Resolution enhancement using an approximate-inverse Savitzky-Golay smooth

Abstract

Any convolution operation, such as the Savitzky-Golay smooth, can be represented as the matrix multiplication of a convolute matrix and a digitized spectrum. The convolute matrix is a band diagonal matrix where the convolute occupies each row, centered on the main diagonal. This convolute matrix can be inverted to provide an inverse convolution operator. In general, these inverse matrices induce large cyclic errors in the data. However, a simple, robust convolute can be extracted from an adjusted inverse matrix. This convolute is not the exact inverse but an approximation. The approximate-inverse Savitzky-Golay convolute provides the inverse of smoothing (i.e. resolution enhancement). This approximate inverse of a smoothing convolute works well, with a significant increase of resolution and minimal decrease in the signal-to-noise ratio of a test NIR spectrum. This resolution enhancement technique, when applied to a set of sample spectra, can improve a multiple-linear-regression model. The derivation of the inverse convolute is shown and the results of its use are discussed.