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

Abstract : 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.
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Henri-Alex Esbelin, Rémy Malgouyres. Sparse convolution-based digital derivatives, fast estimation for noisy signals and approximation results. Theoretical Computer Science, Elsevier, 2016, ⟨10.1016/j.tcs.2015.12.018⟩. ⟨hal-01296759⟩

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