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We propose and study quantitative measures of smoothness which are adapted toanisotropic features such as edges in images or shocks in PDE's. Thesequantities govern the rate of approximation by adaptive finite elements, whenno constraint is imposed on the aspect ratio of the triangles, the simplestexamples of such quantities are based on the determinant of the hessian of thefunction to be approximated. Since they are not semi-norms, these quantitiescannot be used to define linear function spaces. We show that they can be welldefined by mollification when the function to be approximated has jumpdiscontinuities along piecewise smooth curves. This motivates for using them inimage processing as an alternative to the frequently used record variationsemi-norm which does not account for the geometric smoothness of the edges.

Anisotropic smoothness classes : from finite element approximation to image models