  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.
