Smoothed differentiation filters for images

Computation of the derivatives of an image defined on a lattice structure is of paramount importance in computer vision. The solution implies least square fitting of a continuous function to a neighborhood centered on the site where the value of the derivative is sought. We present a systematic approach to the problem involving orthonormal bases spanning the vector space defined over the neighborhood. Derivatives of any order can be obtained by convolving the image with a priori known filters. We show that if orthonormal polynomial bases are employed the filters have closed form solutions. The same filter is obtained when the fitted polynomial functions have one consecutive degree. Moment preserving properties, sparse structure for some of the filters, and relationship to the Marr-Hildreth and Canny edge detectors are also proven. Expressions for the filters corresponding to fitting polynomials up to degree six and differentiation orders up to five, for the cases of unweighted data and data weighted by the discrete approximation of a Gaussian, are given in the appendices.