  We study the problem of sparse signal detection on a spatial domain. We
propose a novel approach to model continuous signals that are sparse and
piecewise smooth as product of independent Gaussian processes (PING) with a
smooth covariance kernel. The smoothness of the PING process is ensured by the
smoothness of the covariance kernels of Gaussian components in the product, and
sparsity is controlled by the number of components. The bivariate kurtosis of
the PING process shows more components in the product results in thicker tail
and sharper peak at zero. The simulation results demonstrate the improvement in
estimation using the PING prior over Gaussian process (GP) prior for different
image regressions. We apply our method to a longitudinal MRI dataset to detect
the regions that are affected by multiple sclerosis (MS) in the greatest
magnitude through an image-on-scalar regression model. Due to huge
dimensionality of these images, we transform the data into the spectral domain
and develop methods to conduct computation in this domain. In our MS imaging
study, the estimates from the PING model are more informative than those from
the GP model.
