Regularized reconstruction of wave height and slope fields from refracted images of water

Refractive imaging of wave fields in an established experimental technique. We consider the associated reconstruction problem and investigate some statistically motivated refinements, including (a) bias correction of local slope estimates, (b) regularization of directional slopes, (c) spatially weighted reconstruction using the estimated variability of local slope estimates, and (d) more accurate estimates of reference light profiles from time sequence data. These refinements are based on a nonparametric observational model for refractive imaging data. Simulation studies show that the refinements can result in substantial improvements in the mean squared error of reconstruction. A computationally efficient algorithm that exploits sparsity is used to evaluate the regularized estimator. Our approach is illustrated by an application to real image data.