Novel Anisotropic Multidimensional Convolutional Filters for Derivative Estimation and Reconstruction

The Savitzky-Golay convolutional filter matches a polynomial to even-spaced, one dimensional data and uses this to measure smoothed derivatives. We re-examine the fundamental concept behind this filter, and generate a formulation approach with multidimensional, heterogeneous, anisotropic basis functions to provide a general smoothing, derivative measurement and reconstruction filter for arbitrary point clouds using a linear operator in the form of a convolution kernel. This novel approach yields filters for a wide range of applications such as robot vision, medical volumetric time series analysis and numerical differential equation solution on arbitrary meshes or point clouds without resampling. The urge to extend polynomial filters to higher dimensions is obvious yet previously unfulfilled. We provide a novel complete, arbitrary-dimensional approach to their construction, and introduce anisotropy and irregularity.