The fundamental problem of smoothing and differentiating of noisy images is an ill-posed problem, and common differentiation filters give very unreliable results. We look at several sources of the errors and show a way to eliminate them. In particular: (1) We show that regularization based filters perform better than the Gaussian, assuming the data changes slowly relative to the noise. (2) Truncation of an infinite filter is very damaging for derivatives, so the common idealized regularization methods cannot be used. We construct finite, discrete regularization based filters using a spline approximation.
Isaac Weiss, Isaac Weiss,
"Image smoothing and differentiation with regularized filters", Proc. SPIE 1610, Curves and Surfaces in Computer Vision and Graphics II, (1 February 1992); doi: 10.1117/12.135152; https://doi.org/10.1117/12.135152