  Surface-based data is commonly observed in diverse practical applications
spanning various fields. In this paper, we introduce a novel nonparametric
method to discover the underlying signals from data distributed on complex
surface-based domains. Our approach involves a penalized spline estimator
defined on a triangulation of surface patches, which enables effective signal
extraction and recovery. The proposed method offers several advantages over
existing methods, including superior handling of "leakage" or "boundary
effects" over complex domains, enhanced computational efficiency, and potential
applications in analyzing sparse and irregularly distributed data on complex
objects. We provide rigorous theoretical guarantees for the proposed method,
including convergence rates of the estimator in both the $L_2$ and supremum
norms, as well as the asymptotic normality of the estimator. We also
demonstrate that the convergence rates achieved by our estimation method are
optimal within the framework of nonparametric estimation. Furthermore, we
introduce a bootstrap method to quantify the uncertainty associated with the
proposed estimators accurately. The superior performance of the proposed method
is demonstrated through simulation experiments and data applications on
cortical surface functional magnetic resonance imaging data and oceanic
near-surface atmospheric data.
