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Anisotropic Smoothness Classes: From Finite Element Approximation to Image Models

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Abstract

We propose and study quantitative measures of smoothness f ↦ A(f) which are adapted to anisotropic features such as edges in images or shocks in PDE’s. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest example being \(A_{p}(f)=\|\sqrt{|\mathrm{det}(d^{2}f)|}\|_{L^{\tau}}\) which appears when approximating in the L p norm by piecewise linear elements when \(\frac{1}{\tau}=\frac{1}{p}+1\). The quantities A(f) are not semi-norms, and therefore cannot be used to define linear function spaces. We show that these quantities can be well defined by mollification when f has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used total variation semi-norm which does not account for the smoothness of the edges.

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Authors and Affiliations

  1. UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, 75005, Paris, France

    Jean-Marie Mirebeau & Albert Cohen

  2. UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, 75005, Paris, France

    Jean-Marie Mirebeau & Albert Cohen

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  1. Jean-Marie Mirebeau
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  2. Albert Cohen
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Correspondence to Albert Cohen.

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Mirebeau, JM., Cohen, A. Anisotropic Smoothness Classes: From Finite Element Approximation to Image Models. J Math Imaging Vis 38, 52–69 (2010). https://doi.org/10.1007/s10851-010-0210-x

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