Multidimensional smoothing using orthogonal expansions

Adaptive smoothing approaches are particularly useful because the amount of smoothing imposed on the data is determined automatically from the statistical characteristics of subsets of the data itself. Although efficient methods are available for performing adaptive smoothings on one-dimensional (1-D) data, extension of these 1-D adaptive smoothing methods to higher dimensions is often difficult because of the lack of a theoretical edifice and prohibitively large computational requirements. Previously, a method was developed that reduces the dimensions of a data function by exploiting its Fourier transform properties and thus achieves an effective multidimensional smoothing by use of low-dimensional smoothing methods. The purpose of this letter is to extend this work and demonstrate the possibility of achieving a higher-dimensional smoothing by applying lower-dimensional smoothing operations on the partial orthogonal expansion coefficients of the data function.