Sparse convolution-based digital derivatives, fast estimation for noisy signals and approximation results

We provide a general notion of a Digital Derivative in 1−dimensional grids, which has real or integer-only versions. From any such masks, a family of masks called skipping masks are defined. We prove general results of multigrid convergence for skipping masks. We propose a few examples of digital derivative masks, including the now well-known binomial mask. The corresponding skipping masks automatically have multigrid convergence properties. We study the cases of parametric curves tangents and curvature. We propose a novel interpretation of digital convolutions as computing points on a smooth curve, the regularity of which depends on the mask. We establish, in the case of binomial and B−spline masks, a close relationship between the derivatives of the smooth curve, and the digital derivatives provided by the mask.