Anisotropic smoothness classes : from finite element approximation to image models

TL;DR

This paper introduces anisotropic smoothness measures tailored for functions with edges or shocks, linking finite element approximation rates to image processing and PDE solutions, emphasizing geometric edge smoothness over traditional semi-norms.

Contribution

It develops new anisotropic smoothness quantities based on the Hessian determinant, applicable to functions with discontinuities, and demonstrates their relevance in adaptive finite element approximation and image analysis.

Findings

Quantitative measures of anisotropic smoothness are defined.

These measures relate to approximation rates in finite element methods.

They offer an alternative to traditional variation semi-norms in image processing.

Abstract

We propose and study quantitative measures of smoothness 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 examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.