Computation of two-dimensional polynomial least-squares convolution smoothing integers

Abstract

The technique of smoothing by polynomial least-squares convolution (PLSC) techniques, commonly known to chemists as Savitzky-Golay smoothing, is arguably the most common digital data processing technique used by analytical chemists with the exception of the fit of a straight line to data to generate a calibration curve ( I ) . This paper presents tools enabling a general extension of the method for smoothing two-dimensional data. The PLSC smoothing technique, simple but effective in both concept and operation (2), can be used to remove noise that is high in frequency relative to the analytical information when data are acquired on an evenly spaced, linear, abscissa axis. The PLSC technique was first pointed out to the chemical community and smoothing coefficients were compiled in Savitzky and Golay’s classic paper (3). Later papers made corrections to these tables ( I , 4), and as digital data acquisition and signal processing became common, these integers saw frequent application. However, analytical techniques increasingly operate in two dimensions: Excitation-emission fluorescence matrices (5) , retention time/absorbance surfaces in liquid chromatography (6 ) , and spatial/spectral maps of emission from atomic spectroscopy sources (7) are but three examples. In these cases the data will often benefit from smoothing in both dimensions. Consequently, when smoothing of two-dimensional data is necessary, techniques operating in two dimensions become imperative. Together with an introduction to smoothing in two dimensions, a limited set of two-dimensional PLSC integers was published by Edwards (8). To compute the integers from the least-squares normal equations, the computer precision required for the computation rapidly overflowed the capability of even a CDC 6600 computer, and the tables were limited to smoothing areas up to a maximum size of 7 by 7 . Because the objective was smoothing of photographic images whose bandwidth was the same in both dimensions, only symmetrical sets of integers were computed. Later, a somewhat more extensive but still limited set of two-dimensional PLSC integers was tabulated, computed by other techniques on a Macintosh computer (2). When applying the original integer tables for linear smoothing, Madden (1) found them insufficient for his needs, limited to smoothing widths up to 25. Additionally, these tables are cumbersome and error-prone. Consequently, algebraic expressions were contributed by which these integers can be computed “on the fly”. Although these formulas appear complex, evaluation on small computers can be easier than manipulating limited tables of data. Furthermore, there is no inherent limit to the smoothing area that can be employed. Unfortunately, no similar set of formulas exists for twodimensional smoothing. In this case the need is even more acute since there is no inherent reason why the smoothing width should be the same in both dimensions. Indeed, as will