Simple filter design for first and second order derivatives by a double filtering approach

Spline filters are usually implemented in two steps, where in the first step the basis coefficients are computed by deconvolving the sampled function with a factorized filter and the second step reconstructs the sampled function. It will be shown how separable spline filters using different splines can be constructed with fixed kernels, requiring no inverse filtering. Especially, it is discussed how first and second order derivatives can be computed correctly using cubic or trigonometric splines by a double filtering approach giving filters of length 7.