High-order differentiation filters that work

Reliable derivatives of digital images have always been hard to obtain, especially (but not only) at high orders. We analyze the sources of errors in traditional filters, such as derivatives of the Gaussian, that are used for differentiation. We then study a class of filters which is much more suitable for our purpose, namely filters that preserve polynomials up to a given order. We show that the errors in differentiation can be corrected using these filters. We derive a condition for the validity domain of these filters, involving some characteristics of the filter and of the shape. Our experiments show a very good performance for smooth functions