MULTIDIMENSIONAL LEAST-SQUARES SMOOTHING USING ORTHOGONAL POLYNOMIALS

The rapid development of multidimensional spectroscopy has created a need for multidimensional noise-filtering techniques. Here we present a general extension of the Savitzky technique for smoothing of multidimensional data arrays. The Savitzky technique achieves a polynomial fit of the experimental data over a predetermined window by applying a moving window weighted average to the data. Here the experimental data are fit to an expansion of products of orthogonal polynomials, which leads to a generalized solution for weighting factors in the moving window average. The resulting solution can be applied to a fitting of arbitrary polynomial order or smoothing window size. These results are then shown to be equivalent to sequential one-dimensional smoothing in each of the dimensions of the data array. The sequential approach represents a significant savings in computation time over the multidimensional smoothing technique. Also, sequential smoothing can be applied to data arrays of arbitrary dimensionality with minimal difficulty. The orthogonal polynomial approach has the added advantage of giving solutions for weighting factors used in smoothing of data points at the edges of the array. The two-dimensional smoothing results are compared to those previously developed with a technique similar to that originally presented by Savitzky. The method developed here is found to have additional terms in the formula for calculation of the weighting factor tables. These additional terms are a direct result of inclusion of cross terms in the derivation. A comparison between the smoothing capabilities of the two techniques reveals that inclusion of these cross terms results in slightly better smoothing characteristics.