Regularized reconstruction of water surfaces from noisy gradient information via plane-wave superposition

Refraction-based imaging systems, used in wave tank facilities, provide measurements of local water surface gradients. The reconstruction of wave height fields from this information is considered. Utilizing the convenient approximation of target wave height fields by superposition of simple plane waves, we explore the possibility of adaptive plane-wave approximation for computation of the regularized solutions to the wave height reconstruction problem. A greedy algorithm is employed. The method developed allows for non-parametric estimation of wave-front shapes and their periodicities. Regularization forces a natural inverse relation between the smoothness of wave-front shapes and their periodicities. A generalized cross-validation statistic based on a novel tomographic approximation to the model degrees of freedom is developed to assess the regularization parameter. The approximation technique would appear to have wider utility in multi-dimensional smoothing via regularization. The methodology is illustrated by application to real and synthetic data associated with an operational imaging system. Generalization of the approach to the nonlinear problem of reconstructing water surfaces from reflectance data is also considered and some preliminary results for Lambertian reflection are provided. The approach is found to offer substantial potential for this class of reconstruction problems.