Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques

This paper introduces and analyzes new approximation procedures for bivariate functions. These procedures are based on an edge-adapted nonlinear reconstruction technique which is an intrinsically two-dimensional extension of the essentially non-oscillatory and subcell resolution techniques introduced in the one-dimensional setting by Harten and Osher. Edge-adapted reconstructions are tailored to piecewise smooth functions with geometrically smooth edge discontinuities, and are therefore attractive for applications such as image compression and shock computations. The local approximation order is investigated both in Lp and in the Hausdorff distance between graphs. In particular, it is shown that for general classes of piecewise smooth functions, edge-adapted reconstructions yield multiscale representations which are optimally sparse and adaptive approximations with optimal rate of convergence, similar to curvelets decompositions for the L2 error.