On simplified application of multidimensional Savitzky-Golay filters and differentiators

I propose a simplified approach for multidimensional Savitzky-Golay filtering, to enable its fast and easy implementation in scientific and engineering applications. The proposed method, which is derived from a generalized framework laid out by Thornley (D. J. Thornley, "Novel anisotropic multidimensional convolution filters for derivative estimation and reconstruction" in Proceedings of International Conference on Signal Processing and Communications, November 2007), first transforms any given multidimensional problem into a unique one, by transforming coordinates of the sampled data nodes to unity-spaced, uniform data nodes, and then performs filtering and calculates partial derivatives on the unity-spaced nodes. It is followed by transporting the calculated derivatives back onto the original data nodes by using the chain rule of differentiation. The burden to performing the most cumbersome task, which is to carry out the filtering and to obtain derivatives on the unity-spaced nodes, is almost eliminated by providing convolution coefficients for a number of convolution kernel sizes and polynomial orders, up to four spatial dimensions. With the availability of the convolution coefficients, the task of filtering at a data node reduces merely to multiplication of two known matrices. Simplified strategies to adequately address near-boundary data nodes and to calculate partial derivatives there are also proposed. Finally, the proposed methodologies are applied to a three-dimensional experimentally obtained data set, which shows that multidimensional Savitzky-Golay filters and differentiators perform well in both the internal and the near-boundary regions of the domain.