametric method to discover the underlying signals from data distributed on complex surface-based domains. The proposed approach involves a penalised spline estimator defined on a triangulation of surface patches, enabling effective signal extraction and recovery. The proposed method offers superior handling of ‘leakage’ or ‘boundary effects’ over complex domains, enhanced computational efficiency, and capabilities for analyzing sparse and irregularly distributed data on complex objects. We provide rigorous theoretical guarantees, including convergence rates and asymptotic normality of the estimators. We demonstrate that the convergence rates are optimal within the framework of nonparametric estimation. A bootstrap method is introduced to quantify the uncertainty in the proposed estimators and to provide pointwise confidence intervals. The advantages of the proposed method are demonstrated through simulations and data applications on cortical surface neuroimaging data and oceanic near-surface atmospheric data.