Local signal detection is useful in many scientific areas such as imaging processing
and speech recognition, for extracting meaningful patterns from noisy signals. In this
dissertation, we study estimation and local signal detection for spatial data distributed
over irregular domains. In particular, we use bivariate splines defined on triangulations
to approximate unknown signals on a complex domain nonparametrically. Subsequently,
we propose a penalized polynomial spline method that simultaneously detects
the null regions with signals and estimates the patterns on non-null regions. A smoothing
proximal gradient (SPG) algorithm is used to find the estimator efficiently. In theory, the
proposed estimator is shown to be consistent in estimating the true underling patterns.
Furthermore, it is also able to detect the null signal region with probability approaching
one. The numeric performance of the proposed method is evaluated through simulation
studies and real data analysis. This validation shows that the proposed method and
algorithm efficiently detect local signals on complex domains.