Smoothed differentiation filters for images

A systematic approach to least square approximation of images and of their derivatives is presented. Derivatives of any order can be obtained by convolving the image with a priori known filters. It is shown 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 the relationship to the Marr-Hildreth and Canny edge detectors are proven.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>