Optimal N-term approximation by linear splines over anisotropic Delaunay triangulations

Anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data, in particular from image data. In previous work, we have proposed a locally adaptive method for efficient image approximation, called adaptive thinning, which relies on linear splines over anisotropic Delaunay triangulations. In this paper, we prove asymptotically optimal N -term approximation rates for linear splines over anisotropic Delaunay triangulations, where our analysis applies to relevant classes of target functions: (a) piecewise linear horizon functions across α Hölder smooth boundaries, (b) functions of Wα,p regularity, where α > 2/p−1, (c) piecewise regular horizon functions of Wα,2 regularity, where α > 1.