Discretized Laplacian Smoothing by Fourier Methods

Abstract An approach to multidimensional smoothing is introduced that is based on a penalized likelihood with a modified discretized Laplacian penalty term. The choice of penalty simplifies computational difficulties associated with standard multidimensional Laplacian smoothing methods yet without compromising mean squared error characteristics, at least on the interior of the region of interest. For linear smoothing in hyper-rectangular domains, which has wide application in image reconstruction and restoration problems, computations are carried out using fast Fourier transforms. Nonlinear smoothing is accomplished by iterative application of the linear smoothing technique. The iterative procedure is shown to be convergent under general conditions. Adaptive choice of the amount of smoothing is based on approximate cross-validation type scores. An importance sampling technique is used to estimate the degrees of freedom of the smooth. The methods are implemented in one- and two-dimensional settings. Some illustrations are given relating to scatterplot smoothing, estimation of a logistic regression surface, and density estimation. The asymptotic mean squared error characteristics of the linear smoother are derived from first principles and shown to match those of standard Laplacian smoothing splines in the case where the target function is locally linear at the boundary. A one-dimensional Monte Carlo simulation indicates that the mean squared error properties of the linear smoother largely match those of smoothing splines even when these boundary conditions are not satisfied. Key Words: FFTImportance samplingIterative smoothing