  A systematic and comprehensive framework for finite impulse response (FIR)
lowpass/fullband derivative kernels is introduced in this paper. Closed form
solutions of a number of derivative filters are obtained using the maximally
flat technique to regulate the Fourier response of undetermined coefficients.
The framework includes arbitrary parameter control methods that afford
solutions for numerous differential orders, variable polynomial accuracy,
centralized/staggered schemes, and arbitrary side-shift nodes for boundary
formulation. Using the proposed framework four different derivative matrix
operators are introduced and their numerical stability is analyzed by studying
their eigenvalues distribution in the complex plane. Their utility is studied
by considering two important image processing problems, namely gradient surface
reconstruction and image stitching. Experimentation indicates that the new
derivative matrices not only outperform commonly used method but provide useful
insights to the numerical issues in these two applications.
