General technique for smoothing multi-dimensional datasets utilizing orthogonal expansions and lower dimensional smoothers

Smoothing methods are often used for discerning underlying patterns and structure concealed by statistical noise within a dataset. 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. In this work, we show theoretically that an effective N-dimensional smoothing can be achieved by utilization of a series of non-identical n-dimensional (n < N) smoothings on the coefficients of a partial orthogonal expansion of the data function. This result provides a framework in which one-dimensional adaptive smoothing methods may be applied to higher-dimensional data. We applied this generalized multi-dimensional data. We applied this generalized image and demonstrate that one can obtain a two-dimensional smoothing effect by applying a series of one-dimensional smoothings to the coefficients of the partial orthogonal expansion of the image.