Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function $f$ of two variables, the algorithm produces a hierarchy of triangulations $(\mathcal {D}_j)_{j\geq 0}$ and piecewise polynomial approximations of $f$ on these triangulations. The refinement procedure consists in bisecting a triangle $T$ in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between $f$ and its piecewise polynomial approximation after $T$ is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the $L^p$ norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of $f$ in the case of $C^2$ functions).